# 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.)
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 .
For every extensive game, there is a corresponding strate- gic game.

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