Game theory summaries 12 15 and 16 part 4 static games of incomplete information and part 5 dynamic games of incomplete information


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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