gameL6 - Extensive Games with Perfect Information There is...

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Extensive Games with Perfect Information There is perfect information if each player making a move observes all events that have previously occurred. Start by restricting attention to games without simulta- neous moves and without nature (no randomness). De fi nition 89.1: An extensive game with perfect in- formation consists of the following components: 1. The set of players, 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 .
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(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 his- tory a member of N . ( P is the player function, with P ( h ) being the player who takes an action after the his- tory, 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 .) 4. For each player, i N , a preference relation º i on the set of terminal histories, Z . If < N, H, P > satis fi es (1)-(3), but preferences are not speci fi ed, it is called an extensive game form with perfect information. (Example: auction rules are speci fi ed, but not preferences over the objects.)
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If the set of histories is fi nite, the game is fi nite. If the longest history is fi nite, the game has a fi nite horizon. If h is a history of length k , then ( h, a ) is the history of length k + 1 consisting of h followed by the action a . After any non-terminal history, h , the player P ( h ) chooses an action from the set A ( h ) = { a : ( h, a ) H } . Note: we could equivalently de fi ne an extensive game as a game tree (a connected graph with no cycles) Each node corresponds to a history, and the connection be- tween two nodes corresponds to an action. A strategy is a plan specifying the action a player takes for every history after which it is his/her turn to move. Note: some simple games have many, many strategies.
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De fi nition 92.1: A strategy of player i N in an ex- tensive game with perfect information h N, H, P, ( º i ) i is a function that assigns an action in A ( h ) to every non- terminal history, h H \ Z for which we have P ( h ) = i . Note: The de fi nition of a strategy requires us to specify an action after histories that are impossible to reach, if the strategy is followed. One could argue that a plan does not have to specify such contingencies. One in- terpretation is that this part of the strategy represents the beliefs that other players have about what the player would do if he/she did not follow the plan. For each strategy pro fi le, s = ( s i ) i N , we de fi ne the outcome of s (denoted O ( s ) ) to be the terminal node resulting when each player i chooses actions according to s i . De fi nition 93.1: A Nash equilibrium of an extensive game with perfect information is a strategy pro fi le s such that for every player i we have O ( s i , s i ) º i O ( s i , s i ) for all s i .
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For every extensive game, there is a corresponding strate- gic game.
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