Sophisticated Sequential Rationality in Extensive Form Games
Abstract: I introduce the notion of sophisticated sequentially rationality for extensive form games. When considering a deviation, sophisticated sequentially rational players dynamically update their beliefs at subsequent information sets, and internalize their subsequent deviations given their updated beliefs. A strategy is sophisticated sequentially rational if it is stable under any such sequence of deviations. In games with perfect recall, sophisticated sequential rationality coincides with ex-ante optimality; however, in games with imperfect recall it need not. I show that a sophisticated sequentially rational behavioral strategy exists for any strategy profile of other players, and a sophisticated sequentially rational equilibrium} in mixed behavioral strategies exists in any finite extensive form game. The set of sophisticated sequentially rational equilibria and the set of Nash equilibria of a game are generally distinct; however, they coincide if players have perfect recall of past information.
Extensive Form Generalized Games and Generalized Perfect Recall
Abstract: I introduce the notion of an extensive form generalized game, a framework for modeling dynamic strategic settings where players' feasible strategies depend on those chosen by others. Extensive form generalized games are motivated by models of limited strategic complexity and endogenous information acquisition, and nest extensive form games, generalized games, games played by finite automata, and games with rational inattention. I determine tight sufficient conditions for existence of an equilibrium in behavioral strategies in finite extensive form generalized games. The most salient of which, generalized perfect recall, is a generalized convexity condition which is equivalent to perfect recall in extensive form games. Feasibility correspondences describing rational inattention are shown to satisfy generalized perfect recall, and a simple sufficient condition for generalized perfect recall is provided.
Convexity and Perfect Recall in Extensive Form Games
Abstract: While it is well known that Nash equilibria in behavioral strategies may not exist in games with imperfect recall, the reason has remained an open question. In this paper, I answer this question by providing a geometric characterization of perfect recall. In particular, I show that in essentially any extensive form game a player has perfect recall if and only if his set of feasible mixed strategies are convex. In addition, I introduce the notion of behavioral convexity, a natural notion of generalized convexity for behavioral strategies. Players' expected utility is ``linear" in behavioral convex combinations of behavioral strategies, and behavioral convexity of players' behavioral strategies is equivalent to convexity of their feasible mixed strategies, and is therefore equivalent to perfect recall in essentially any extensive form game.
Extensive Form Generalized Games with Generalized Imperfect Recall
Works in Progress:
Totally Sequential Equilibrium in Extensive Form Generalized Games
Games with Costly Information