Incorporating different frameworks to interpret imperfect recall

When computing the expected payoffs of a player at an information set with absentmindedness, there are many issues regarding the belief system that the player forms on the probability measures over histories in that information set. One of those is the issue raised by the circle “strategy- belief- strategy". The circle “strategybelief- strategy" indicates the belief at an information set is formed by the strategy, and the decision maker needs to choose an optimal strategy in the premise of the belief, which implies a problematic logic. To solve this problem, we interpret a finite decision problem with absentmindedness into a psychological multiself game form. A self called planner represents the decision maker before the decision problem starts. A doer represents the decision maker at each node during execution. The planner could choose a full strategy for the decision problem based on his different beliefs on how the doers behave. A confident planner believes the doers will follow his chosen strategy. A knowledgable planner believes the doers will reevaluate the decision problem themselves and choose conditional optimal one-shot behaviour. The doers behave the same way as described in the knowledgable planner’s belief. Thus, we transfer a decision problem with absentmindedness into a twostage psychological game. The existence of a psychological multiself equilibrium with a confident planner indicates that an ex-ante optimal strategy is not only modified multiself consistent (defined in Piccione & Rubinstein (1997a)) but stable in terms of multiple one-shot deviations. It indicates if doers do not cooperate, they cannot reach a better situation than the situation if they execute the ex-ante optimal strategy. The existence of a psychological multiself equilibrium with a knowledgable planner presents the situation if the planner expects potential deviations. In conventional decision theory, it is not much different whether a decision maker knows future imperfect recall, except there is an extra requirement of strategy choose if the decision maker knows. It is interesting to explore how a decision maker behaves if he knows he has imperfect recall and prepares for it in the way that he takes how he will evaluate the decision problem at the future information set into consideration when he reassesses the decision problem currently. We call such a decision maker sophisticated. We develop the sophisticated recursive calculation rule to describe the process of reconsideration at an information set. The resulted expected payoff function is called IS expected payoff function. The IS expected payoff function is used to examine the time consistency of a strategy at an information set in a decision problem and the sequential rationality of a strategy profile at a collection of information sets in a game. Then, it concludes a strategy is ex-ante optimal if and only if it is IS time consistent and an IS sequential equilibrium exists in any finite game

Remarks on the absentminded driver

Piccione and Rubinstein (1997) present and analyse the sequential decision problem of an “absentminded driver”. The driver's absentmindedness (imperfect recall) leads him to time-inconsistent strategy evaluations. His original evaluation gets replaced by a new one under impact of the information that the circumstances have changed, notwithstanding the fact that this change in circumstances has been expected by him all along. The time inconsistency in strategy evaluation suggests that such an agent might have reason to renege on his adopted strategy. As we shall see, however, this danger is only apparent. There is no serious problem of dynamic inconsistency in this case. My diagnosis of the case under consideration is in many respects similar to the one provided by Aumann, Hart and Perry (1997), but the analysis leading to this diagnosis is not quite the same