paradox of the absentminded driver
Paradox of the absentminded driver An individual is sitting late at night in a bar planning his midnight trip home. In order to get home he has to take the highway and get off at the second exit. Turning at the first exit leads into a disastrous area payoff 0. Turning at the second exit yields the highest reward payoff 4. If he continues beyond the second exit, he cannot go back and at the end of the highway he will find a motel where he can spend the night payoff 1. The driver is absentminded and is aware of this fact. At an intersection, he cannot tell whether it is the first or the second intersection and he cannot remember how many he has passed one can make the situation more realistic by referring to the 17th intersection.
While sitting at the bar, all he can do is to decide whether or not to exit at an intersection. We exclude at this stage the possibility that the decision maker can include random elements in his strategy. Example 1 describes this situation. Planning his trip at the bar, the decision maker must conclude that it is impossible for him to get home and that he should not exit when reaching an intersection. Thus, his optimal plan will lead him to spend the night at the motel and yield a payoff of 1. Now, suppose that he reaches an intersection. If he had decided to exit, he would have concluded that he is at the first intersection. Having chosen the strategy to continue, he concludes that he is at the first intersection with probability 1r2. Then, reviewing his plan, he finds that it is optimal for him to leave the highway since it yields an expected payoff of 2. Despite no new information and no change in his preferences, the decision maker would like to change his initial plan once he reaches an intersection! Note that if the decision maker now infers that he would have exited the highway had he passed the first intersection, his reasoning becomes circular; he must conclude that he is at the first intersection and that it is optimal to continue.
— On the Interpretation of Decision Problems with Imperfect Recall (Piccione, Rubinstein, 1997)