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# gameL8 - Extensive Games with Imperfect Information In...

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Extensive Games with Imperfect Information In strategic games, players must form beliefs about the other players’ strategies, based on the presumed equilib- rium being played. In Bayesian games, players must form beliefs about the other players’ strategies and their types, based on the probability distribution over types and the presumed equi- librium being played. In extensive games with perfect information, there is the possibility that a player will face a situation that is in- consistent with the presumed equilibrium being played. However, subgame perfection takes care of this issue by requiring a form of sequential rationality, even o ff the equilibrium path.

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In extensive games with imperfect information, when a player faces a situation that is inconsistent with the pre- sumed equilibrium being played, she may also be forced to form beliefs about the other players’ past behavior. These beliefs are often crucial in evaluating whether the ensuing play is rational. Put another way, games with imperfect information may have few subgames, possibly only one. How then are we to eliminate Nash equilibria that involve threats that are not credible?
De fi nition 200.1: An extensive game consists of the following components: 1. The set of players (assumed to be a fi nite set), N . 2. A set, H , of sequences (histories of actions) satisfying the following properties: (2a) The empty sequence, , is an element of H . (2b) If ( a k ) k =1 ,...,K H , (where K may be in fi nity) and L < K , then ( a k ) k =1 ,...,L H . (2c) If an in fi nite sequence, ( a k ) k =1 ,..., satis fi es ( a k ) k =1 ,...,L H for every positive integer L , then ( a k ) k =1 ,..., H . 3. A function, P , that assigns to each non-terminal history a member of N { c } . ( P is the player func- tion, with P ( h ) being the player who takes an action after the history, h . If P ( h ) = c holds, then chance or nature takes the action after the history, h . A history, ( a k ) k =1 ,...,K , is terminal if it is in fi nite or if there is no a K +1 such that ( a k ) k =1 ,...,K +1 H .)

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4. A function, f c , that associates with each h H for which P ( h ) = c , a probability measure on A ( h ) , denoted by f c ( · | h ) . Each such measure is assumed to be independent of every other such measure. 5. For each player, i N , a partition = i of { h H : P ( h ) = i } with the property that A ( h ) = A ( h 0 ) when- ever h and h 0 are in the same element of the partition. For I i = i , we denote by A ( I i ) the set A ( h ) and by P ( I i ) the player P ( h ) for any h I i . 6. For each player, i N , a preference relation º i on lotteries over the set of terminal histories, Z .
I i = i is called an information set of player i . A player cannot distinguish two histories (nodes) in one of her information sets. Notice the requirement that the set of available actions must be the same for any two histories in an information set, or else the player would be able to distinguish between the histories.

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gameL8 - Extensive Games with Imperfect Information In...

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