game theory tadelis textbook summaries


part 1 prologue Rational Decision Making
Chapter 1 The Single-Person Decision Problem 3 1.1 Actions, Outcomes, and Preferences 4 1.1.1 Preference Relations 5 1.1.2 Payoff Functions 7 1.2 The Rational Choice Paradigm

1.3 Summary . A simple decision problem has three components: actions, outcomes, and preferences over outcomes. . A rational player has complete and transitive preferences over outcomes and hence can always identify a best alternative from among his possible actions. These preferences can be represented by a payoff (or profit) function over outcomes and the corresponding payoffs over actions. . A rational player chooses the action that gives him the highest possible payoff from the possible set of actions at his disposal. Hence by maximizing his payoff function over his set of alternative actions, a rational player will choose his optimal decision. . A decision tree is a simple graphic representation for decision problems.

Chapter 2 Introducing Uncertainty and Time 14 2.1 Risk, Nature, and Random Outcomes 14 2.1.1 Finite Outcomes and Simple Lotteries 15 2.1.2 Simple versus Compound Lotteries 16 2.1.3 Lotteries over Continuous Outcomes 17 2.2 Evaluating Random Outcomes 18 2.2.1 Expected Payoff: The Finite Case 19 2.2.2 Expected Payoff: The Continuous Case 20 2.2.3 Caveat: It’s Not Just the Order Anymore 21 2.2.4 Risk Attitudes 22 2.2.5 The St. Petersburg Paradox 23 2.3 Rational Decision Making with Uncertainty 24 2.3.1 Rationality Revisited 24 2.3.2 Maximizing Expected Payoffs 24 2.4 Decisions over Time 26 2.4.1 Backward Induction 26 2.4.2 Discounting Future Payoffs 28 2.5 Applications 29 2.5.1 The Value of Information 29 2.5.2 Discounted Future Consumption

2.7 Summary . When prospects are uncertain, a rational decision maker needs to put structure on the problem in the form of probability distributions over outcomes that we call lotteries. . Whether the acts of Nature evolve over time or whether they are chosen once and for all, a rational player cares only about the distribution over final outcomes. Hence any series of compound lotteries can be compressed to its equivalent simple lottery. . When evaluating lotteries, we use the expected payoff criterion. Hence every lottery is evaluated by the expected payoff it offers the player. . Unlike certain outcomes in which only the order of preferences matters, when random outcomes are evaluated with expected payoffs, the magnitude of payoffs matters as well. The difference in payoff values between outcomes will also be related to a player’s risk preferences. . Rational players will always choose the action that offers them the highest expected payoff. . When decisions need to be made over time, a rational player will solve his problem “backwards” so that early decisions take into account later decisions. . Payoffs that are received in future periods will often be discounted in earlier periods to reflect impatience, costs of capital, or uncertainty over whether future periods will be relevant.

PART II Static Games of Complete Information
Chapter 3 Preliminaries 43 3.1 Normal-Form Games with Pure Strategies 46 3.1.1 Example: The Prisoner’s Dilemma 48 3.1.2 Example: Cournot Duopoly 49 3.1.3 Example: Voting on a New Agenda 49 3.2 Matrix Representation: Two-Player Finite Game 50 3.2.1 Example: The Prisoner’s Dilemma 51 3.2.2 Example: Rock-Paper-Scissors 52 3.3 Solution Concepts 52 3.3.1 Assumptions and Setup 54 3.3.2 Evaluating Solution Concepts 55 3.3.3 Evaluating Outcomes 56

3.4 Summary . A normal-form game includes a finite set of players, a set of pure strategies for each player, and a payoff function for each player that assigns a payoff value to each combination of chosen strategies. . Any two-player finite game can be represented by a matrix. Each row represents one of player 1’s strategies, each column represents one of player 2’s strategies, and each cell in the matrix contains the payoffs for both players. . A solution concept that proposes predictions of how games will be played should be widely applicable, should restrict the set of possible outcomes to a small set of reasonable outcomes, and should not be too sensitive to small changes in the game. . Outcomes should be evaluated using the Pareto criterion, yet self-enforcing behavior will dictate the set of reasonable outcomes.

