Characterizations of perfect recall

This paper considers the condition of perfect recall for the class of arbitrarily large discrete extensive form games. The known definitions of perfect recall are shown to be equivalent even beyond finite games. Further, a qualitatively new characterization in terms of choices is obtained. In particular, an extensive form game satisfies perfect recall if and only if the set of choices, viewed as sets of ultimate outcomes, fulfill the Trivial Intersection property, that is, any two choices with nonempty intersection are ordered by set inclusion

Reduced Normal Forms Are Not Extensive Forms

Fundamental results in the theory of extensive form games have singled out the reduced normal form as the key representation of a game in terms of strategic equivalence. In a precise sense, the reduced normal form contains all strategically relevant information. This note shows that a difficulty with the concept has been overlooked so far: given a reduced normal form alone, it may be impossible to reconstruct the game’s extensive form representation

Kuhn's Theorem for Extensive Form Ellsberg Games

The paper generalizes Kuhn's Theorem to extensive form games in which players condition their play on the realization of ambiguous randomization devices and use a maxmin decision rule to evaluate the consequences of their decisions. It proves that ambiguous behavioral and ambiguous mixed strategies are payoff and outcome equivalent only if the latter strategies satisfy a rectangularity condition. The paper also discusses dynamic consistency. In particular, it shows that not only the profile of ambiguous strategies must be appropriately chosen but also the extensive form must satisfy further restrictions beyond those implied by perfect recall in order to ensure that each player respects her ex ante contingent choice with the evolution of play

Ex-Post Regret Learning in Games with Fixed and Random Matching: The Case of Private Values

In contexts in which players have no priors, we analyze a learning process based on ex-post regret as a guide to understand how to play games of incomplete information under private values. The conclusions depend on whether players interact within a fixed set (fixed matching) or they are randomly matched to play the game (random matching). The relevant long run predictions are minimal sets that are closed under 'the same or better reply' operations. Under additional assumptions in each case, the prediction boils down to pure Nash equilibria, pure ex-post equilibria or pure minimax regret equilibria. These three paradigms exhibit nice robustness properties in the sense that they are independent of beliefs about the exogenous uncertainty of type spaces. The results are illustrated in second-price auctions, first-price auctions and Bertrand duopolies

Iterated elimination procedures

We study the existence and uniqueness (i.e.,order independence) of any arbitrary form of iterated elimination procedures in an abstract environment. By allowing for a transfinite elimination, we show a general existence of the iterated elimination procedure. Inspired by the seminal work of Gilboa, Kalai and Zemel (1990), we identify a fairly weak suffcient condition of Monotonicity* for the order independence of iterated elimination procedure. Monotonicity* requires a monotonicity property along any elimination path. Our approach is applicable to different forms of iterated elimination procedures used in (in)finite games, for example, iterated elimination of strictly dominated strategies, iterated elimination of weakly dominated strategies, rationalizability, and soon. We introduce a notion of CD* games, which incorporates Jackson's (1992) idea of "boundedness", and show the iterated elimination procedure is order independent in the class of CD* games. In finite games, we also formulate and show an "outcome" order-independence result suitable for Marx and Swinkels's (1997) notion of nice weak dominance