On the Expected Value of Games with Absentmindedness

Piccione and Rubinstein have argued that a seemingly paradoxical form of time inconsistency can arise in games of imperfect recall. At the heart of this argument is a calculation of the expected value of a game, from the standpoint of a player who is in the middle of play. We claim that this concept is not well defined in games with absentmindedness (that is, games where two nodes in the same path can be in one information set). To make it well defined, additional assumptions must be made. We show that there are reasonable assumptions under which no time inconsistency arises. It is also possible to make assumptions that validate Piccione and Rubinstein's calculations, but the nature of these asssumptions is generally such as to remove the appearance of paradox.