Chapter 4 Rationality and Common Knowledge 59 4.1 Dominance in Pure Strategies 59 4.1.1 Dominated Strategies 59 4.1.2 Dominant Strategy Equilibrium 61 4.1.3 Evaluating Dominant Strategy Equilibrium 62 4.2 Iterated Elimination of Strictly Dominated Pure Strategies 63 4.2.1 Iterated Elimination and Common Knowledge of Rationality 63 4.2.2 Example: Cournot Duopoly 65 4.2.3 Evaluating IESDS 67 4.3 Beliefs, Best Response, and Rationalizability 69 4.3.1 The Best Response 69 4.3.2 Beliefs and Best-Response Correspondences 71 4.3.3 Rationalizability 73 4.3.4 The Cournot Duopoly Revisited 73 4.3.5 The “p-Beauty Contest” 74 4.3.6 Evaluating Rationalizability

4 Summary . Rational players will never play a dominated strategy and will always play a dominant strategy when it exists. . When players share common knowledge of rationality, the only strategies that are sensible are those that survive IESDS. . Rational players will always play a best response to their beliefs. Hence any strategy for which there are no beliefs that justify its choice will never be chosen. . Outcomes that survive IESDS, rationalizability, or strict dominance need not be Pareto optimal, implying that players may not be able to achieve desirable outcomes if they are left to their own devices.

Chapter 5 Pinning Down Beliefs: Nash Equilibrium 79 5.1 Nash Equilibrium in Pure Strategies 80 5.1.1 Pure-Strategy Nash Equilibrium in a Matrix 81 5.1.2 Evaluating the Nash Equilibria Solution 83 5.2 Nash Equilibrium: Some Classic Applications 83 5.2.1 Two Kinds of Societies 83 5.2.2 The Tragedy of the Commons 84 5.2.3 Cournot Duopoly 87 5.2.4 Bertrand Duopoly 88 5.2.5 Political Ideology and Electoral Competition

5.3 Summary . Any strategy profile for which players are playing mutual best responses is a Nash equilibrium, making this equilibrium concept self-enforcing. . If a profile of strategies is the unique survivor of IESDS or is the unique rationalizable profile of strategies then it is a Nash equilibrium. . If a profile of strategies is a Nash equilibrium then it must survive IESDS and it must be rationalizable, but not every strategy that survives IESDS or that is rationalizable is a Nash equilibrium. . Nash equilibrium analysis can shed light on phenomena such as the tragedy of the commons and the nature of competition in markets and in politics.

Chapter 6 Mixed Strategies 101 6.1 Strategies, Beliefs, and Expected Payoffs 102 6.1.1 Finite Strategy Sets 102 6.1.2 Continuous Strategy Sets 104 6.1.3 Beliefs and Mixed Strategies 105 6.1.4 Expected Payoffs 105 6.2 Mixed-Strategy Nash Equilibrium 107 6.2.1 Example: Matching Pennies 108 6.2.2 Example: Rock-Paper-Scissors 111 6.2.3 Multiple Equilibria: Pure and Mixed 113 6.3 IESDS and Rationalizability Revisited 114 6.4 Nash’s Existence Theorem

6.5 Summary . Allowing for mixed strategies enriches both what players can choose and what they can believe about the choices of other players. . In games for which players have opposing interests, like the Matching Pennies game, there will be no pure-strategy equilibrium but a mixed-strategy equilibrium will exist. . Allowing for mixed strategies enhances the power of IESDS and of rationalizability. . Nash proved that for finite games there will always be at least one Nash equilibrium.

