Perfect recall and the informational contents of strategies in extensive games
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This dissertation consists of five chapters on the informational contents of strategies and the role of the perfect recall condition for information partitions in extensive games. The first, introductory, chapter gives basic definitions of extensive games and some results known in the game theory literature. The questions that will be investigated in the remaining chapters and their significance in the literature are also described.
In the second chapter it is shown that strategies defined as contingent plans may contain some information that is additional to what the information partition describes. Two types of additional information that strategies may contain when perfect recall is violated are considered. Both behavior and mixed strategies contain the first type of information, but only mixed strategies contain the second type. Addition of either type of information, however, leads to a refinement of the information partition that satisfies perfect recall. The perfect recall condition is found to be significant in demarcating the roles of strategies and information partitions in extensive games.
In the third chapter the full informational contents of mixed strategy spaces is explored. The informational content of mixed strategy spaces is found to be invariant over a range of information partitions. A weakening of the perfect recall condition called A-loss is obtained and found to be necessary and sufficient for the information contained in mixed strategies to be equivalent to that of a game with perfect recall. An implication of this result is that a player whose information partition satisfies A-loss can play "as-if" he has perfect recall and a player without A-loss can't. In other words, if an information partition satisfies A-loss, every mixed strategy makes up for any lack of perfect recall described by the information partition. For behavior strategies, we never obtain informational equivalence between distinct information partitions. A-loss turns out to also be a necessary condition for a game without chance moves to have a Nash equilibrium in pure strategies for all payoff assignments.
In the fourth chapter the role of the perfect recall condition in preserving some information in the transformation from an extensive game to its agent normal form is discussed. If we interpret a player as a team of agents (one at each information set) then the essential difference between an extensive game and the associated agent normal form game is that in the former the agents act cooperatively while in the latter they act independently. The perfect recall condition is shown to be necessary and sufficient for the perfect equilibria of an extensive game to coincide with those of the associated agent normal form game for all payoff assignments. The contribution of this result is necessity; sufficiency is already known. Since this is proved using pure strategies for the player with imperfect recall in question, one subtle implication is obtained: a perfect equilibrium of the agent normal form game where each agent effectively knows the actions taken and information acquired by his preceding agents, may not be a perfect equilibrium in the original extensive game. This means that perfect recall implies more than just effective knowledge of what happened previously.
Chapter 5 concludes.
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