Mixed and behavioural security levels without perfect recall ($v_{\mathrm{behavioural}}=1/4,v_{\mathrm{mixed}}=1/2$)

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.