PART III Dynamic Games of Complete Information Chapter 7 Preliminaries 129 7.1 The Extensive-Form Game 130 7.1.1 Game Trees 132 7.1.2 Imperfect versus Perfect Information 136 7.2 Strategies and Nash Equilibrium 137 7.2.1 Pure Strategies 137 7.2.2 Mixed versus Behavioral Strategies 139 7.2.3 Normal-Form Representation of Extensive-Form Games 143 7.3 Nash Equilibrium and Paths of Play

7.4 Summary . In addition to the set of players, their possible actions, and their payoffs from outcomes, the extensive-form representation captures the order in which they play as well as what each player knows when it is his turn to move. Game trees provide a useful tool to describe extensive-form games. . If a player cannot distinguish between two or more nodes in a game tree then they belong to the same information set. Care has to be taken to specify correctly each player’s information sets. . In games of perfect information, every player knows exactly what transpired before each of his turns to move, so each information set is a singleton. If this is not the case then the players are playing a game of imperfect information. . A pure strategy defines a player’s deterministic plan of action at each of his information sets. A mixed strategy is a probability distribution over pure strategies, while a behavioral strategy is a plan of probability distributions over actions in every information set. . Every extensive-form game has a unique normal-form representation, but the reverse is not necessarily true.

Chapter 8 Credibility and Sequential Rationality 151 8.1 Sequential Rationality and Backward Induction 152 8.2 Subgame-Perfect Nash Equilibrium: Concept 153 8.3 Subgame-Perfect Nash Equilibrium: Examples 159 8.3.1 The Centipede Game 159 8.3.2 Stackelberg Competition 160 8.3.3 Mutually Assured Destruction 163 8.3.4 Time-Inconsistent Preferences

8.4 Summary . Extensive-form games will often have some Nash equilibria that are not sequentially rational, yet we expect rational players to choose sequentially rational strategies. . In games of perfect information backward induction will result in sequentially rational Nash equilibria. If there are no two payoffs at terminal nodes that are the same then there will be a unique sequentially rational Nash equilibrium. . Subgame-perfect equilibrium is the more general construct of backward induction for games of imperfect information. . At least one of the Nash equilibria in a game will be a subgame-perfect equilibrium.

Chapter 9 Multistage Games 175 9.1 Preliminaries 176 9.2 Payoffs 177 9.3 Strategies and Conditional Play 178 9.4 Subgame-Perfect Equilibria 180 9.5 The One-Stage Deviation Principle

9.6 Summary . Multistage games are defined by a series of normal-form stage-games that are played in sequence, in which players obtain payoffs after every stage-game and future payoffs are discounted. . Any sequence of stage-game Nash equilibrium play can be supported as a subgame-perfect equilibrium in the multistage game, regardless of the discount factor. . Players can use credible threats and promises in later stages to provide longterm incentives for short-term actions that may not be self-enforcing in the earlier-period stage-games. . Because future payoffs are discounted, the effectiveness of long-term incentives will depend on how patient the players are. . The set of outcomes that can be supported by a subgame-perfect equilibrium will often depend on the discount factor.

Chapter 10 Repeated Games 190 10.1 Finitely Repeated Games 190 10.2 Infinitely Repeated Games 192 10.2.1 Payoffs 193 10.2.2 Strategies 195 10.3 Subgame-Perfect Equilibria 196 10.4 Application: Tacit Collusion 201 10.5 Sequential Interaction and Reputation 204 10.5.1 Cooperation as Reputation 204 10.5.2 Third-Party Institutions as Reputation Mechanisms 205 10.5.3 Reputation Transfers without Third Parties 207 10.6 The Folk Theorem: Almost Anything Goes

10.7 Summary . If a stage-game that has a unique Nash equilibrium is repeated for finitely many periods then it will have a unique subgame-perfect equilibrium regardless of the discount factor’s value. . If a stage-game that has a unique Nash equilibrium is repeated infinitely often with a discount factor 0 <δ< 1, it will be possible to support behavior in each period that is not a Nash equilibrium of the one-shot stage-game. The “carrot” and “stick” incentives are created by bootstrapping the repetition of the stage-game’s unique equilibrium, which becomes a more potent threat as the discount factor approaches 1. . The folk theorem teaches us that as the discount factor approaches 1, the set of average payoffs that can be supported by a subgame-perfect equilibrium of the infinitely repeated game grows to a point that almost anything can happen. . Repeated games are useful frameworks to understand how people cooperate over time, how firms manage to collude in markets, and how reputations for good behavior are sustained over time even when short-run temptations are present.

