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Consider a zero-sum game in which a player chooses a binary action, forgets it, and chooses again, while an opponent makes one hidden binary choice. The first player's payoff is one if all three choices agree and zero otherwise. The later information set merges both initial-action histories. This violates perfect recall without revisiting the same information set, unlike an absent-minded decision problem.

A behavioural strategy chooses the first and second actions with independent left probabilities $p,r$. Its security level payoff is $\min (pr,(1-p)(1-r))$, since the opponent selects the less favorable target action. This minimum is at most $\sqrt{p(1-p)r(1-r)}\le 1/4$, attained at $p=r=1/2$.

A mixed strategy instead samples the all-left or all-right complete plan with equal probabilities, guaranteeing $1/2$. Against an opponent who chooses each target equally often, each complete constant plan earns $1/2$ and each nonconstant plan earns zero, so no mixed strategy can guarantee more. The separation comes from correlation between a sampled plan's actions; fresh independent randomizations cannot retain it after the initial action is forgotten.

This is an absent-minded decision problem: the two decision nodes belong to the same information set. A behavioural strategy must therefore use the same probability at both visits. Put $p=\mathrm{Pr}(D)$ and $1-p=\mathrm{Pr}(R)$; fresh randomization occurs independently at each visit.

The terminal probabilities are $1-p$ for stopping immediately, $p(1-p)$ for going down once and then right, and $p^2$ for going down twice. Thus the expected payoff isThis expression gives the unique global maximum on $[0,1]$:

The independence of repeated randomizations is essential. A mixed strategy choosing once between the two deterministic plans “always $R$” and “always $D$” gives at most $1$, since their payoffs are $0$ and $1$. It cannot arrange “$D$ first, then $R$” as a deterministic plan respecting this information set. The failure of perfect recall is exactly why mixed strategies and behavioural strategies need not have the same attainable outcome distributions here.