10 DynamicIncompleteInfoGames1(2).pdf

10 DynamicIncompleteInfoGames1(2).pdf - Lecture 10 Dynamic...

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Lecture 10: Dynamic games with incomplete information I November 1, 2016 1 Overview In the Bayesian Nash equilibrium of a static game with incomplete information: 1. Each player correctly forms belief (i.e., use Bayes’ rule when needed) about every player’s types (including the Nature’s type - the true state). 2. Based on these beliefs, players calculate their expected utilities and choose best responses to maximize these expected utilities. This is similar to Nash equilibrium for static games with complete information, except that we’ve added step 1 for players to calculate their expected utilities before best responding. Recall: An information set for a player is a collection of decision nodes satisfying: (i) the player has the move at every node in the information set, and (ii) when the play of the game reaches a node in the information set, the player with the move does not know which node in the information set has been reached When the game is dynamic and has incomplete information, the equilibrium concept we use is called perfect Bayesian equilibrium . It requires that, for every player, 1 and 2 hold at every information set , even at the information sets that are not reached in equilibrium. This is similar to subgame-perfect Nash equilibrium for dynamic games with com- plete information, except that each player is required to correctly update beliefs and calculate expected utilities. In other words, perfect Bayesian equilibrium requires that every player is always maximizing expected utility in every scenario, even in scenarios that didn’t actu- ally arise in equilibrium. Let’s look at an example. 1
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2 Application: Informational cascades (/“herding”) The theory of informational cascades is a theory that explains fads, fashion, custom and cultural changes from the perspective of observational learning. The question is simple: why do people follow suit and cluster on the same (occupa- tional, cultural, consumption) choice? There are many possible answers: similarity in taste, social conformity, positive net- work externality... Bikhchandani, Hirshleifer, Welch (1992) looks at the particular angle of observational learning and they offer a rational justification of people’s incentive to abandon private opinion and follow the crowd, even if the crowd’s choice is not necessarily correct. 2.1 Set up There are two new restaurants in town, A and B. Which restaurant is better is uncertain with equal prior probability, i.e., true state is either A or B with Pr( A ) = Pr( B ) = 0 . 5 .
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