Chapter 12 Bayesian Games 241 12.1 Strategic Representation of Bayesian Games 246 12.1.1 Players, Actions, Information, and Preferences 246 12.1.2 Deriving Posteriors from a Common Prior: A Player’s Beliefs 247 12.1.3 Strategies and Bayesian Nash Equilibrium 249 12.2 Examples 252 12.2.1 Teenagers and the Game of Chicken 252 12.2.2 Study Groups 255 12.3 Inefficient Trade and Adverse Selection 258 12.4 Committee Voting 261 12.5 Mixed Strategies Revisited: Harsanyi’s Interpretation

12.6 Summary . In most real-world situations players will not know how much their opponents value different outcomes of the game, but they may have a good idea about the range of their valuations. . It is possible to model uncertainty over other players’ payoffs by introducing types that represent the different possible preferences of each player. Adding this together with Nature’s distribution over the possible types defines a Bayesian game of incomplete information. . Using the common prior assumption on the distribution of players’ types, it is possible to adopt the Nash equilibrium concept to Bayesian games, renamed a Bayesian Nash equilibrium. . Markets with asymmetric information can be modeled as games of incomplete information, resulting in Bayesian Nash equilibrium outcomes with inefficient trade outcomes. . Harsanyi’s purification theorem suggests that mixed-strategy equilibria in games of complete information can be thought of as representing pure-strategy Bayesian Nash equilibria of games with heterogeneous players.

Chapter 15 Sequential Rationality with Incomplete Information 303 15.1 The Problem with Subgame Perfection 303 15.2 Perfect Bayesian Equilibrium 307 15.3 Sequential Equilibrium

15.4 Summary . Because games of incomplete information have information sets that are associated with Nature’s choices of types, it will often be the case that the only proper subgame is the whole game. As a consequence, subgame-perfect equilibrium will rarely restrict the set of Bayesian Nash equilibria to those that are sequentially rational. . By requiring that players form beliefs in every information set, and requiring these beliefs to be consistent with Bayes’ rule, we can apply the concept of sequential rationality to Bayesian games. . In a perfect Bayesian equilibrium, beliefs are constrained on the equilibrium path but not off the equilibrium path. It is important, however, that beliefs off the equilibrium path support equilibrium behavior. . In some games the concept of perfect Bayesian equilibrium will not rule out play that seems sequentially irrational. Equilibrium refinements, such as sequential equilibrium, have been developed to address these situations.

Chapter 16 Signaling Games 318 16.1 Education Signaling: The MBA Game 319 16.2 Limit Pricing and Entry Deterrence 323 16.2.1 Separating Equilibria 324 16.2.2 Pooling Equilibria 330 16.3 Refinements of Perfect Bayesian Equilibrium in Signaling Games

16.4 Summary . In games of incomplete information some types of players would benefit from conveying their private information to the other players. . Announcements or cheap talk alone cannot support this in equilibrium, because then disadvantaged types would pretend to be advantaged and try to announce “I am this type” to gain the anticipated benefits. This strategy cannot be part of an equilibrium because by definition players cannot be fooled in equilibrium. . For advantaged types to be able to separate themselves credibly from disadvantaged types there must be some signaling action that costs less for the advantaged types than it does for the disadvantaged types. . Signaling games will often have many perfect Bayesian and sequential equilibria because of the flexibility of off-the-equilibrium-path beliefs. Refinements such as the intuitive criterion help pin down equilibria, often resulting in the least-cost separating equilibrium.

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