§ 4.2 Strategies in extensive-form games

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This is a chapter from the graduate-level game-theory course I taught circa 1992–1997. Check out the preface and Table of Contents. I no longer maintain, update, or correct these notes. However, I do appreciate hearing from people who download these notes and find them useful.

Abstract

We define a strategy for a player in an extensive-form game as a specification for each of her information sets of the (pure or mixed) action she would take at that information set. One such strategy for each player constitutes a strategy profile for the extensive-form game.

Every extensive-form game can be expressed as a strategic-form game.

We incorporate uncertain exogenous events into the extensive form by introducing Nature as a nonstrategic player who acts randomly. We learn how to construct the strategic-form of an extensive-form game when Nature takes a turn at bat.

We study two different types of randomized strategies in extensive-form games. The behavioral strategy specifies randomizations at each information set independently. The mixed strategy specifies randomizations over pure extensive-form strategies. Behavior strategies do not permit correlations across information sets that mixed strategies allow. However, we show that, in extensive-form games satisfying perfect recall, mixed and behavior strategies can be used interchangeably in a precise sense. We learn how to convert back and forth between the two types of randomized strategies.

We define how to restrict an extensive-game strategy to a particular subgame.

Navigate between chapters of the game-theory course

§ 4.1 Intro to extensive-form games

§ 4.3 Extensive-form solution concepts

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