= Mixed and behavioural security levels without perfect recall
{title2=$v_{\rm behavioural}=1/4,\quad v_{\rm mixed}=1/2$}
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 $\sqrt{p(1-p)r(1-r)}\le1/4$, 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.
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