4.2.Strategies InExtensive Form Games.1.0(博弈论讲-Arizona State University)

ÙStrategies in Extensive-Form Games

Recapitulation____________________________________________________________________1Strategy profiles of extensive-form games______________________________________________2The strategic form of an extensive-form game___________________________________________5When Nature takes her turn at bat____________________________________________________7Randomized strategies in extensive-form games_________________________________________9Behavior strategies_______________________________________________________________9Mixed strategies_________________________________________________________________12Mixed strategy § behavior strategy: the role of perfect recall_____________________________13Mixed strategy § behavior strategy: computing the conditional probabilities_________________16Behavior strategy § mixed strategy_________________________________________________18Many mixed strategies give rise to the same distribution over outcomes______________________19Kuhn’s theorem: The equivalence of behavior and mixed strategies_________________________20The restriction of a strategy to a subgame_____________________________________________21Recapitulation

We are now studying dynamic games—in which some decisions are made after others. We use theextensive form because it allows us to make explicit this temporal structure as well as to define thegame’s information structure. (The results of previous decisions by one player may not be observableimmediately, if ever, to her opponents.)

We introduced the game tree as the supporting framework for the extensive form. Choices are madeat decision nodes, which belong to the set X. One of these, viz. O, is designated the initial node, at whichthe game begins. It is assigned either to one of the players in the player set I={1,…,n} or to Nature;Nature would make a random choice representing any exogenous uncertainty by some or all of theplayers. Play progresses from node to node based on the players’ decisions. The game ends when anyone of the terminal nodes, which belong to the set Z, is reached, at which point the players are awardedthe payoffs corresponding to that node. Player i’s preferences over terminal nodes are represented by hervon Neumann-Morgenstern utility function µi:ÙZ§Â; namely, if terminal node z˙Z is reached, herutility is µiªzº.

When called upon to move, a player might not know precisely at which of her nodes she is located.Her uncertainty is modeled by information sets. An information set h˙H is a set of nodes, all belongingto the same player and at all of which the same set of actions is available. When called upon to act, aÙ© 1997 by Jim Ratliff , <jim@http://www.wendangwang.com>, <http://www.wendangwang.com/gametheory>.

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