MSc Global Energy Management

Games of Strategy Identify a real-world environment in which there is strategic interdependence and commitment can play a central role for a decision-maker in improving their outcome from a strategic interaction. After describing the features of the environment, formally model it as a game explaining the assumptions you make; consider an analysis of the environment using the tools of game theory; consider the credibility of threats/promises that players might like to make; and carefully analyse the efficacy of at least one commitment mechanism. In addition, critically reflect on the use of game theory in understanding the real world.
Instruction:
Some Advice
Throughout your work you should seek to demonstrate your understanding
of the conceptual framework of game theory and commitment in particular
(the ‘science’), and how it is applied to real world settings (the ‘art’). Essays
are unlikely to be successful if they do not demonstrate full understanding
of the theoretical framework that will be achieved by detailed explanation
of equilibrium concepts and careful analysis and explanation of the strategic
environments studied.
Your work should evidence exploration of the literature that goes beyond
the basic recommended readings in the class to demonstrate independent
study and thought. Remember: your work should be your own individ-
1
ual account (this is not group work) and all sources should be referenced
appropriately.
References
Below are some sources that you might find useful in completing your essay.
• A good starting place, but by no means sufficient in the context of the
essay, is Dixit, A., S. Skeath and D. Riley; Games of strategy; Norton,
2015 (4th ed) and Dixit, A. and B. Nalebuff; The art of strategy: a
Game Theorist’s guide to success in business and life; Norton; 2008.
• Some other texts on game theory that I like:
– Binmore, K.; Playing for real; OUP; 2007,
– Osborne, M.; An introduction to Game Theory; OUP; 2004, and
– Watson, J.; Strategy: an introduction to Game Theory; Norton,
2013 (3rd ed).
• There are lots of useful sources of information on the internet. In particular,
check out http://www.gametheory.net/. If you have a lot of
time, Benjamin Polak’s lectures on Game Theory all the way from Yale
are available at https://www.youtube.com/playlist?list=PL70344AC73AE276E3.
For inspiration, try searching “game theory” and “commitment” on
Google News.
• Don’t forget Thomas Schelling!

Harvard style

EC935 Games of Strategy SBS Spring School Assessment Essay 2016/17 You are required to submit one assessment essay that contributes 75% to the final mark for the class. Your assessment essay must be entirely your own work and be fully referenced where appropriate. The maximum word limit is 3,000 words, although students are encouraged to write concisely in 2,000 words. The submission deadline is 5pm on Monday 24th April 2017. Feedback and marks will be available by Monday 15th May 2017. The Question Identify a real-world environment in which there is strategic interdependence and commitment can play a central role for a decision-maker in improving their outcome from a strategic interaction. After describing the features of the environment, formally model it as a game explaining the assumptions you make; consider an analysis of the environment using the tools of game theory; consider the credibility of threats/promises that players might like to make; and carefully analyse the efficacy of at least one commitment mechanism. In addition, critically reflect on the use of game theory in understanding the real world. Some Advice Throughout your work you should seek to demonstrate your understanding of the conceptual framework of game theory and commitment in particular (the ‘science’), and how it is applied to real world settings (the ‘art’). Essays are unlikely to be successful if they do not demonstrate full understanding of the theoretical framework that will be achieved by detailed explanation of equilibrium concepts and careful analysis and explanation of the strategic environments studied. Your work should evidence exploration of the literature that goes beyond the basic recommended readings in the class to demonstrate independent study and thought. Remember: your work should be your own individ- 1 ual account (this is not group work) and all sources should be referenced appropriately. References Below are some sources that you might find useful in completing your essay. • A good starting place, but by no means sufficient in the context of the essay, is Dixit, A., S. Skeath and D. Riley; Games of strategy; Norton, 2015 (4th ed) and Dixit, A. and B. Nalebuff; The art of strategy: a Game Theorist’s guide to success in business and life; Norton; 2008. • Some other texts on game theory that I like: – Binmore, K.; Playing for real; OUP; 2007, – Osborne, M.; An introduction to Game Theory; OUP; 2004, and – Watson, J.; Strategy: an introduction to Game Theory; Norton, 2013 (3rd ed). • There are lots of useful sources of information on the internet. In particular, check out http://www.gametheory.net/. If you have a lot of time, Benjamin Polak’s lectures on Game Theory all the way from Yale are available at https://www.youtube.com/playlist?list=PL70344AC73AE276E3. For inspiration, try searching “game theory” and “commitment” on Google News. • Don’t forget Thomas Schelling! 2 Assessment and Submission Guidelines: • This assignment contributes 75% of the marks for this class. Each student is required to submit an essay of not more than 3,000 words on the question given. • You must submit a pdf of your assignment electronically via myplace following the instructions on the EC935 Games of Strategy myplace page. • Normal academic conventions on citation of the works of others and web resources used apply. It is imperative therefore that you very carefully study the University Rules relating to plagiarism at http://www.strath.ac.uk/plagiarism/. • A penalty of 2% per day will be levied on late submissions. Extensions will only be granted in exceptional circumstances and must be agreed by the Programme Director prior to the submission deadline. 3

GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Games of Strategy SBS Spring School Alex Dickson University of Strathclyde Spring 2017 GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Games of Strategy Welcome! to Games of Strategy. 10 credit SBS MSc class. Alex Dickson. Duncan Wing 5.08, alex.dickson@strath.ac.uk. ‘Intensive’ class: combination of lectures and tutorials. Organization. ‘Learning by doing’ and participation. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Games of Strategy Assessment: 25% participation and group presentations; 75% individual assignment. Participation: zero for no participation; 30% for sub-standard participation; 50% for standard participation; 80% for continued active participation; 100% if your contribution to the class is deemed excellent. Peer assessment of other group members. Submit by 5pm the day after the class finishes; failure to do so will result in your participation mark being capped at 30%. Assignment: The instructions for the assignment (3,000 words max, but you are encouraged to write concisely in 2,000 words) as well as information on deadlines will be distributed during the class. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Objectives of this Introduction Game Theory 1 Understand the motivation for studying Game Theory. 2 Understand the sorts of environments where Game Theory is useful. 3 Understand the differences between types of games. 4 Appreciate the importance and uses of Game Theory. 5 Be motivated to study Game Theory and foresee the benefits! GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Reading Reading Dixit, Skeath and Reiley; ch 1, ch 2. Dixit and Nalebuff; Introduction and ch 1. Dufwenberg, M.; “Game Theory”; WIREs Cognitive Science (available on myplace). GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Decisions Decisions, decisions! We are regularly called upon to make decisions. Seek to make the best choice from the set of available alternatives… according to some criteria. Decision-making is ubiquitous in everyday life, as it is in business and politics. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Decision Making Choose amongst your available actions, each of which has well-defined outcomes. Decision making can be simple. May also be very complex… Outcomes may be uncertain. Outcomes may be unknown. How actions map into outcomes may be complicated. This decision-making process may be inter-twined with others… But: so far considering a decision made by you that affects your outcome. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Decisions vs. Games What happens if your decisions influence the outcome for others, and others’ decisions influence the outcome for you? You will become very interested in what others are doing. And others will become very interested in what you are doing. Decisions have an element of inter-dependency. A game! Ubiquitous: we live, work, play (and evolve!) in a society, not in isolation! GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Games This inter-dependence of decisions reveals a rich tapestry of strategic interactions. Game Theory is all about the study of decision making in such strategic environments. In many decision environments, the outcome individuals receive is not determined solely by their own actions. Sometimes, the effects are small and can be assumed away. If not, effective decision makers must think about the game that is being played. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Games Acting na¨ıvely, as though you are in isolation, may result in mis-predicting the outcome of your actions. May also result in you making sub-optimal decisions. Must think about the actions your opponents might take and how your actions (or the actions your opponents expect you to take) might provoke re-actions in your opponents. And realize that your opponents will be thinking in the same way about you! Study of Game Theory will give you the tools recognize such decision environments and to think, and act, strategically! GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Some Examples Any action by a firm in a market. Investment decisions. Mitigating pollution. Ebay. Attempts to motivate employees. Countries in international negotiations. Hitler and Stalin. Kruschev and Kennedy (Cuban missile crisis). Others? Important applications in Economics; Business; Environment; etc. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Classifying Strategic Situations There are different types of strategic situations, and different tools that should be applied depending on the strategic situation. What is important in defining the game? Players, actions, payoffs from combinations of actions. The order that actions are taken: Simultaneous: without observing others’ actions (imperfect information). Sequential: others’ actions observed before choices are made (perfect information). Does everyone know everything about the rules of the game? Is information complete or incomplete? GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Games of Strategy Schedule for Games of Strategy: GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Cooperative vs. Non-cooperative Non-cooperative Game Theory focuses on situations in which players choose actions independently. Cannot form legally-binding contracts with other players in the game. In cooperative Game Theory, players can form coalitions and agree to split the proceeds in some way. Our focus in this class will mainly be on non-cooperative Game Theory. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk A Simple Example Alice and Bob have to decide between a skiing holiday or a beach holiday. Alice prefers skiing, Bob prefers the beach, but both would prefer to go on holiday together. Sequential moves: Bea Ski A Bea 1,1 Ski 3,2 B Bea 2,3 Ski 1,1 B Simultaneous moves: Alice Bob Ski Bea Ski 3,2 1,1 Bea 1,1 2,3 GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk On Abstraction We will not be taking a ‘case study’ approach to studying Game Theory. Abstraction from reality. Think of the simplest setting possible that gives rise to a particular phenomenon. Learn the principles from these stylized examples. Knowledge is then portable: apply it whenever you see the basic features of the example. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Games in the Real World The ‘real world’ is a complicated place. Need to abstract from reality to say anything meaningful. Consider the setting that you want to analyze. Distill the crucial information required to define the game. Decide on the boundaries of the game. Construct reasonable payoff representations for the strategic actors in the game. Have half an eye on the ‘robustness’ of your conclusions to changing these boundaries/payoff representations. The Science and Art of Game Theory. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Some Motivating Examples Splitting the bill. Consider yourself in a restaurant with 6 friends. You order, you pay the bill for anything you consume. [A decision] “My my, that fillet steak sounds nice but I couldn’t possibly justify spending £30 when I could have the chicken for £16.” Now suppose you are splitting the bill… If I do order the fillet steak, I only have to pay an extra £2. What a bargain! [But a game!] …except everyone is thinking in the same way! Gneezy, U., E. Haruvy and H. Yafe; 2004; “The inefficiency of splitting the bill”; The Economic Journal; 114, pp 265-80. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Some Motivating Examples UK Government and international negotiations; Financial Services Tax; banking regulation. Financial services tax and banking regulation in the face of the 2007-8 financial crisis. Gov’t is not acting in isolation…other countries making similar decisions. Large banks/financial services firms affected by it. Govt’s ‘payoff’ depends on the actions, and re-actions of all these players. Brexit negotiations will depend on agreement from many stakeholders. Each affected by UKs actions, and will respond accordingly. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Some Motivating Examples Common access resources and ‘social dilemmas’. Fishermen extract resources from fishing grounds. Each cares about their own profit. Catching fish imposes costs on others. Over-fishing, relative to the efficient level. Individual fishermen don’t take into account the negative effects of their actions on others. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Some Motivating Examples The Market Environment. Businesses compete in markets. The market environment affects all businesses in that market. Actions to better your business may have adverse consequences on others in the market. Are your competitors likely to react? Actions might also have a complementary effect in the market. Thought needs to be quite deep in strategic situations. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk There’s been a Murder… Prisoners’ Dilemma Alice and Bob committed a murder. There’s enough evidence to convict them of GBH, but not of the murder. Sherlock Holmes is questioning Alice in one room and Dr Watson is questioning Bob in another room. Alice and Bob have to choose independently whether to deny the murder or confess to it. If both deny, they go to jail for 3 years. If one confesses and the other denies, the confessor gets an easy ride (2 years) whilst the liar gets punished heavily (10 years). If they both confess, they each go down for 8 years. Alice Bob deny confess deny 3,3 10,2 confess 2,10 8,8 GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Incentives A key point that emerges from these examples is the following: Rule of Strategy Players are guided by the incentives they face. It is the incentives a player faces that determines the actions she chooses to take, so we need to carefully consider the incentives individuals face, which will depend on the environment in which individuals are acting. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Uses of Game Theory Explanation of why things happen. Prediction of outcomes of strategic environments. Advice to participants of games. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Why Study Game Theory? You survived so far without it, why now? Start to think about how others think about making decisions. Expand the domain of your thoughts when making decisions. If other people are trying to outmaneuver you, you don’t want to be the proverbial sitting duck! Indeed, your choice of studying Game Theory could itself be modeled as a game! GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Learning Objectives of GoS 1 Understand the difference between decisions and games and learn to identify when a decision environment should be analyzed as a game. 2 To distinguish between different types of game and apply the appropriate tools to analyze the game. 3 To understand the concept of equilibrium in games and use it to predict the outcome. 4 Appreciate the power of commitment in strategic environments. 5 Abstraction and problem-solving skills; communication skills. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Summary Introduction to the class and the joys to come. Simple decisions vs. playing games. Strategic environments. Think about others, and realize they are thinking about you. Motivating examples.

GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Games of Strategy SBS Spring School Alex Dickson University of Strathclyde Spring 2017 GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Simultaneous-Move Games Strategic environments where decisions are made simultaneously. Players make decisions without knowledge of others’ decisions. ‘Strategic uncertainty’. How do players make decisions? What is a likely prediction of the outcome of the game? Array of games that represent many strategic situations. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Aims and Objectives 1 Understand when a strategic decision environment features simultaneous-moves. 2 Understand what features of the environment are necessary to define the game. 3 Understand how to represent games. 4 Understand the concepts of dominant and dominated strategies. 5 Learn what a best response is, and what a Nash equilibrium is. 6 Become familiar with some elementary games, and recognise their features in environments that you are familiar with. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Reading Reading Dixit, Skeath and Reiley; ch 4, ch 5. Dixit and Nalebuff; ch 4. Rothschild, R.; 1995; “Ten simple lessons in strategy from the games firms play”; Management Decision; 33, pp 24-9. David Levine’s “What is Game Theory?” page: http://www.dklevine.com/general/whatis.htm. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Simultaneous-Move Games Consider yourself in a decision environment: faced with a set of choices. You want to make a choice/decision that leads to the best possible outcome. But the outcome of each choice depends on what your adversaries do. Your optimal choice may very well depend on your adversary’s actions. Decisions are made at the same time: no decision-maker observes the actions of any other before making their decision. Have to think about how others are thinking: get into their minds. Want a ‘forecast’ of their choices, since you will probably want to tailor your decision to what you expect them to do. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Simultaneous-Move Games If no player has the opportunity to observe the actions of any other player when they need to make a decision, the game is a simultaneous-move game. Players are subject to strategic uncertainty. The basic environment in which we analyze strategic decision-making. The defining features of the environment are the players, their available strategies and the payoffs to each player. Payoffs can be seen as a monetary equivalent of the outcome. Once we have this information, we can construct the ‘strategic form’ (aka ‘normal form’) of the game, and analyze it. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk The Strategic Form Focus on two-player games. The strategies of players are simply the actions that are available to them. The strategic (or normal) form is a payoff matrix detailing the payoffs the two players receive from the various combinations of strategies the players might use. Row Column L R U 3,4 4,2 D 1,6 10,1 Row player chooses the rows. Column player chooses the columns. Each cell corresponds to an outcome. First entry is the row player’s payoff. Second entry is the column player’s payoff. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk The Strategic Form Players will not necessarily only have two strategies. 1 2 X Y Z A 4,0 2,1 3,2 B 2,2 3,4 0,1 C 2,3 1,2 0,3 Players can have an unequal number of strategies. A B left middle right up 3,2 4,6 1,1 down 9,3 1,6 3,8 GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Making Decisions Making a ‘simple decision’ is a relatively straightforward task. Look at all the available options, choose the one that gives you the highest payoff. You (row player) quite like the look of the payoff of 10 in the previous game. Likely to be achieved? Your opponent’s behavior is not fixed! Think carefully about their decision-making process. Na¨ıve decisions can have serious consequences! Need to think about what the other player might do, and consider your best course of action given your beliefs. Also realize that your opponent is undertaking the same thought process. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Best Responses Players form beliefs about what the other player might do, and choose the strategy that gives them the highest payoff. Players best respond to their beliefs. This is the fundamental notion of rationality. For each strategy of the other player, find the strategy of the player in question that gives them the highest payoff. Method of underlining. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Best Responses How to find best responses: 1 a) Pick a player. b) Fix the strategy of the other player. c) Find the strategy that maximizes the player in question’s payoff. Underline payoff. 2 Repeat for all strategies of the other player. 3 Repeat for the other player. Row Column L R U 3,4 4,2 D 1,6 10,1 GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Playing the Game… Suppose you actually came to play this game. Would need to think about the other player. Would also need to think about what the other player is thinking about you. Each player’s belief needs to be consistent with their actions. No player would want to change, given the actions of the other player. Both players must be using a best response, at the same time. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Nash Equilibrium Equilibrium [ee-kwuh-lib-ree-uhm]: a state of rest or balance. Unless players are both using best responses to the strategies of the other, the strategies cannot form an equilibrium… …at least one player could make a unilateral deviation and make themselves better off. If both players are using best responses, each is doing the best they can, given the choices of the other player. Nash equilibrium [John Forbes Nash, Jr. (1928-2015)]. Players use ‘mutually consistent’ best responses. Prediction of the outcome in the game. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Nash Equilibrium Row Column L R U 3,4 4,2 D 1,6 10,1 Nash equilibrium is (U,L) with payoffs (3,3). Row is doing the best they can in the game, but can’t achieve the magic 10. Rule of Strategy When your decision environment is a game, your best outcome is not always achievable. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk The Glasgow Business Schools Marketing Game SBS and UGBS are each deciding on the intensity of their marketing strategies for their MBA programmes. Marketing is costly, and its effectiveness is dependent on the intensity of marketing undertaken by both institutions. SBS UGBS I M L I 2,2 3,1 3,3 M 1,3 5,5 7,2 L 3,3 2,7 6,6 Intense, Medium or Low? Nash equilibrium: (M,M) with payoffs (5,5) GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Nash Equilibrium – Discussion Players are rational and best respond to their beliefs. Players think about the way others think. Common knowledge of rationality. I am rational. You know that I am rational. I know that you know that I am rational. You know that I know that you know that I am rational. Etc… Nash equilibrium (in ‘pure strategies’) may not exist. Might be many Nash equilibria. Lots of work on relaxing the CKR assumption. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Dominant Strategies In some games, players may have one strategy that is always better than any of the other strategies available to them, for every choice of strategy that may be chosen by their opponent. If so, the player is said to have a dominant strategy. Rule of Strategy If you have a dominant strategy it is better for you than any other available strategy regardless of what your opponent does, so use it! If both players (in a 2 player game) have a dominant strategy, then they should both use them, and the game has a dominant strategy equilibrium. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Dominant Strategies Dominant strategy equilibrium is a very strong solution concept: only requires players to be rational. SBS UGBS I M L I 2,2 3,4 3,3 M 4,3 5,5 7,2 L 3,3 2,7 6,6 SBS has a dominant strategy: M. For each strategy choice of UGBS, M gives a higher payoff than any other strategy. Likewise, M is a dominant strategy for UGBS. There is a dominant strategy equilibrium. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Dominant Strategies It may be the case that only one player has a dominant strategy. Can be sure that they will use it! Other player needs only to think about best responding to that dominant strategy. In our introductory example: Row Column L R U 3,3 4,2 D 1,6 10,1 Rule of Strategy If your opponent has a dominant strategy they will always use it, so don’t waste your time trying to achieve anything that involves them doing anything else. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Dominated Strategies Not all games will have dominant strategies… …in fact most don’t! Players may, however, have strategies that are dominated. A strategy is a dominated strategy if there is another strategy available that is always better for you, irrespective of the choice of your opponent. Rational players will not play dominated strategies, so if they exist they can be eliminated from consideration. Players infer that if their opponent has a dominated strategy they will not use it – can shape decisions. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Dominated Strategies Iterated elimination of dominated strategies. If any player has a dominated strategy, eliminate it from consideration. Successively eliminate dominated strategies. SBS UGBS I M L I 2,7 3,1 3,3 M 1,3 5,5 7,4 L 3,3 4,7 6,6 If strategies can be iteratively eliminated, the game is said to be dominance solvable. Sherlock Holmes (Arthur Conan Doyle): “…when you have eliminated the impossible, whatever remains, however improbable, must be the truth.” GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Dominated Strategies Not all games are dominance solvable, even if there are some dominated strategies. SBS UGBS I M L I 2,2 3,1 3,3 M 1,3 5,5 4,4 L 3,3 4,4 6,6 Nevertheless, dominated strategies can be eliminated from consideration, to leave a simpler game. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Games of Pure Conflict In some games, one player’s gain is another’s loss. The sum of the players payoffs is zero (or constant). Game of pure conflict: harming the other player is good for you! Example: splitting a pie of size 10 The game: Alice Bob L M R U 3,7 4,6 1,9 M 7,3 6,4 8,2 D 9,1 3,7 5,5 Relative to equal split: Alice Bob L M R U -2,2 -1,1 -4,4 M 2,-2 1,-1 3,-3 D 4,-4 -2,2 0,0 GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Competitive vs Cooperative ‘Pie splitting’ games can represent many interesting strategic situations. Zero sum games are purely competitive. Dedicate effort to splitting a pie of a fixed size: any gain for you is a loss for the other player. But efforts in competing over sharing the pie could also increase the size of the pie. A gain for you no longer necessarily implies a loss for the other player. What sort of environments are we talking about here? Cooperative competition: Brandenburger, A. and B. Nalebuff; 1997; Coopetition; Profile. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Some Classic Games There are a handful of games that represent many common strategic situations that we come across. 1 Pure Coordination: Alice Bob Opera Football Opera 1,1 0,0 Football 0,0 1,1 Both players simply want to coordinate their actions. Two Nash equilibria, and the potential for mis-coordination. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Some Classic Games 2 Assurance (Stag Hunt): Alice Bob Stag Hare Stag 2,2 0,1 Hare 1,0 1,1 Coordination again. Two Nash equilibria, (stag, stag) and (hare, hare). Hunt a stag if others are hunting a stag, but otherwise best to hunt a hare. Need assurance that others are going for the big gains. Ubiquitous in business. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Some Classic Games 3 Battle of the Sexes: Alice Bob Opera Football Opera 2,1 0,0 Football 0,0 1,2 Again, two Nash equilibria. Players want to coordinate, but each has a preferred outcome. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk An Interluding Lesson In coordination games, can you do anything in an attempt to get your most preferred outcome? Rule of Strategy In games where there is no opportunity to agree a course of action you should select (and if possible signal your commitment to) a clear strategy to encourage your opponent to behave in a manner congenial to yourself. Put another way… it’s not always best to play your cards close to your chest! GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Some Classic Games 4 Chicken: Alice Bob Chicken Tough Chicken 1,1 0,2 Tough 2,0 -3,-3 Each wants to be the tough guy whilst the other is a chicken. Players want to avoid actions with the same labels, especially Tough. Entry into a small market. Intensive marketing campaigns. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Some Classic Games 5 Prisoners’ Dilemma: Alice Bob Cooperate Defect Cooperate 3,3 1,4 Defect 4,1 2,2 Each player has a dominant strategy to defect on cooperation. But there are mutual gains to cooperating. Cannot be realized, since there is always an incentive to cheat. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Some Classic Games 6 Matching Pennies: Alice Bob H T H 1,-1 -1,1 T -1,1 1,-1 Zero sum game of pure conflict. No Nash equilibrium (at least in pure strategies). Need to play your cards very close to your chest to avoid being exploited by your opponent! GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Three-Player Games? 3 Contributes 1 2 C D C 5,5,5 3,6,3 D 6,3,3 4,4,1 3 Doesn’t contribute 1 2 C D C 3,3,6 1,4,4 D 4,1,4 2,2,2 1 chooses rows, 2 chooses columns, 3 chooses tables. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Summary When players make decisions without observing the actions of other players, moves are simultaneous. Key information required: players, available strategies, payoffs. Define boundaries of the game. Construct the strategic, or normal, form. Apply the tools of analysis to the strategic form. Dominant strategies, dominated strategies, Nash equilibrium. Think about your opponent. Understand that they are trying to out-manoeuver you. Attempt to out-manoeuver them. Don’t be strategically na¨ıve!

GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Games of Strategy SBS Spring School Alex Dickson University of Strathclyde Spring 2017 GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Dilemmas and Cooperation Individuals make self-interested choices, knowing that others are doing the same. Choose the best possible course of action, given your conjecture about the actions of others. In the context of society, what are the implications? Basic game: Prisoners’ Dilemma. Individual self-interest can lead to ‘bad’ outcomes from the perspective of society. Any way to regain what’s best for society? GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Aims and Objectives 1 Understand the basic motivation of players in games that have the features of a prisoners’ dilemma. 2 Understand the implications this has for expected outcomes. 3 Consider the responses to these conclusions, as players and as policy makers. 4 Appreciate the ubiquity of prisoners’ dilemmas. 5 Develop an understanding of how cooperation can emerge. 6 Have an awareness of alternative ways of thinking about such games and the experimental evidence. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Reading Reading Dixit, Skeath and Reiley; ch 10. Dixit and Nalebuff; ch 3. Game theory.net news articles. Axelrod, R.; “The Emergence of Cooperation among Egoists”; The American Political Science Review; 75, pp 306-18 GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk The Prisoners’ Dilemma Alice and Bob committed a murder. There’s enough evidence to convict them of GBH, but not of the murder. Sherlock Holmes is questioning Alice in one room and Dr Watson is questioning Bob in another room. Alice and Bob have to choose independently whether to deny the murder or confess to it. If both deny, they go to jail for 3 years. If one confesses and the other denies, the confessor gets an easy ride (2 years) whilst the liar gets punished heavily (10 years). If they both confess, they each go down for 8 years. Alice Bob deny confess deny 3,3 10,2 confess 2,10 8,8 GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk The Prisoners’ Dilemma The general form of the prisoners’ dilemma: Alice Bob cooperate defect cooperate C,C L,H defect H,L D,D H>C>D>L Each player has a dominant strategy to ‘defect’ on ‘cooperation’, but there are gains from mutual cooperation. When players act according to their incentives they exert ‘negative externalities’ on the other player. Strong ‘positive externalities’ from cooperating. Since players will use dominant strategies if they have them, dominant strategy equilibrium is Defect, Defect. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk The Prisoners’ Dilemma Each player has a personal incentive to do something that ultimately leads to a result that is bad for everyone. If you like: private rationality leads to collective stupidity! Implies that the outcome could be better for everyone if they could somehow coordinate on cooperation. Communication doesn’t work: ‘cheap talk’. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Examples There are many many illustrations of the prisoners’ dilemma. Formation of cartels to restrict output/raise prices. Price wars. Advertising. Working vs shirking in a partnership (free riding). Carbon emissions mitigation. Contribution to a common resource. ‘Tragedy of the commons’. Collective action problems: there are benefits to acting collectively but an individual incentive to breach these collective agreements. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Example – Price War The prisoners’ dilemma is a classic game that appears ubiquitously throughout life and in business. A B high low high 600,600 170,1000 low 1000,170 400,400 GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Example – Cartel Formation Firms in a market forming an agreement in a cartel to restrict output/raise prices to the detriment of consumers. A B cartel defect cartel 5m,5m 2m,7m defect 7m,2m 3m,3m Individual incentive to break cartel agreements, but collectively can rule the market. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Example – Advertising Recall the SBS-UGBS advertising game… SBS UGBS I M L I 2,2 1,1 3,3 M 1,1 5,5 7,2 L 3,3 2,7 6,6 I dominated for each player. Once removed, a prisoners’ dilemma! GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Example – Doping in Cycling Cyclists considering whether to use performance-enhancing drugs. A B clean dope clean 6,6 2,8 dope 8,2 4,4 Dominant strategy to take drugs! See “The Doping Dilemma”; Scientific American. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Example – Emissions Reductions Two countries engaging in activity to reduce emissions. A B cut don’t cut -1,-1 -20,0 don’t 0,-20 -12,-12 Collectively better to reduce emissions, but individual incentive not to spend resources on abatement. See “How to Save the Planet: Be Nice, Retaliatory, Forgiving and Clear”. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Just Because You’re in a Dilemma… Note that the games we have been looking at so far have been symmetric. Both you and your opponent face the same incentives. If the game is asymmetric, there may not be a dilemma! A B R&D No R&D R&D -2,-2 -2,0 No R&D 0,-2 -0.8,-2.4 Rule of Strategy Don’t presume that your opponent faces the same incentives as you just because you’re playing the same game. If their payoffs differ from yours, their incentives may be different. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk The Prisoners’ Dilemma The general form of the prisoners’ dilemma: A B coop def coop C,C L,H def H,L D,D Dominant strategy to defect. Negative ‘externalities’ from following individual incentives. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk An Interpretation via Externalities Prisoners’ dilemma is a simple case of more general issues: an important idea! One’s actions confer externalities on others when they influence others well-being. Positive externalities; negative externalities. People don’t internalise the externalities they generate: they only care about their private costs and benefits, not social costs and benefits. Positive externalities: don’t see the benefits conferred to others, so too little is done and everyone can be better off by doing more. Negative externalities: don’t see the costs exerted on others, so too much of the harmful activity is done and everyone can be better off by doing less. Ostrom, E.; 1990; Governing the Commons; CUP. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Cooperation You will, at some point, find yourself in a prisoners’ dilemma. Mutual benefits from achieving cooperation between you and your adversary. There may not be an opportunity to write a formal contract, enforceable by law. Communication unlikely to induce cooperation: ‘cheap talk’ uninformative. What else can be done? Can the mutually beneficial cooperative outcome be achieved by overcoming the incentives to defect? GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Rewards… Could promise to reward people for cooperation. The promise might not be credible: there is nothing to prevent you not giving the reward. The reward required to prevent defection might not be affordable! But see Andreoni, J. and H. Varian; 1999; “Preplay contracting in the Prisoners Dilemma”; Proc. Natl. Acad. Sci; 96, pp 10933-8. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk …and Punishments Could institute a post-game penalty on any lone defector in the game. Alice Bob cooperate defect cooperate C,C L,H-P defect H-P,L D,D Must be large enough to make cooperating the better option: costly punishment erodes the benefits from cooperation. Could also penalize any use of the defect strategy. Perhaps requires third party enforcement. (Mafia?) Since cooperation might harm ‘society’ may not be able to implement punishments. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Internal and External Norms of Behavior Players may be constrained by internal and/or external norms of behavior. Acting in contrast with those norms gives rise to a psychological cost. If large enough, can temper people’s behavior in defecting on cooperation. Internal norms – fairness; external norms – social norms. Alice Bob cooperate defect cooperate C,C L,H-K defect H-K,L D,D GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Repetition of the Game In many situations the game is not just played once. Game is repeated on a continuous basis. Finite number of times, for T periods. Indefinitely. Possibility for dynamic relationships to emerge. Future periods, in which players may expect defection to be ‘punished’. Players might adopt strategies that are contingent on past behavior: ‘trigger strategies’. Start by cooperating. If your opponent defects then you stop cooperating to punish your opponent. Can cooperation be sustained? In an entirely tacit (implicit) way? GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Dynamic Punishment Remember the pricing game. A B high low high 600,600 170,1000 low 1000,170 400,400 Begin by cooperating. If your opponent defects, you can levy a ‘punishment’ by defecting next period. The opportunity to punish implies a cost to defecting on cooperation. Can mutual cooperation can be sustained through the ‘fear’ of being punished by your opponent? If you end up both defecting, no opportunity for punishment. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Finite Repetition Suppose you get to the last period. Whatever has happened up to now, you have a dominant strategy, so should use it. Outcome is (defect,defect) in the final period. Now think about the penultimate period. The outcome next period will be mutual defection, so there is no opportunity for your opponent to punish you. Then you may as well try to screw your opponent over this period. Outcome is mutual defection. In the period prior to this, the same reasoning applies. Since no future punishments can be levied, same is true in all periods. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Finite Repetition In finitely repeated games, it is unlikely that mutual cooperation will be achieved through tacit means. The presence of a definite end point gives incentives for defection, which unwind through all periods. Even though there is a dynamic relationship, no opportunity for punishment. If you’re operating in time-limited environments playing games can be a tough business! GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Infinitely Repeated Games Suppose now the game continues infinitely. (Or at least there is a positive probability that the game will continue to the next period – indefinite repetition.) May allow us to get around the issue of non-cooperation in the finitely repeated game. Before, players infer that defection already occurs therefore opponent has no opportunity to punish you! Now there is no final period, so there is always the opportunity for punishment of non-cooperation. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Trigger Strategies We will consider contingent, or trigger strategies: a player’s action in the following period depends on what their opponent did this period. ‘Tit-for-tat’ trigger strategy Start by cooperating. Then play as opponent played in the last round. ⋆ Cooperation followed by cooperation. ⋆ Defection followed by defection. Start by being nice, then “Do unto others as they have done to you”. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Trigger Strategies ‘Grim’ trigger strategy Start by cooperating. Then: ⋆ If opponent cooperated in the previous round, cooperate in this round. ⋆ If opponent defected in the previous round, defect in this round, and forever afterwards. Start by being nice, then if someone is nasty to you screw them over forever more! GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Trigger Strategies TFT and Grim are two ends of the extremes of trigger strategies. TFT is very forgiving, but might not be enough to induce cooperation. Grim is the least forgiving, but is likely to induce cooperation. If grim works, cooperation is possible. If TFT works, cooperation is easy. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk A Tool – Discounting Need to evaluate a stream of payoffs. In today’s terms, money in the future worth less than today – present value of future cash. Interest rate is r. Invest £1 today, grows to £1(1 + r) tomorrow. £1 tomorrow worth £ 1 1+r today. Value today (period 0) of £1 in the future? Period 0 1 2 3 . . . T PV 1 1 1+r 1 (1+r) 2 1 (1+r) 3 . . . 1 (1+r) T £1 in period t worth 1 (1+r) t today. £X in period t worth 1 (1+r) t X today. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk A Tool – Discounting r is the discount rate. δ = 1 1+r is the discount factor. r = 1/10 (10%), δ = 10/11. 1 (1+r) t = δ t . So £X in period t worth £δ tX today. δ close to 1: patient; δ small: impatient. Cashflow stream: X0, X1, X2, X3, . . . , XT . P V = X0 + δX1 + δ 2X2 + δ 3X3 + · · · + δ T XT . GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Discounting Infinite Streams of Cash Suppose we have £1 this period and in every period for ever more. PV is 1 + δ + δ 2 + δ 3 + · · · + δ t + · · · . Takes a while using your calculator! Infinite geometric series. Mathematical fact: 1 + δ + δ 2 + δ 3 + · · · = 1 1 − δ GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Discounting Infinite Streams of Cash 1 + δ + δ 2 + δ 3 + · · · = 1 1 − δ Payoff of 7 in every period from now till forever? P V = 7 + 7δ + 7δ 2 + 7δ 3 + · · · = 7(1 + δ + δ 2 + δ 3 + · · ·) = 7 1 1 − δ Payoff of 10 this period, 2 next period and 2 forever more? P V = 10 + 2δ + 2δ 2 + 2δ 3 + · · · = 10 + 2δ(1 + δ + δ 2 + δ 3 + · · ·) = 10 + 2δ 1 1 − δ GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Grim Trigger Strategy Suppose you believe your opponent is playing grim. Should you defect on cooperation? A B high low high 600,600 170,1000 low 1000,170 400,400 A: Cooperate, and your opponent will – get 600 forever. 600 + 600δ + 600δ 2 + 600δ 3 + · · · = 600 1−δ . B: Cheat: get 1000 this period, but only 400 in all subsequent periods. 1000 + 400δ + 400δ 2 + 400δ 3 + · · · = 1000 + 400δ 1−δ . Cooperate ⇔ 600 1 − δ ≥ 1000 + 400δ 1 − δ ⇔ δ ≥ 2 3 GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Nash Equilibrium? Grim strategy: play cooperate so long as opponent cooperates. Is grim vs. grim a NE? With opponent playing grim, can any player deviate and get a higher payoff? Deviation: whilst opponent is still cooperating, choose defect. Then your opponent will defect forever (and so will you). Lower payoff so long as δ ≥ 2 3 (sufficiently patient). If δ ≥ 2 3 grim vs. grim is a NE. And cooperation is sustained. Dilemma resolved! GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Tit-for-Tat Grim yields massive punishment for deviation from cooperation. Anything a little kinder? Tit-for-tat: start by cooperating, then take the same action as opponent in the previous period. A B high low high 600,600 170,1000 low 1000,170 400,400 Defect followed by defect. But, cooperation can be restored after defection. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Tit-for-Tat A B high low high 600,600 170,1000 low 1000,170 400,400 Suppose you believe your opponent is playing TFT. Should you defect? Defect forever – same as grim: won’t if r ≤ 50% Defect once: gain 400 this period, but must lose 430 next period to restore cooperation. Will continue to cooperate if 400 ≤ 430δ ⇒ δ ≥ 40 43 . If δ ≥ 40 43 (sufficiently patient) cooperation will be sustained. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Nash Equilibrium Is TFT vs TFT a Nash equilibrium? Suppose you believe your opponent is playing TFT. Started cooperating, so will be cooperating. Can you do better? Must defect. Lower payoff so long as δ ≥ 40 43 . Yes? NE. And cooperation sustained. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Changes to The Game Increased benefits of defection: A B high low high 600,600 170,1000 low 1200,170 400,400 B: as before. A: Grim requires δ > 3/4. A: TFT requires δ > 60/43 > 1!. Cooperation can’t be sustained by TFT. Larger punishment: A B high low high 600,600 100,1000 low 1000,170 400,400 B: as before. A: Grim is as before. A: TFT requires δ > 4/5. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Aside – Infinitely Repeated? The game doesn’t actually have to be repeated an infinite number of times. Just need for there to be a chance that the game will continue in the future. Prob of game continuing is ρ. (I.e. prob. of game ending is 1 − ρ) Effective discount factor is δ = ρ 1 + r . GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Equilibria in Infinitely Repeated Games If players form beliefs that their opponent will punish them from defecting, cooperation may be sustained. Potential for Grim vs. Grim and TFT vs. TFT to both be NE if the players have appropriate discount factors. TFT vs TFT harder to satisfy: cooperation may not be easy! But in many instances, cooperation can be sustained. Out of the prisoners’ dilemma. Repeated interaction can sustain collusion. Note: this may be a cause for concern, depending on the game! GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Factors Conducive to Cooperation Factors that affect the likelihood of sustaining cooperation. Size of the gain from defecting. Size of the punishment that can be levied. Ability to detect defection. Ability to levy punishments, and the expectation of being punished. Stability of the game. Establish systems to detect and punish cheating: punishment should be strong, swift and certain. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Summary An incredibly simple game has an abundance of content! Prisoners’ Dilemma: mutual gains from cooperation, but an individual incentive to defect on cooperation. Dominant strategies. Players defect. Can cooperation be sustained? Repeated games. Implications for doing the best you can in dilemmas.

GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Games of Strategy SBS Spring School Alex Dickson University of Strathclyde Spring 2017 GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Mixed Strategies So far, considered players using only pure strategies. Undertake a certain course of action, for sure. In some games, you may want to keep your opponent guessing… Randomize between pure strategies. Equilibrium when we allow for randomization? How to outdo your opponent, and not be outdone by your opponent? Prediction of outcome? In other games, there may be coordination issues… Mixed strategies can provide a reasonable prediction of the outcome. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Aims and Objectives 1 Understand the meaning of a mixed strategy. 2 Appreciate the value of randomizing over your choice of strategies to keep your opponent guessing. 3 Understand how to deduce the equilibrium in mixed strategies of 2×2 games. 4 To be able to identify when a game will have an equilibrium in mixed strategies. 5 To be able to offer a probabilistic prediction of the outcome of the game. 6 To understand the implications of mixed strategy equilibria for behavior. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Reading Reading Dixit, Skeath and Reiley; ch 7. Dixit and Nalebuff; ch 5. Tim Harford’s “World cup game theory”, available at http://tinyurl.com/6r2l3po. An academic paper on applying mixed strategies to soccer. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Example Undertaking a particular course of action with certainty may mean that you can never achieve what’s best for you, as your opponent will always prevent you from achieving it. An employee has to decide whether to Work or Shirk, and a manager has to decide whether to monitor the employee or not. Employee Manager Monitor Not Work 1,1 0,2 Shirk -1,0 2,-1 The same is true in any game with no PSNE (e.g. matching pennies). Question: How should we play in such games, and what is the likely outcome? GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Mixed Strategies Your adversary will exploit you if you undertake a certain course of action for sure. Mixing things up a little can serve to guard against this exploitation. Introduce ‘unpredictability’ into games. Randomization over strategies, but not ‘random’. Mixed strategy: specifies the probability with which players use each of their pure strategies. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Mixed Strategies Mixed strategy: specifies the probability with which you play each of your pure strategies. In a 2×2 game, it is sufficient to specify a single probability for each player. Employee Manager Monitor (q) Not (1 − q) Work (p) 1,1 0,2 Shirk (1 − p) -1,0 2,-1 We then look for a Nash equilibrium in mixed strategies: a pair of probabilities such that no player can make a profitable unilateral deviation. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Payoffs? When considering mixed strategies, a player’s payoff is their expected payoff: probability-weighted average of payoffs. Expected payoff: 50% chance of £10, 50% chance of £100, expected payoff is 0.5 × 10 + 0.5 × 100 = 55. 20% chance of -£10, 80% chance of £50, expected payoff is 0.2 × −10 + 0.8 × 50 = 38. If manager is playing q = 0.5, expected payoff to employee from working is 1 × 0.5 + 0 × 0.5 = 0.5. If q = 0.8, Expected payoff from working is 1 × 0.8 + 0 × 0.2 = 0.8. Leaving q general, the expected payoff to the employee from working is 1q + 0(1 − q). GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Mixed Strategies in Football You’re not very good at football and when taking a penalty can either shoot left (L) or right (R). Likewise, the goalkeeper can choose either L or R. If you both match, GK wins. If you mismatch, you win. You Goalkeeper L (q) R (1 − q) L (p) -1,1 1,-1 R (1 − p) 1,-1 -1,1 NO pure strategy Nash equilibrium (PSNE). Analogous to matching pennies. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Penalty Taking Consistent mixed strategies? Suppose the goalkeeper plays L 3/4 of the time and R 1/4 of the time. Then if you play L you get a goal only 1/4 of the time and if you go R you get a goal 3/4 of the time. You should play R ALL the time, but then the goalkeeper will dive R all the time. No mutually consistent mixed strategies involving GK playing L 3/4 of the time. Is there an equilibrium? GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Penalty Taking Suppose the GK goes L 1/2 the time and R 1/2 the time. Now if you play L you get a goal 1/2 the time… … and if you play R you get a goal 1/2 the time. You are indifferent between your pure strategies, so any probability of L is as good as any other. But not any is equilibrium! You must choose L 1/2 the time and R 1/2 the time. Otherwise the 50:50 mix that the GK makes is not optimal. (why?) GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Penalty Taking If the GK plays L 1/2 the time and R 1/2 the time, it is optimal for you to play L 1/2 the time and R 1/2 the time. Likewise, if you play L 1/2 the time and R 1/2 the time it is optimal for the GK to play L 1/2 the time and R 1/2 the time. Wait a minute!!! Mutually consistent probabilities… An equilibrium! GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Mixed Strategy Nash Equilibrium To be in equilibrium, your mixed strategy must make your opponent indifferent between her pure strategies. Otherwise, your opponent can use a pure strategy to make herself better off, but then your mixed strategy would no longer be best for you. I.e. your opponent cannot exploit your choice of mixed strategy by pursuing a pure strategy. Rule of Strategy When using mixed strategies, your mix should be such that your opponent cannot exploit your choice by pursuing a pure strategy, meaning you must make your opponent’s expected payoff from using each of their pure strategies exactly the same. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk How to Compute Mixed Strategies Easily GK plays L with prob. q, R with prob 1 − q. You need to be indifferent between your pure strategies: Play L get q · (−1) + (1 − q) · 1 = 1 − 2q. Play R get q · 1 + (1 − q) · (−1) = 2q − 1. Need 1 − 2q = 2q − 1 ⇒ q = 1/2. If q < 1/2, L better. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk How to Compute Mixed Strategies Easily You play L with prob p, R with prob 1 − p. GK needs to be indifferent between pure strategies: Play L get p · 1 + (1 − p) · (−1) = 2p − 1. Play R get p · (−1) + (1 − p) · 1 = 1 − 2p. Need 2p − 1 = 1 − 2p ⇒ p = 1/2. If p < 1/2, R better. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk The Equilibrium MSNE: p = q = 1/2. Probability of observing outcomes You Goalkeeper L (1/2) R (1/2) L (1/2) 1/4 1/4 R (1/2) 1/4 1/4 Expecetd payoff: 0 to you, 0 to GK. (why?) GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Understand Your Talents! Suppose now that your left side is weak: only score 1/2 the time. You Goalkeeper L (q) R (1 − q) L (p) -1,1 1 ⁄2,- 1 ⁄2 R (1 − p) 1,-1 -1,1 Alters the equilibrium mix: p = 4/7, q = 3/7. Even though your left side is weak, use it more! Otherwise your opponent will exploit your weakness. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Mixed Strategies - Discussion To successfully keep your opponent guessing, you need to make her indifferent between her pure strategies. When you come to play the game, choose pure strategy randomly, with a given probability. Note: not alternating between strategies. If patterns are detected they will be (should be!) exploited. You may suffer bad outcomes. Probabilistic prediction of equilibrium. In games with no pure strategy equilibrium, there is always a mixed strategy equilibrium. In games with more than one pure strategy equilibrium, there will also be a mixed strategy equilibrium. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Mixed Strategies in Coordination Games Consider a game in which two firms A and B decide whether to offer a price promotion. A B D (q) No D (1 − q) D(p) -2,-2 2,0 No D (1 − p) 0,2 1,1 Chicken-style game: two (pure strategy) Nash equilibria. How should the players play? Possibility for mis-coordination and bad outcomes without ’focal points’. In any game with more than one Nash equilibrium, there will also be a mixed strategy equilibrium. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Aside on Coordination Whenever there is more than one Nash equilibrium, there is an issue of coordination. Difficult to make a prediction of the outcome. Schelling-type salience to break the coordination problem? Schelling, T. The strategy of conflict. 1960, Harvard. Label salience. Payoff salience. MSNE can offer a prediction of the outcome in games with multiple PSNE. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Mixed Strategies in Coordination Games If B promotes with probability q then: A promotes, it gets q · (−2) + (1 − q) · 2 = 2 − 4q. A doesn’t promote, it gets q · 0 + (1 − q)1 = 1 − q. Indifference requires 2 − 4q = 1 − q ⇒ q = 1/3. q < 1/3, promoting better. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Mixed Strategies in Coordination Games Similar analysis for firm B... If A promotes with probability p then: B promotes, it gets p · (−2) + (1 − p) · 2 = 2 − 4p. B doesn’t promote, it gets p · 0 + (1 − p) · 1 = 1 − p. Indifference requires 2 − 4p = 1 − p ⇒ p = 1/3. p < 1/3, promoting better. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Mixed Strategies in Coordination Games MSNE: p = q = 1/3. Probability of outcomes? A B D (1/3) No D (2/3) D (1/3) 1/9 2/9 No D (2/3) 2/9 4/9 Expected payoffs? Both firms simultaneously promote very infrequently. Explains why we see infrequent promotions? GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Manager Monitoring Game Mixed strategy equilibrium in the monitoring game? Employee Manager Monitor Not Work 1,1 0,2 Shirk -1,0 2,-1 GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Summary Keep your opponent guessing, otherwise they can exploit your choice: choose strategy randomly. Probabilistic selection of strategy when playing the game. Mixed strategy equilibrium. Exists whenever there is no PSNE or multiple PSNE. Leads to probabilistic outcomes. By playing mixed strategies, certainly not guaranteed the best outcome. Indeed, may sometimes suffer the worst outcome. But... protected against being exploited by your opponent. Go forth and apply! GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Games of Strategy SBS Spring School Alex Dickson University of Strathclyde Spring 2017 GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Signaling and Cheap Talk Again in a world of private information: informed and uninformed players, and nature. Uninformed players make decisions using the best available information they have: prior beliefs. If uninformed players observe the actions of informed players, can they infer anything about their type? If only one type of player would undertake a particular action, and this action is observed, then yes! For the informed player, taking this action signals their type to the uninformed player. Two scenarios in which information might be revealed: signaling games; cheap talk games. Signaling games: signaling is costly, but beneficial. Cheap talk: messages are costless, but still might be informative. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Aims and Objectives 1 Understand the nature of signaling games, and appreciate that uninformed players can gain information about players types through the actions they take. 2 Understand the concept of ‘perfect Bayesian equilibrium’. 3 Appreciate the characteristics of strategic environment that make signaling easier, and so understand in what sort of real world environments signaling is likely to be effective. 4 Comprehend the idea of ‘cheap talk’ and develop an understanding of cheap talk games to reveal information and to reveal intentions. 5 Understand the difference between signaling and cheap talk. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Reading Reading Dixit, Skeath and Reiley; ch 8. Dixit and Nalebuff; ch 8. Watson ch 27-29. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Sequential-Move Games with Private Information In the last lecture we looked at the case where some information was private and moves were made simultaneously, or the informed party moved last. Uninformed players only have their prior beliefs to make decisions. If the informed player moves first, the uninformed player might be able to learn something from their actions. If the informed player reveals something about her type, the uninformed player can use this information to make a better decision. But we have to think quite carefully about the information content of actions. Should only ‘update beliefs’ if actions/messages do indeed ‘signal’ a player’s type... ...other types of player might want to try to imitate. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Sequential-Move Games with Private Information General form of signaling games. Two players, Sender and Receiver. Sender’s type chosen by Nature and revealed to Sender, but not to Receiver. Sender’s type is private information (but common prior beliefs). Sender moves first choosing an action, after observing this the Receiver chooses their action. Receiver’s optimal action will depend on Sender’s type, but only has prior beliefs on which to base this assessment. Senders don’t always have the incentive to truthfully reveal their type, if asked. Is there any information content in the Sender’s action that the Receiver can use to update her beliefs about the Sender’s type. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Sequential-Move Games with Private Information If it is optimal for a Sender of a particular type to undertake a particular action, AND it is not optimal for Senders of other types to choose this particular action, then when the Receiver observes this action they can learn the type of the Sender. Senders separate: can infer Sender’s type from actions. If not, no new information can be learned: Senders pool. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Signaling Games Manager and a trainee. Trainee is either lazy or industrious, but this is private information. Manager’s prior belief (which is common knowledge) is that 75% of trainees are lazy and 25% are industrious. Manager wants to permanently hire trainee if industrious, not if lazy. Induction period in which trainee has to decide on the number of hours to work. Personal cost of effort Type 40hrs 60hrs 80hrs Prior Value to M Lazy 50 75 120 0.75 25 Industrious 30 50 80 0.25 100 Trainee wants to be permanently hired: payoff if hired is 130; if not hired is 70. If the manager doesn’t hire the trainee her payoff is 60; the Manager only wants to hire an industrious trainee. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Signaling Games Timing of signaling games: 1) Nature chooses the Sender’s type; 2) Sender learns type and chooses an action; 3) Receiver observes Sender’s action, modifies their beliefs in light of this action, and chooses an action themselves. An important step here is the Receiver modifying her beliefs in light of the Sender’s action. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Manager-Trainee Game The extensive form. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Manager-Trainee Game Should the manager just hire any trainee? 75% of the time the trainee is lazy; 25% the trainee is industrious. The average value of a trainee is 0.75 × 25 + 0.25 × 100 = 43.75. Without any further information, the manager should not hire. Is there any information content in the number of hours that a trainee works during the production period? If it is optimal for different types of trainee to choose different work patterns (given the future actions of the manager) then the manager can deduce the type of trainee by observing their work pattern. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Manager-Trainee Game Consider the following scenario: an industrious Trainee chooses 80hrs; a lazy Trainee chooses 40hrs; the Manager believes that only industrious Trainees work 80hrs and only lazy Trainees work 40hrs and hires any Trainee that chooses 80hrs and fires any Trainee that works 40hrs or 60hrs. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Manager-Trainee Game Given that a manager that observes 80hrs hires but if she observes less than 80hrs fires, are the two types of trainee doing the best they can? Industrious: 80hrs followed by being hired ⇒ payoff of 50, which is better than 20 (60hrs) or 40 (40hrs). Lazy: 40hrs followed by being fired ⇒ payoff of 20, which is better than −5 (60hrs) or 10 (80hrs). Note, very importantly, that a lazy Trainee finds it in her own best interests not to work 80hrs even though she would be hired afterwards. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Manager-Trainee Game Given this, the Manager should update her beliefs accordingly. If Manager observes 40hrs should update belief about worker: only a lazy Trainee would choose 40hrs. If Manager observes 80hrs should update belief; only an industrious Trainee would choose 80hrs. [If observe 60hrs, no updating of prior belief.] GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Manager-Trainee Game Is Manager’s strategy consistent with updated beliefs? If observe 40hrs believe lazy for sure so prefers to fire (60 > 25). If observe 60hrs no updating of beliefs, so prefers to fire as 60 > 43.75. If observe 80hrs believe industrious for sure so prefers to hire (100 > 60). Manager’s action is optimal for each possible action of the trainee. Each type of Trainee is doing the best they can given the actions of the Manager; and the Manager’s beliefs are consistent and she is doing the best she can given the actions of each type of trainee. Separating equilibrium in which Senders of different types choose different actions. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Signaling Games Perfect Bayesian Equilibrium. Sequential nature of the game means we have a bit of backward induction type stuff (sequential rationality) going on, but we also have updating of beliefs. The appropriate solution concept in sequential-move games with incomplete information is perfect Bayesian equilibrium. PBE requires that a) players obey sequential rationality: each player’s strategy must prescribe an optimal action given their beliefs about what other players will do; and b) players’ beliefs are consistent: players should update their prior beliefs in light of observed actions, taking into account that all players act in their own best interests. …and relax! GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Manager-Trainee Game In the Manager-Trainee game there is a separating equilibrium. Lazy workers work 40hrs and get fired, industrious workers work 80hrs and get hired. An industrious worker has to distort their behaviour from what would otherwise be optimal in order to distinguish herself from Lazy workers and consequently get hired. It is VERY IMPORTANT that a lazy worker does not find it in her best interests to ‘mimic’ an industrious worker. If so, signaling breaks down. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Manager-Trainee Game Suppose the option of working 80hrs is removed. Consider the following strategies: lazy works 40hrs; industrious works 60hrs; Manager fires if she observes 40hrs and hires if she observes 60hrs. Industrious Trainee would choose to work 60hrs on the basis that she would get hired. BUT: lazy Trainee would find it in her best interests to work 60hrs as well, given that she will be hired. Observing 60hrs has no information content: both types of Trainee choose it. Not optimal for the manager to hire after observing 60hrs because 75% of the time get a lazy worker and 43.75 < 60. Signaling breaks down. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Signaling Games Even if the Manager cannot observe a Trainee’s type, if she understands about games of incomplete information she can screen trainees. Give them a task to do that involves choosing between several options. The cost of undertaking this task must be more costly for bad trainees than good ones. There must be at least one option that good trainees would want to choose. Of these options, there must be at least one that bad trainees do not want to choose, even though a reward will follow from choosing it. Trainees ‘self-select’ into groups depending on their type: they are screened according to the option they choose. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Signaling Games Effective screening requires: 1 good and bad types to have sufficiently different characteristics; 2 a choice amongst various options to be made; 3 the cost of choosing one particular option must not be so high that the good types don’t not choose it; 4 the cost of this same option must be high enough that the bad types don’t choose it. Observing this option being chosen then tells you a lot about the type of the person that has chosen it! GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Signaling Games In the original Manager-Trainee game the separating equilibrium is not the only equilibrium. Consider the following scenario: Trainees of both types work 40hrs, the Manager doesn’t update her prior beliefs, and fires trainees regardless of the number of hours they worked. One can check that this is also an equilibrium: no type of Trainee can profitably deviate and the manager does best given her beliefs, which are consistent with the trainees’ actions. Senders pool on a strategy and no information is revealed about their type: Pooling equilibrium. In the second variant of the Manager-Trainee game (where the 80hrs option is removed) pooling is the only equilibrium. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Signaling Games There may be several different types of player. Separation into types may not be so clean-cut. Semi-separating: a subset of types choose the same strategy. How should beliefs be updated? GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Signaling Games In the final scenario, if you observe F you know the player must be a Generous type. If you observe C, you don’t know much, but you do know something: not Generous. Can update beliefs about probability of other types given this. Probability that C is chosen is 0.7. Of these 0.7 times, 0.2 of the time player will be a Greedy type. If observe C update belief of Greedy to 0.2/0.7=0.29. Pr(Greedy|C) = Pr(C|Greedy)∗Pr(Greedy) Pr(C) . In general, Pr(A|I) = Pr(I|A)∗Pr(A) Pr(I) . This is Bayes’ Rule. So there you go! GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Signaling Games This is really cool! Any other applications? Yes, you’re doing it! Workers’ ability is unknown to employers, who only have a prior belief about ability. Workers have either a high or low productivity (H-type or L-type). Employer could base wage on average productivity (pooling). Suppose education is available. Education is costly (in terms of effort) and must be more costly for L-types than H-types. H-types undertake education; L-types do not; if an employer sees a certificate of education they believe the worker is H-type, if they don’t they believe the worker is L-type and the employer pays a high wage if they see a certificate of education, otherwise a low wage. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Signaling Games Given that a high wage will be paid following education, H-types must not find it too costly to undertake education (vs not, and being paid the low wage). Education cannot be too tough. L-types must find it too costly to undertake education, even though they would receive a high wage (vs not doing it and being paid a low wage). H-types self-select into education; L-types self-select into no education; the employer’s belief will be correct and so long as they make more this way than just paying the average wage we have a separating equilibrium. The signaling role of education. Spence (1973) “Job market signaling”. And this is why you and I are here; and why I make your life difficult, but not that difficult! GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Job Market Signaling Example Worker H-type or L-type; chooses Education or Not; Firm employs in Managerial or Clerical job. Separating equilibrium in which L-types don’t undertake education and are employed in clerical jobs, and H-types do undertake education and are employed in managerial jobs; and the Firm’s belief is that education is only undertaken by high ability workers. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Signaling Games A multitude of other applications... Used cars and warranties. Conspicuous consumption to find a wife. Advertising to signal quality when consumers make repeat purchases. Etc. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Cheap Talk In signaling games, actions of informed players conveyed information to uninformed players because they were costly and the cost of undertaking the action varied with the player’s type. In a separating equilibrium, only one type of player chose a particular action, so observing that action signaled the player’s type. The question we want to ask here is: can costless actions or messages (‘cheap talk’) also serve a signaling role? Pre-play stage to a game where communication can occur: costless messages can be sent to convey information (in incomplete information games) or convey intentions (in imperfect information games). GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Cheap Talk Since there is no cost to sending misleading messages, players should not presume messages received are truthful. ‘Talk is cheap’: should messages be believed? Messages convey information only when there is a lack of desire to deceive. In equilibrium, players will not be deceived. What are the conditions under which information/intentions can be conveyed? GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Communicating Information Consider the following incomplete information framework. Two players: Sender and Receiver. Sender has private information. A cheap talk game has three stages: 1) nature chooses Sender’s type; 2) Sender learns her type and chooses a message; 3) Receiver observes Sender’s message, updates her belief in light of the information, and chooses an action. Does the message convey any information about the Sender’s type that allows the receiver to update her belief? Often...no way! Senders have an incentive to deceive. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Communicating Information Question: should expert advice be believed? Patient goes to a Doctor with a symptom. Doctor evaluates patient and learns whether an expensive test will be beneficial. Patient has a prior belief about whether the test will be beneficial. Patient decides whether to follow the Doctor’s advice. Doctor cares about the patient, but also favors testing to avoid malpractice allegations (value of this is a). Cost to the patient of having the test is 5. Benefit if it is useful is 10 (so net benefit is 5). Can the doctor convey the information she has learned to the patient? GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Communicating Information Extensive form: If a > 0 a potential conflict of interest: doctor may prefer that the test is conducted even when it is not beneficial. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Communicating Information Consider the following scenario: the Doctor always recommends the test whether or not it is beneficial; the Patient ignores the Doctor’s recommendation and maintains the same beliefs about whether the test is beneficial. Message from the doctor is uninformative, so expected payoff from taking the test when it is recommended is 1/3 × 5 + 2/3 × −5 = −5/3 < 0. Since patient will not take the test, Doctor’s payoff is zero whether or not she recommends the test, so cannot do better. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Communicating Information Expert’s advice is worthless as it doesn’t depend on the true state of affairs. Babbling equilibrium: expert’s message is no more informative than inane babble. In every cheap talk game there is a babbling equilibrium. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Communicating Information Is there a ‘separating equilibrium’ in which messages actually contain information? Consider the following scenario: the Doctor recommends the test only if it is beneficial to the Patient; if the doctor recommends the test the Patient believes it is beneficial and if the Doctor doesn’t recommend the test the patient believes it is not beneficial, and the patient follows the Doctor’s recommendation. Doctor recommends what is best for the patient and the patient trusts the doctor: is this an equilibrium? Since the test is recommended only when it is beneficial, the patient can do no better than follow the doctor’s advice. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Communicating Information The Doctor optimally recommends the test when it is beneficial, but does she ONLY recommend it when it is beneficial? Given that the Patient follows the recommendation, this requires a − 5 ≤ 0 or a ≤ 5, otherwise there is an incentive to recommend the test when it is not beneficial. If a ≤ 5 there is a separating equilibrium in which the Doctor’s message is informative. If a > 5 the only equilibrium is the babbling equilibrium. Patient trust in Doctors depends on the incentives Doctors face: strong incentive to avoid malpractice allegations means messages become uninformative. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Signaling Intentions Back to the old friend: simultaneous-move game of complete information. Complete but imperfect information: there is strategic uncertainty. Suppose there is a pre-play stage in which all or a subset of players can costlessly communicate with other players. Pre-play communication. If there is a unique pure strategy Nash equilibrium there isn’t much to say (as we deduced in prisoners’ dilemmata). Otherwise, can players effectively communicate their intentions? GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Signaling Intentions Assurance game: Alice Bob Stag Hare Stag 2,2 0,1 Hare 1,0 1,1 Interests are perfectly aligned. If either or both players are allowed to communicate they will signal their intention to play Stag, which will not be disregarded. Cheap talk is informative. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Signaling Intentions Matching Pennies game: Alice Bob H T H 1,-1 -1,1 T -1,1 1,-1 Interests are conflicting. If either player has the opportunity to communicate their intention, the message should not be believed. There is an incentive to deceive. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Signaling Intentions Battle of the Sexes: Alice Bob Opera Football Opera 3,2 0,1 Football -1,0 2,3 Partially aligned interests. If ONE player has the opportunity to communicate their intention, should signal that they intend to play their preferred strategy, which should be believed. Pre-play communication by a single player resolves coordination issues. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Signaling Intentions Mixed strategy Nash equilibrium in BoS. A B O (q) F (1 − q) O (p) 3,2 0,1 F (1 − p) -1,0 2,3 p = 3/4, q = 1/3. A B O (1/3) F (2/3) O (3/4) 3/12 6/12 F (1/4) 1/12 2/12 Expected payoffs: 1 for Alice, 3/2 for Bob. In the MSNE coordinate 5/12 ≈ 42% of the time. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Signaling Intentions In a pre-play stage, each player can costlessly communicate their intention to the other player. Merely say: ‘Opera’ or ‘Football’. If messages match, undertake the action that corresponds to the message, if not play `a la MSNE (Alice chooses opera 3/4 of the time, Bob chooses football 2/3 of the time) with expected payoffs of 1 for Alice and 3/2 for Bob. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Signaling Intentions Messaging game: A B O (b) F (1 − b) O (a) 3,2 1,1.5 F (1 − a) 1,1.5 2,3 A B O (1/3) F (2/3) O (3/4) 3/12 6/12 F (1/4) 1/12 2/12 MSNE of messaging game: a = 3/4, b = 1/3. Players coordinate in messages 5/12 of the time. The other 7/12 of the time they play mixed strategies, where they also coordinate 5/12 of the time. Coordination rate increases to 5/12 + 7/12 × 5/12 ≈ 66%. Cheap talk via pre-play communication helps players to coordinate their actions. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Summary Games where players have an opportunity to either signal information about themselves or to signal their intentions. Signaling games: if an action is available that is chosen by one Sender-type but not by others, that action is informative to Receivers. Separating equilibrium in which Senders of different types are treated differently by Receivers as a result of their actions. Very widely applicable; important deductions to be taken from this: how to deduce the ‘known unknowns’. Cheap talk games with costless messages. Does cheap talk work for communication of intentions in games with imperfect information? Depends on incentives to deceive. If incentives are partially aligned communication can serve to increase coordination rates.

GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Games of Strategy SBS Spring School Alex Dickson University of Strathclyde Spring 2017 GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Games with Private Information So far assumed complete information: all players know the rules of the game and this is common knowledge. Incomplete information: some players don’t know some features of the game. Interesting scenario: an uninformed player does not know the payoffs of her opponent. Has some prior belief, but does not know for sure. Decision-making process somewhat different…for both players. How do we analyze such games and what is our prediction of the outcome? Principal-Agent game and moral hazard problems. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Aims and Objectives 1 Understand the difference between games of complete and incomplete information. 2 Understand the concept of introducing ‘nature’ into the game as a player. 3 Be able to construct the ‘Bayesian normal form’ for a simultaneous-move game of incomplete information. 4 Be able to analyze sequential-move games where the informed player moves last. 5 Comprehend the nature of the Principal-Agent framework and understand the general implications. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Reading Reading Dixit, Skeath and Reiley; ch 8, 13. Dixit and Nalebuff; ch 8, 13. Watson ch 24-26. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Games with Incomplete Information Complete information: all players know everything there is to know about the rules of the game (including payoffs) and this is common knowledge. It is not inconceivable that this is not true. Examples? In particular, players might not know their adversaries’ payoffs. Information is no longer complete: game of incomplete information. Some players may be perfectly informed, but others are not. Some information in the game is ‘private’ to particular players. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Games with Incomplete Information Players can take one of several ‘types’. Different types of player have different payoffs. ‘Nature’ draws players’ types according to some known prior distribution. Gives the probability with which each type is drawn. Important that ALL players know this (‘common prior’). If nature informs all players about all types then we have complete information. For some players, nature might inform only that player of their type. Two player game; everything is known by both players about player A; only player B knows player B’s type. Informed and uninformed players: player B has private information. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Interesting Questions When faced with this uncertainty through a lack of information, how should players make strategic decisions? Uninformed player needs to think about the consequences of actions for each type of opponent they might face. Different decision-making problem. Informed player knows the uninformed player is uninformed…needs to incorporate this in beliefs of what opponent might do. How do we analyze such games? What is the appropriate notion of equilibrium? What is our prediction of the outcome? GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Example: Market Behaviour There is an established company (E) in the market whose operating costs are well-known. There is also a new entrant (N). Each company has to decide simultaneously whether to be aggressive (A) or passive (P) in competing with each other. If N has a high cost (H-type), the game is represented by the following strategic form: E N A P A 6,1 5,2 P 5,4 4,3 The Nash equilibrium is (A,P). GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Example: Market Behaviour If N has a low cost (L-type) then the game is as follows. E N A P A 1,5 8,3 P 3,7 3,4 (P,A) is the Nash equilibrium. Two very different games. If E knows which type of N she is playing against: easy! GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Example: Market Behaviour What happens if E does not know N’s ‘type’ (H or L)? Prior belief that the probability N is either H or L is 50%. N knows her type, and E’s prior belief. N is L: prob=0.5 E N A P A 1,5 8,3 P 3,7 3,4 N is H: prob=0.5 E N A P A 6,1 5,2 P 5,4 4,3 GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Extensive Form 6,1 5,2 5,4 4,3 1,5 8,3 3,7 3,4 GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Strategies and Equilibrium A strategy of a player should specify the action that each possible type of that player should use. If a player has multiple types, think of them as separate players in the game. For E this is just an action: she only has one ‘type’. For N this is an action if she is an L-type and an action if she is an H-type. An equilibrium (Bayesian Nash equilibrium) is a set of strategies such that each type of each player is choosing the action that maximises their (expected) payoff given the action of the other player. The familiar ‘no player can make a profitable unilateral deviation’, but profitable in the sense of expected payoff, and true for all possible types of a player. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Example: Market Behaviour N is L: prob=0.5 E N A P A 1,5 8,3 P 3,7 3,4 N is H: prob=0.5 E N A P A 6,1 5,2 P 5,4 4,3 For an L-type N, P is dominated so can never be part of an equilibrium strategy. Check to see if other strategy combinations constitute an equilibrium. A,(A,A): H-type N can do better playing P. A,(A,P): E can do better playing P (exp. payoff of 3.5 > 3). P,(A,A): This is it! Neither E or either type of N can do any better. P,(A,P): H-type N can do better playing A. Bayesian Nash equilibrium is P,(A,A). Note the contrast to if E assumed N was H-type! GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Bayesian Normal Form This was a bit laborious… Bayesian normal form is a payoff matrix that lists all strategies for both players, and the expected payoffs (given prior beliefs). N is L: prob=0.5 E N A P A 1,5 8,3 P 3,7 3,4 N is H: prob=0.5 E N A P A 6,1 5,2 P 5,4 4,3 E N (A,A) (A,P) (P,A) (P,P) A 3.5,3 3,3.5 7,2 6.5,2.5 P 4,5.5 3.5,5 4,4 3.5,3.5 Use the standard method of underlining using expected payoffs to find the Bayesian Nash equilibrium. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Bayesian Nash Equilibrium Recap: One player has private information about their type. Once this becomes known, they know the payoffs. The uninformed player doesn’t. Form beliefs about the actions of each type of their opponent, and evaluate actions using their prior belief about their opponent’s type. Given beliefs, uninformed player chooses action that maximizes expected payoff. The informed player infers this, which is used in guiding their beliefs. Mutually consistent behaviour of all players and all player types: regardless of whether N is L-type or H-type, she plays A; E plays P. E doesn’t play A because of the consequences if N is L-type. Very different to the outcome if N is H-type for sure. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Battle of the Sexes? Battle of the Sexes-type game where Alice doesn’t know whether Bob likes her or not. Bob likes Alice: prob=0.5 A B O F O 2,1 0,0 F 0,0 1,2 Bob doesn’t like Alice: prob=0.5 A B O F O 2,0 0,2 F 0,1 1,0 Bayesian normal form A B (O,O) (O,F) (F,O) (F,F) O 2, 1/2 1, 3/2 1,0 0,1 F 0,1/2 1/2,0 1/2, 3/2 1,1 GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk We’re on a Stag Hunt…Maybe Consider the following game: A B S H S x,3 0,1 H 1,0 1,1 x = 3 with probability p; x = 0 with probability 1 − p. Stag hunt, or a very different game? Nature chooses the value of x; player A becomes informed of her payoff; player B remains uninformed. Strategy for player A should a type-0 action and type-3 action. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk We’re on a Stag Hunt…Maybe Suppose p = 1/2: probability that the game is stag hunt is 1/2. A B S H S x,3 0,1 H 1,0 1,1 A B S H (S,S) 3/2,3 0,1 (S,H) 1/2, 3/2 1/2,1 (H,S) 2, 3/2 1/2,1 (H,H) 1,0 1,1 Two Bayesian Nash equilibria: (H,S),S and (H,H),H GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk We’re on a Stag Hunt…Maybe Consider different prior beliefs: p = 1/4 A B S H S x,3 0,1 H 1,0 1,1 A B S H (S,S) 3/4,3 0,1 (S,H) 1/4, 9/4 1/4,1 (H,S) 6/4, 3/4 3/4,1 (H,H) 1,0 1,1 Now a single Bayesian Nash equilibrium: (H,H),H. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Uncertain Market Entry Game Recall the game of entering a £20m market with a £15m investment cost. A B Enter Stay out Enter -5,-5 5,0 Stay out 0,5 0,0 Suppose that A moves first and, after observing A’s move, B moves second. Suppose further that with probability p the game is as above, but with probability 1 − p the payoff to B from entering when A enters is +5 (e.g. B can use aggressive behavior to signal aggression elsewhere). Informed player moves last. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Uncertain Market Entry Game If A enters, standard will SO but signaller will E: expected payoff is 5p − 5(1 − p) = 10p − 5. If A SO, payoff is zero. Compare exp. payoffs: E if 10p − 5 > 0, i.e. p > 1/2. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Principal-Agent Framework Interesting model to capture conflicting incentives. Principal contracts with an agent to complete a task. If the agent accepts the contract decides whether to ‘work’ or ‘shirk’. Principal’s interests best served by the Agent choosing to work; Agent has an incentive to shirk. Nature determines outcomes: ‘good’ or ‘bad’. Working(shirking) means good(bad) outcomes more likely, but a worker(shirker) might get unlucky(lucky). Principal only observes outcomes, can’t deduce Agent’s effort, which is private information. Effort can’t be verified: could ask the Agent, but she has an incentive to lie! Principal-Agent problem: how can the Principal contract with the Agent to incentivize the Agent to work hard? GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Principal-Agent Framework Game: Principal decides on the contract; Agent decides whether to accept or reject, and if accept whether to work or shirk; Nature decides the outcome. If the Principal can observe effort, or if effort is verifiable from observing outcomes (i.e. nature does not take a role), there is a relatively straightforward solution. Reward hard work, punish shirking. Incentivizes the Agent to work hard. But effort is very often not verifiable. Q: When effort cannot be verified, can the Principal incentivize the Agent to work hard, and is there any cost of doing this? GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Principal-Agent Framework These situations are ubiquitous. Manager working for the owner. Parent company and its subsidiaries. Executives working on behalf of equity owners. Advertising agents working on behalf of managers. Other examples? Q: How should the Principal contract with the Agent to motivate her to work hard, and is there any cost of doing this? A: Condition payments on outcomes, ensuring a) the Agent accepts the contract, and b) the Agent is exposed to sufficient risk to incentivize her to work hard. There is a cost associated with exposing agents to risk. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Aside: Risk in Uncertain Environments We will want to make payments depend on outcomes: Agents will be exposed to risk. How do Agents evaluate risky payments? Consider a gamble: 50% chance of winning £100 and 50% chance of winning 0. Expected payoff is 0.5 × 0 + 0.5 × 100 = £50. Question: would you prefer £50 or the gamble? You are risk averse if you prefer to take the expected payoff for sure than to take on the gamble. Individuals tend to be risk averse. Firms tend to be risk neutral: indifferent between the gamble and its expected payoff. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Aside: Risk in Uncertain Environments Offered two jobs: 1) a salary that pays £100,000 in a boom (50% of the time) and £0 in a bust (50% of the time); 2) £50,000 for sure. Risk averse individuals that are exposed to risk need compensating for bearing risk. To induce an individual to take the risky salary, need to increase the expected wage. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Aside: Risk in Uncertain Environments Expected Utility Individuals care not about wealth, but the utility of wealth u(w). When faced with risks they care about expected utility – probability-weighted average of utility in each ‘state’ – and choose the alternative that gives the highest expected utility. Individuals seek to maximize their expected utility. Risk averse individuals have concave utility functions: utility of the expected value higher than the expected utility. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Aside: Expected Utility A utility function that looks like this is u(w) = √ w: a transformation of wealth to allow us to evaluate uncertain environments. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Aside: Risk in Uncertain Environments What are the implications of the presence of risk averse individuals? They value risky ‘prospects’ at less than their expected value. 50% chance of success, where salary is £40k and 50% chance of failure, where salary is £10k. Vs salary of £25k? EU is 0.5 × √ 40 + 0.5 × √ 10 ≈ 4.74 vs √ 25 = 5. Risk averse individuals prefer certain salary of £25k. To induce individuals to choose the risky option, would need to increase the expected utility of the salary. This increases the expected salary making it higher than £25k. Must compensate risk averse individuals for bearing risk. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Principal-Agent Framework Consider a risk neutral Principal who wants to contract with a risk-averse Agent. Principal creates contract specifying how the agent is paid. Agent then decides first whether to accept the contract, then whether to Work or Shirk. The outcome can be either Good or Bad (for the Principal): determined partly by nature. If the Agent works(shirks) the probability of the good(bad) outcome occurring is higher than if she shirks(works). Principal observes outcome, but effort is unverifiable; private info. for Agent. The profit generated for Principal is higher in the Good outcome than in the Bad outcome. Other things equal, the Principal wants to incentivize the Agent to Work, not Shirk. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Principal-Agent Framework Suppose u(w) = √ w, and working involves putting in 50 effective hrs a week whilst shirking involves putting in 5 hrs. The payoff to the Agent is u(w)−effort: Agent likes money and dislike effort. E.g. if the Agent ends up being paid £22,500 and works 50 hrs, her payoff is √ 22500 − 50 = 100. If the outcome is Good (the project is a success), the Principal gets £80,000, whilst if the outcome is Bad (the project fails) she loses £50,000. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Principal-Agent Framework If A works nature dictates that the outcome will be Bad with probability 0.2 and Good with probability 0.8. If A shirks the probability of the Bad outcome occurring is 0.8 and the probability of the Good outcome occurring is 0.2. Hence, the Principal cannot infer effort from outcome: effort is unverifiable. The A has an outside option that pays £5625, giving utility of 75. A contract will involve hiring the Agent and specifying wages wB and wG for Good and Bad outcomes, respectively. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Moral Hazard Suppose the Principal is a trusty soul who thinks that all Agents will work hard. Pay an unconditional wage wˆ. Must be such that the worker prefers this to the outside option. Need √ wˆ − 50 ≥ 75, or wˆ ≥ £15, 625. Since the principal wants to minimize the wage bill, pays exactly wˆ = £15, 625. Principal naively expects to make profits of 0.2 × −50000 + 0.8 × 80000 − 15625 = £38, 375. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Moral Hazard Upon receiving the wage, the Agent has two options: Work or Shirk. Working gives a payoff of √ 15625 − 50 = 75. Shirking gives a payoff of √ 15625 − 5 = 120. Shirking is obviously better for the Agent. But it is surely not in the best interests of the Principal: makes losses of 0.8 × −50000 + 0.2 × 80000 − 15625 = −24000 − 15625 = −£39, 625. There is a moral hazard problem: the Agent has incentives to undertake actions after the contract has been signed that alter the value of the contract to the Principal. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Observable Effort Suppose that effort is perfectly verifiable (i.e. observable). The Principal would rather walk away than accept that the worker will shirk: loses £24k. To incentivize the agent to work hard… If shirking observed, pay zero (i.e. fire the Agent). If working observed, pay a wage that minimizes the wage bill (0.2wB + 0.8wG) and at the same time ensures that the Agent chooses to work for you (over the outside option). If you pay wB, wG, the expected payoff to working is 0.2 √ wB + 0.8 √ wG − 50. Thus, make sure that 0.2 √ wB + 0.8 √ wG − 50 ≥ 75. This is called the participation constraint. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Observable Effort Since the Agent is risk averse, if the principal makes her wage risky she will need to reward her for taking on this risk. To satisfy the participation constraint, this will require a higher expected wage. But this reduces the Principal’s expected profit. Principal should therefore try to pay a certain wage to the Agent: wB, wG = ˜w. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Observable Effort Principal doesn’t want to pay more than she has to, so satisfies the participation constraint with equality. √ w˜ − 50 = 75, so w˜ = £15, 625. P’s expected payoff is 0.2 × −50000 + 0.8 × 80000 − 15625 = £38, 375. Principal writes a contract that pays the Agent zero if she shirks, and £15,625 if she works, regardless of whether the Good or Bad outcome occurs. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Unobservable Effort What happens if effort is unobservable/unverifiable? As we noted, the Principal cannot infer effort from the outcome: the bad outcome doesn’t necessarily mean the worker shirked. Since the outcome is the only thing that can be observed, have to make the wage conditional on this. Contract will specify wB, wG. How should the Principal write the contract in order to get the Agent to work hard, in the best interests of the Principal? GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Unobservable Effort This is a good question! The Principal wants to minimize her expected wage bill: 0.2wB + 0.8wG. Presuming the Agent works hard, the Principal must make sure that the Agent chooses to work over the outside option. Must satisfy the participation constraint: 0.2 √ wB + 0.8 √ wG − 50 ≥ 75, i.e. 0.2 √ wB + 0.8 √ wG ≥ 125. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Unobservable Effort Upon accepting the contract wB, wG, the A has two options: Work: with payoff 0.2 √ wB + 0.8 √ wG − 50, Shirk: with payoff 0.8 √ wB + 0.2 √ wG − 5. The Principal must make sure the contract is such that Working is better than Shirking. That is, the contract must satisfy the incentive compatibility constraint: 0.2 √ wB + 0.8 √ wG − 50 ≥ 0.8 √ wB + 0.2 √ wG − 5, i.e. −0.6 √ wB + 0.6 √ wG ≥ 45. Note, importantly, that wB = wG = ˜w = £15, 625 DOES NOT satisfy the ICC! GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Unobservable Effort To minimize expected wage bill, the Principal will just satisfy these constraints: 0.2 √ wB + 0.8 √ wG = 125 (P) −0.6 √ wB + 0.6 √ wG = 45 (IC). (P) multiplied by 3 0.6 √ wB + 2.4 √ wG = 375 Add to (IC): 3 √ wG = 420 so wG = 19, 600 From (P) √ wB = 125 − 0.8 √ wG 0.2 = 65 so wB = 4, 225 Optimal contract provides a base salary of £4,225 and a bonus linked to good outcomes of £15,375. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Unobservable Effort Optimal contract: wB = 4, 225, wG = 19, 600. To induce the Agent to work hard, the Principal must make the wage conditional on the outcome. The Principal faces an expected wage bill of 0.2 × 4225 + 0.8 × 19, 600 = 16, 525. This is £900 higher than when the wage can be conditioned on effort. This is called an agency cost. What is the source of the agency cost? GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Unobservable Effort To induce the Agent to work hard, the Principal has to make the wage dependent on the outcome. Makes the outcome risky for the Agent. To induce the Agent to take the contract over the outside option (participate) the expected wage must be higher, since the Agent is risk averse. Agency cost. If the Principal can undertake monitoring of effort for less than this, should do it and revert to a contract that conditions on effort. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Recap Principal contracting with Agent. If effort can be observed, P does best to pay the A the same salary regardless of the outcome, which must be high enough to induce the A to participate. If effort is unobservable, to incentivize the A to work hard, the P must condition the payment on the outcome. Satisfy both the participation constraint, and the incentive compatibility constraint. Basic salary, plus bonus for good outcomes: a carrot to incentivize the agent to choose the option that makes receiving the carrot more probable. But, since the outcome for the agent is now risky, have to pay a higher expected wage. Unobservability of effort induces an agency cost, up to which the P would be willing to pay to monitor effort. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Some Implications P-A problems: How to make people do stuff that is in your best interests, even if it is not in their best interests to do so. It may be that the optimal contract calls for the Agent to be punished for Bad outcomes (stick). Sometimes this is infeasible. Have to make the payment for good outcomes even larger: large bonuses! The more risk the Agent is exposed to, the larger the agency cost. In situations where the Agent must be exposed to a substantial amount of risk (e.g. correlation between effort and outcome is not high) the cost of contracting might erode all the benefits of incentivizing the agent. Contracting is infeasible. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Summary Games with private information in which nature takes a role. Players in a game might not know the type of their adversary. Instead have some prior belief over which type they are. How should players behave in such environments? Influences both the uninformed and the informed players. What is the prediction of equilibrium? Simultaneous move games, and sequential-move games where the informed player player moves last. Principal-Agent game; Agent has private information about effort since outcomes are determined by nature. How to incentivize Agents to work in the best interests of the Principal even though effort can’t be verified.

GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Games of Strategy SBS Spring School Alex Dickson University of Strathclyde Spring 2017 GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Strategic Moves Players in a game might find that the outcome is not to their liking. Can anything be done in an attempt to alter the outcome of the game? Issue threats/promises. Unless these are warnings or assurances, such threats/promises are incredible. When called upon to follow through with threat/promise, the player would not, so should not be believed by the adversary. How to make threats/promises credible? Welcome to the wonderful world of commitment. The Edge Annual Question: what is your favorite deep, elegant, or beautiful explanation? Richard Thaler: Commitment. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Aims and Objectives 1 Understand that players in a game may want to alter the outcome to be more favorable to them. 2 Comprehend the notion of incredible threats and promises, and understand that they should not be believed. 3 Develop an appreciation of actions to commit oneself to make a threat/promise credible. 4 Appreciate the ‘paradox of commitment’. 5 Begin to recognize situations in which strategic moves may be enacted, and the implications for ‘strategy’. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Reading Reading Dixit, Skeath and Reiley; ch 9, 14. Dixit and Nalebuff; ch 6, 7. Schelling, T; 1960; The Strategy of Conflict. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Undesirable Outcomes In many strategic situations, the predicted outcome of the game may not be to your liking. Defending an Island. 1 2 Retreat Fight Att 3,0 -1,-1 No Att 0,3 0,3 GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Undesirable Outcomes Entering a £20m market with a £15m investment cost. A B Enter Stay out Enter -5,-5 5,0 Stay out 0,5 0,0 GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Undesirable Outcomes A partnership game and the prisoners’ dilemma. You Partner Work Shirk Work 3,3 0,4 Shirk 4,0 2,2 GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Manipulating Games Question: can you take actions to manipulate the game to make yourself better off? Undertake actions to guarantee a particular outcome from the game? If you like, to ‘win’ the game. Undertake strategic actions in a pre-play stage to alter the game in your favor. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Threats and Promises Players could just issue threats or promises to their adversaries. If you enter the market, I will enter too making it unprofitable for you to enter. If you attack the island, I will fight you, you… If you work hard in our partnership, I promise that I will work hard too. Threats and promises seek to deter or compel our adversaries to undertake a particular course of action. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Credibility The problem is that ‘talk is cheap’, in fact free, and very easy. If you talk in this class I will launch a nuclear attack; on your head. Am I bluffing? Such threats and promises lack credibility. In the sense that if a rational player was ever called upon to carry out the threat or promise, they would never do it. Because when it gets to it, it just doesn’t make any sense! Credibility of threats and promises is the key to their value, but is difficult to come by. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Credibility NA 0,3 A 1 F -1,-1 R 3,0 2 “If you attack, we will fight”. If army 1 attacks, will army 2 ever fight? No, and army 1 knows this. Army 1 invades and army 2 retreats. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Credibility SO E B SO 5,0 E -5,-5 A SO 0,0 E 0,5 A A threatens; “if you enter, I will enter as well”. If your opponent does enter, you should enact your threat. But you will never enter having observed your opponent enter. Are you crazy? No, and your opponent knows this! GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Credibility S W P S 1,4 W 3,3 Y S 2,2 W 4,1 Y Promise: “if you work, I will work as well”. If your opponent does work, you should carry through your promise. Will you? No, and your opponent knows this! GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Credibility Threats and promises are easy to make. Conditional strategic moves. They are only effective if it is the best interests of the issuer to enact them when called upon to do so. In which case threats are called warnings and promises assurances. Alice Bob Stag Hare Stag 2,2 0,1 Hare 1,0 1,1 H S A H 0,1 S 2,2 B H 1,1 S 1,0 B GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Credibility If threats or promises lack credibility they will never be enacted. Sequential rationality via backward induction. Incredible threats should not be believed. And therefore will not work. Well then, how can we manipulate the game to our advantage? GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Schelling Thomas Schelling; 1960; The strategy of conflict. NA 0,3 A 1 F -1,-1 R 3,0 2 Threat; “if you attack, we will fight”. If army 1 attacks, will army 2 ever fight? No, and army 1 knows this. Army 1 invades and army 2 retreats. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Commitment Thomas Schelling; 1960; The strategy of conflict. When army 2 occupies the island, it chooses whether to ‘burn the bridge’: no means of escape. Burn Leave 2 N 0,3 A 1 F -1,-1 R 3,0 2 N 0,3 A -1,-1 1 If 2 commits to a course of action – fighting if army 1 invades – best for army 1 not to invade. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Commitment Army 2 has managed to bring about a better outcome for itself… much better! By ‘burning its bridge’. Committing to fight if army 1 attacks. Tying its hands by reducing the options available to it. Limiting the actions available to you can be beneficial. There is value in inflexibility. Note that army 1 has a high vested interest in maximizing the options for army 2! GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Commitment Rule of Strategy There can be value in taking actions to reduce the options available to you. ‘Paradox of commitment’. Why? Alters rivals expectations about you; commits you to an inferior decision later in the game. The commitment action must be communicated to the other player, and be irreversible. In order for the commitment action to work, it must be credible. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Commitment Through Investment Consider the game of entering the £20m market with a £15m investment cost. A B Enter Stay out Enter -5,-5 5,0 Stay out 0,5 0,0 Threat of ‘if you enter, I will enter as well’ does not work. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Commitment Through Investment Suppose you consider making a £7.5m upfront investment. After making the investment, the payoffs in the game change A B Enter Stay out Enter -5,-5 5,0 Stay out -7.5,5 -7.5,0 It now becomes a dominant strategy to enter the industry, whatever the opponent does. After the investment has been taken, the threat of ‘if you enter, I will enter as well’ becomes credible. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Commitment Through Investment Not Invest A SO E B SO 5,-7.5 E -5,-5 A SO 0,-7.5 E 0,5 A SO E B SO 5,0 E -5,-5 A SO 0,0 E 0,5 A GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Commitment By undertaking appropriate actions to alter the payoffs in the game, you can dramatically alter the outcome to be in your favor. Investment makes the threat of entry credible. But, the investment must be large enough. Why would £3m not work? GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Price Matching Guarantees Consider firms setting prices in competition with each other. Very often, there is an incentive (a dominant strategy, even) to undercut your rivals. Price war. A B H L H 5,5 2,7 L 7,2 3,3 L H A L 2,7 H 5,5 B L 3,3 H 7,2 B GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Price Matching Guarantees Now consider an early commitment to a price-matching guarantee: A will match any price of its competitor. No G Match A L H A L 3,3 H 5,5 B L 3,3 H 7,2 B L H A L 2,7 H 5,5 B L 3,3 H 7,2 B GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Price Matching Guarantees Price matching guarantee sounds great for consumers! But it is actually a commitment device to sustain high prices! By committing to match a low price, A changes the payoffs such that it is no longer beneficial for B to undercut, since the big payoff cannot be realized. Both firms end up charging high prices. Note: out of the (pricing) prisoners’ dilemma! Economist Article on PMG. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Commitment Commitment: central idea. An unconditional strategic move. Can be achieved by reducing your set of available actions – tying your hands, or burning your bridges. Doing something that looks stupid can have benefits in the context of the game. Schelling: “The power to constrain an adversary depends upon the power to bind oneself”. Paradox of commitment. Or, by undertaking actions that manipulate your payoffs. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Self-Control We all have problems of self control: “the best laid plans of mice and men often go awry” (Burns). Procrastination is a wonderful thing! Model of dual selves: long term planning self playing a game with short-term impulsive selves. Long-term self cares about the bigger picture, short term selves care only about instantaneous decisions. Long-term self makes plans for you to study 3 hours every night. Monday-, Tuesday-, Wednesday- etc. selves have different plans! GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Self-Control Simple Illustration: Your night time self is playing a game with your morning self on whether to get up… Night GU 10,0 LI 0,10 Morning GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Self-Control Can your long-term self take actions to commit your future selves to do the thing that’s in your long-term best interest? Set A No A N GU 10,0 LI 0,10 M GU 8,-1 LI -2,-5 M By setting the alarm your night self makes a commitment for your morning self to get up. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Self-Control Your long-term ‘current’ self can take actions to commit your future self or selves to a particular course of action. Gym memberships: pay as you go vs tied-in contract. Other stuff? StickK and ‘commitment contracts’. Freakonomics page on ‘personal commitment’. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Lessons in Acquiring Credibility For self-study, in conjunction with the discussion in ch 9 of DSR. In order to make strategic actions credible, you need to work on two dimensions: A: Limit the options available to you. Limit your ability to back out of a commitment. B: Change your payoffs. Turn a threat into a warning, or a promise into an assurance. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk A: Limiting Your Available Options 1 Leave actions beyond your control. Institute an automata to carry out pre-decided actions for you (Doomsday devices). 2 Delegation. Delegate to a negotiating agent who is required to follow certain rules and procedures. 3 Burn your bridges. Tie your hands to a particular course of action. 4 Cut off communication. Communicate your commitments then cease communication to avoid counter-commitments. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk A: Limiting Your Available Options—Doomsday Devices In making a threat you want the other player to believe that you will act irrationally later. You know you won’t, and more importantly your adversary knows that you won’t: threat is incredible. One solution: irrevocably take the decision out of your hands. Burning the bridge was one example of this. Pre-specify actions that cannot be reneged upon. Appoint negotiators to take actions on your behalf, and cease communication. Set up mechanistic rules. Toddlers and terrorists have the upper hand in negotiations, since they are immune to reason. The rest of us are rational in the eyes of our adversaries! GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk B: Changing Your Payoffs 1 Reputation. In repeated settings, acquire a reputation for carrying out threats and delivering promises. You may have to incur the expense of short term losses. 2 Play in parts. Try to divide a single game into a sequence of smaller games, and establish reputation. 3 Teamwork. Teams of players monitor each other: monitor whether threats and promises carried out and punish if not. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk B: Changing Your Payoffs 5 Irrationality. Gain a reputation for being irrational! 6 Contracts. Make it costly to fail to carry out a threat by signing a contract, but the counterparty must be incentivised, and the contract must not be renegotiable. 7 Brinkmanship. Threats may be too large to be credible: instead initiate a situation of risk as a threat. Increase the risk if it doesn’t work. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk The Adversary Our focus has been on the actions that we can take to credibly commit to an action. Your adversary might be undertaking similar considerations. How do we counter our opponent’s strategic moves? GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk The Adversary 1 Signal irrationality: I will not give in to any threat. 2 Cut off communication so threats can’t be communicated. 3 Leave escape routes open. 4 Make secret deals, so your opponent doesn’t have to uphold her reputation. 5 Salami tactics: take actions in the face of a threat one slice at a time. Massive mutually harmful punishment for a small transgression? GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk In an Uncertain World? There is value in committing to a course of action early in the game. In an uncertain world, there is value in delaying actions to resolve some of the uncertainty. (Real) Option value of waiting to make investment decisions, for example. Tension in a competitive world: commitment vs flexibility. This is an incredibly interesting business problem. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Summary There are often possibilities to alter the outcome of a game. Unless a threat (promise) constitutes a warning (assurance) it is not credible. Incredible threats should not be believed. Threats can be made credible through commitment. Restrict your actions, alter payoffs, to change the outcome in the game. Keep your own commitment options open… Limit your opponent’s ability to commit.

GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Games of Strategy SBS Spring School Alex Dickson University of Strathclyde Spring 2017 GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Sequential-Move Games In many decision environments, players observe the decisions of their adversaries before they make their own decisions. When moves are sequential, what else do we need to think about? Later movers have more information, but earlier movers may be able to alter the play of the game. How to represent and analyze such games. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Aims and Objectives 1 To be able to recognize when a strategic situation involves sequential moves. 2 Understand how to represent sequential-move games using the extensive form. 3 Understand the concepts of backward induction and sequential rationality. 4 Understand the notion of strategy in sequential-move games. 5 Be able to derive predictions of outcomes when moves are sequential. 6 Appreciate that there may be advantages to players, depending on the order of moves. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Reading Reading Dixit, Skeath and Reiley; ch 3. Dixit and Nalebuff; ch 2. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Sequential-Move Games In simultaneous-move games, all players make decisions without observing the decisions of other players. Players have to think about how other players are thinking and make conjectures about their behavior. A game has sequential moves if certain players observe the choices of other players before they have to make a decision. There is a specified order of play. The need for making conjectures about what opponents might do is limited: some players observe what others do. And other players know this. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Sequential-Move Games To define the game, still need to specify the players, their available actions and their payoffs from different combinations of actions. Must also specify the order of moves. This defines what information players have when they make their decisions. If, whenever a player is called upon to play they know the history of play, then it is a game of perfect information. We represent sequential-move games using the extensive form. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Venture Capitalist The extensive form represents the players, their actions, the payoffs and the order of moves. Don’t 0,0 Inv VC S -1,2 W 3,1 Ent Initial node; decision nodes; terminal nodes. Payoffs. Actions are branches. Sequences of choices trace a path of play. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Sequential Pricing Tesco sets its price first, after which its smaller local rival set its price. L H T L 2,7 H 5,3 LS L 3,1 H 10,-1 LS’ GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Strategy in Sequential-Move Games In simultaneous-move games, strategy was synonymous with action. In sequential move games, what does “do action X” mean for later movers? In a sequential move game, a strategy is a complete plan of action, that specifies the action a player should take at each decision node they may be called upon to play at in the game. For T: H,L. For LS: (H,H); (H,L); (L,H); (L,L) If your adversary does something unexpected, you have a plan of action. Strategies in sequential-move games can be complicated! GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Supply Chain Game N U M D C R N 100,350 A 180,300 M’ N 40,120 A 120,280 M” D 130,150 C 60,380 R’ M starts the game and if it chooses U must make another decision later in the game. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Backward Induction Players that move later in the game observe the actions of earlier movers. Then choose the action that maximizes their payoff given their observations. Later movers have more information than earlier movers. But, an early-mover knows that later movers have this information. And know that later-movers will also act in their own best interest. Can predict which course of action will be taken following actions taken earlier in the game. Players in sequential-move games look forward and reason back – sequential rationality. Solve sequential move games by backward induction. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Backward Induction Venture Capital Game: Don’t 0,0 Inv VC S -1,2 W 3,1 Ent Ent will optimally shirk once the investment has been undertaken. If the VC invests, it knows that then Ent will optimally shirk. Investment gives a payoff of -1. Equilibrium: (Don’t;S) GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Backward Induction The pricing game: L H T L 2,7 H 5,3 LS L 3,1 H 10,-1 LS’ Equilibrium: (L; (L,L)) Equilibrium path of play: the sequence of actions taken by players found using backward induction. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Backward Induction Supply chain game: N U M D C R N 100,350 A 180,300 M’ N 40,120 A 120,280 M” D 130,150 C 60,380 R’ Equilibrium: (U,(N,A);(D,D)). GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Backward Induction The process of backward induction – looking forward in the game, realizing your opponents are rational, and using this information to inform your decisions, gives us another rule of strategy. Rule of Strategy Look forward and reason back. Anticipate what your opponents will do, i.e. where your decisions will ultimately lead, and use this information to calculate your best choice. Backward induction can be thought of in terms of sequential rationality. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Sequential-Move Games with Imperfect Information Sequential-move games might be combined with simultaneous-move components. Enter Not E H 0,3 L 0,4 I2 L H I1 H 2,1 L 0,3 H 1,2 L -1,4 E After I1 : E I L H L 0,3 -1,4 H 2,1 1,2 Information Set: if two nodes are connected by an information set the player does not know at which of the two nodes they are making their decision GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Subgame Perfect Nash Equilibrium A subgame is any part of the game branching from a decision node not connected by an ‘information set’. The appropriate solution concept for sequential-move games is SPNE: a set of strategies that constitute a Nash equilibrium in every subgame of the game. In sequential-move games with perfect information backward induction, which ensures sequential rationality, gives precisely this. In sequential-move games with imperfect information (that combine simultaneous moves) we need the full definition of a Nash equilibrium in every subgame. Look forward and deduce what the Nash equilibrium will be in the subgame where simultaneous decisions are made, thereby predicting the outcome for you. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Dominant Strategies May No Longer Be Dominant… F1 F2 Agg Pass Agg 26,8 34,9 Pass 31,12 37,11 Pass Agg 1 Pass 34,9 Agg 26,8 2 Pass 37,11 Agg 31,12 2 DS of Pass for F1 not part of the equilibrium path in the sequential game. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Sequential Assurance Remember the assurance (stag hunt) game? Simultaneous choices: A B Stag Hare Stag 2,2 0,1 Hare 1,0 1,1 If A is the leader: H S A H 0,1 S 2,2 B H 1,1 S 1,0 B GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Enron Prosecution Game Prosecutors “dealed” their way up the corporate ladder GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Bargaining Ultimatum Game Two individuals, a Proposer (A) and a Receiver (B), are bargaining over their share of a pie. Proposer makes an offer of a split; receiver decides accept or reject. If receiver accepts, the split is made; if the receiver rejects both leave with nothing. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Bargaining If proposer has a choice of three offers: L H M A R 0,0 A 1 − sH,sH B R 0,0 A 1 − sM,sM B R 0,0 A 1 − sL,sL B Responser will accept any acceptable offer, so proposer should offer the smallest acceptable share. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Bargaining Proposer can make any offer. Proposer looks forward and deduces that B will accept any acceptable offer, however small. If B has no ‘outside option’ A proposes the smallest possible denomination (s ≈ 0), and B will accept. A gets ≈ 1 and B gets ≈ 0. Power to the proposer! If receiver has an outside GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Alternating Offers Bargaining Suppose now that, if B rejects the initial offer, she gets the opportunity to make a counter-offer. If this offer is rejected in the last round the game ends and the players leave with nothing. Players are impatient: they discount future payoffs δA, δB is A,B’s discount factor. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Alternating Offers Bargaining In second period, B only needs to offer A a token amount, keeping the whole pie for herself. But the present value of this is δB: ‘fall-back’ position of rejecting the first offer Inferring this, A needs to offer δB to B this period, leaving her with 1 − δB. The deal is struck immediately, A offers B δB, and B accepts. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Alternating Offers Bargaining The ability to make a counter-offer does a lot for B! Tempers A’s desire to capture the entire pie. Gives B an alternative to accepting the initial agreement, that she did not have before. The structure of bargaining environments is very important! Outcome for B depends on δB; more precisely A’s perception of it! What is important for the outcome of bargaining: having strong outside options; appearing to be patient. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Holdup Game Partnership between an Inventor and a Retailer. Inventor chooses between H (cost=10), L (cost=1) or No investment, and Retailer chooses whether to take the product to market or not. If H can realize sales of 18, if L can realize sales of 4. No 0,0 H L I N -10,0 P tH − 10,18 − tH R N -1,0 P tL − 1,4 − tL R Retailer, entering the game after the inventor has made their investment decision, makes a take-it-or-leave-it offer over how to share the surplus. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Holdup Game Surplus after H is 18, surplus after L is 4. Ultimatum bargaining: tH = tL ≈ 0, Equilibrium involves No investment. No 0,0 H L I N -10,0 P +ǫ − 10,18 − ǫ R N -1,0 P +ǫ − 1,4 − ǫ R Efficient (maximizes joint value) to undertake H investment. R extracts significant surplus since can threaten to ‘hold up’ production. I can’t extract full benefit of investment, so under-invests. Check: even if R ‘splits the surplus’ equilibrium is L investment. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Order Advantages Players that move later in the game maintain flexibility over their decisions. But are guided down a path of play by players that move earlier, who have the opportunity to commit to a course of action. What has the greater value: commitment or flexibility? Relative to simultaneous play, do you get a higher payoff to pre-empting your opponent and forcing them to move second? First mover advantage. Or is it better to let your opponent move first, maintain flexibility over your decision until you have observed what they have done, and move second? Second-mover advantage. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk First-Mover Advantage Two firms ACME and Bo Selecta! are deciding on the capacity to install to serve a market. Simultaneous choices: ACME Bo Selecta! L H NE L 5,5 3,7 7,0 H 7,3 2,2 9,0 NE 0,7 0,9 0,0 ACME goes first: NE L H A NE 7,0 L 5,5 H B 3,7 NE 9,0 L 7,3 H B 2,2 NE 0,0 L 0,7 H B 0,9 GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Second-Mover Advantage Technology adoption game. Pico (small) and Mega (large) are deciding on one of two technologies to adopt. Pico prefers A, but Pico can only make substantial profit if it adopts the same technology as Mega, in contrast to Mega! Simultaneous choices: Pico Mega A B A 8,4 2,5 B 1,6 5,3 Pico second: B A M B 6,1 A 4,8 P B 3,5 A 5,2 P’ For Pico, there is a second-mover advantage: benefits from being able to mimic what Mega does. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Order Advantages Considering a strategic situation and the presence of order advantages is very important. Should you keep your cards close to your chest. Should you delay decisions as long as possible? Or should you make a decision quickly, and announce it as loudly as possible. All depends on whether commitment or flexibility has the greater value. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Timing Matters Rule of Strategy The order of moves in the game matter and can alter the way you behave in the game. Be sure to understand the information all players will have when they come to make their choices. If you presume the game is simultaneous-move but actually you are the leader, you may be missing a trick! And you may shoot yourself in the foot if you think you are the leader but actually moves are simultaneous. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Analyzing Sequential-Move Games Recognize a game as sequential whenever there is a player in the game that observes the actions of another player. You now have the tools to analyze pretty much any sequential-move game. Identify the players, their available actions, and the order of moves. Construct the extensive form. Apply backward induction to determine the equilibrium path. GoS A Dickson Introduction SimultaneousMove Games Mixed Strategies Dilemmas and Cooperation SequentialMove Games Strategic Moves Games with Private Info Signaling and Cheap Talk Summary Games when moves are sequential. Extensive form representation. Actions vs strategies. Backward induction: look forwards and reason back. Equilibrium path of play. Commitment vs flexibility.

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