11 DynamicIncompleteInfoGames2.pdf

# 11 DynamicIncompleteInfoGames2.pdf - Lecture 11 Dynamic...

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Lecture 11: Dynamic games with incomplete information II October 24, 2016 1 Overview 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. 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. 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. 2 Application: Persuasion games 2.1 Cheap talk 2.1.1 Cheap talk with misaligned incentives Suppose a seller is recommending a product to a potential buyer. The true value of the product is b = 1 or 7 with equal probability. The buyer is willing to buy if and only if the expected value of the product is above 5 (e.g. the price of the product is 5). Only the seller sees the true value. Observing the true value, the seller sends a message m ( b ) ∈ { 1 , 7 } to the buyer. I.e., the seller says: “This product has value m .” 1

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Observing only the message m , the buyer decides whether to buy or not based on m . Seller gets payoff 1 if the buyer buys and 0 otherwise. In this game, since the message of the seller is not verifiable (it contains no hard evidence), we call it “cheap talk”. Unique perfect Bayesian equilibrium: The seller always sends the same message (for example, m ( b ) = 7 ) for both b = 1 , 7 . (I.e., the seller always says “my product has excellent value.”) The buyer never buys the product. Proof. The given messaging strategy is called a “pooling” strategy because the seller is always “pooling the two possible states together” by sending the same message unconditionally. If the seller is using a pooling strategy, then his message is completely uninformative to the buyer. Therefore, the buyer relies on his prior belief to make a decision. In this game, since b = 1 or 7 with equal probability, the expected value of the product is 4 < 5, and the buyer chooses not to buy it.
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