of a disjoint union of events.) This process is essentially a discrete version of “backwards induction.”

The positive prefixes are locally optimal by definition. If a prefix $p$ has no (proper) prefix flattenings that are positive, then $p$ must be globally optimal as well.

Consider the tree shown in Figure \[f:treeex\] for $N = 4$ at ${\theta}= 1$. The strike probabilities ${\mathcal{S}}(p)$ are illustrated in the figure. Since ${\mathcal{S}}(123) =
3/4 \geq 1/4 = {\mathcal{S}}(123^{\circ})$, we have that $123$ is a positive prefix. Similarly, ${\mathcal{S}}(12) = 6/12 \geq 5/12 = {\mathcal{S}}(12^{\circ})$ so $12$ is also a positive prefix. On the other hand, ${\mathcal{S}}(1) = 6/24 < 11/24 = {\mathcal{S}}(1^{\circ})$ so $1$ is a negative prefix. The optimal strike set consists of $12$ (contributing $6/12$), $213$ (contributing $3/4$), $3124$ (contributing $1/1$), and $3214$ (contributing $1/1$), together with all of the prefixes of size $4$ that aren’t already related to one of these, each contributing a strike probability of $0/1$. The optimal probability is the $\oplus$-sum of these contributions, namely $11/24$.

Prefix equivariance and positional strategies {#s:tilt}
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While any game of best choice has an optimal strike strategy, the classical game (where $f(\pi) = 1/N!$ uniformly) is optimized by a [**positional 