Games in Extensive Form

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This chapter deals with games in extensive form. Here an explicit evolution of the interaction is given, describing precisely when each player plays, what actions are available and what information is available to each player when he makes a decision. We start with games with perfect information (such as chess) and prove Zermelo’s theorem for finite games. We then consider infinite games a la Gale–Stewart: we show that open games are determined and that under the axiom of choice, there exists an undetermined game. Next we introduce games with imperfect information and prove Kuhn’s theorem, which states that mixed and behavioral strategies are equivalent in games with perfect recall. We present the standard characterization of Nash equilibria in behavioral strategies and introduce the basic refinements of Nash equilibria in extensive-form games: subgame-perfection, Bayesian perfect and sequential equilibria, which impose rational behaviors not only on the equilibrium path but also off-path. We prove the existence of sequential equilibrium (Kreps and Wilson). For normal form games as in Chap. 4 we introduce the standard refinements of Nash equilibrium: perfect equilibrium (Selten) and proper equilibrium (Myerson). We prove that a proper equilibrium of a normal form game G induces a sequential equilibrium in every extensive-form game with perfect recall having G as normal form. Finally we discuss forward induction and stability (Kohlberg and Mertens).