 Solution

There is no perfect recall. Player 1's later information set joins histories following both of her initial actions, so she has forgotten whether she chose $L$ or $R$. It also hides player 2's intervening action, but failure to observe another player's action is imperfect information, whereas forgetting one's own earlier action is the particular violation of perfect recall. Player 2's two nodes form a single information set because player 2 does not observe the initial choice.

For a behavioural strategy, let $p$ be player 1's initial probability of $L$ and $r$ her later probability of $L$. These are separate information sets, so $p$ and $r$ need not be equal; their random draws are independent. Name player 2's unlabeled branches left and right, and let $q$ be the probability of taking the left branch. The only unit-payoff paths are left–left–left and right–right–right. ConsequentlyBecause this is a zero-sum game, the natural meaning of the claimed best behavioural strategy is its maximum guaranteed expected payoff. Player 2 can choose either branch, so the guarantee at $(p,r)$ isFor nonnegative numbers, their minimum is at most their geometric mean. Also $p(1-p)\le 1/4$ and $r(1-r)\le 1/4$, henceTaking $p=r=1/2$ makes both possible branch-conditioned success probabilities $1/4$, so this upper bound is attained against every $q$. Thus the optimal guaranteed behavioural strategy success probability is $1/4$.

A mixed strategy can instead select the complete plan $LL$ with probability $1/2$ and the complete plan $RR$ with probability $1/2$. The two choices within a sampled plan are correlated. If player 2 chooses left, exactly the sampled plan $LL$ succeeds; if player 2 chooses right, exactly $RR$ succeeds. Therefore the success probability isagainst every opposing mixed strategy. It is optimal among mixed strategies too: player 2's uniform mixture gives success probability $1/2$ to each of $LL,RR$ and zero to $LR,RL$, so no mixture of the four plans can exceed $1/2$ against that opponent. This establishes mixed and behavioural security levels without perfect recall directly.

The guarantee qualification matters: if player 2's branch were known to be left, the deterministic plan $LL$ would succeed with probability one. The number $1/4$ is not an upper bound against every fixed opposing choice separately.