Absent-minded decision problem

A decision problem in which the same information set can be reached more than once along a single path, so the decision-maker does not know which visit is occurring. It violates perfect recall. A behavioural strategy uses a common probability at each visit and independent random draws, making terminal probabilities nonlinear in that probability.

For instance, let stopping at the first visit pay zero, continuing and then stopping pay eight, and continuing twice pay one. With continuation probability $p$, expected payoff is $8p(1-p)+p^2=16/7-7(p-4/7{)}^2$. Its maximum $16/7$ exceeds the best deterministic-plan payoff of one. Randomizing once between the deterministic plans cannot reproduce the two fresh decisions.