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We obtain a valid game run by terminating a game early of course,
We may consider the adversaries ${\cal A}_n$ obtained by terminating
$\cal A$ after the first $n$ refresh attempts $R_C$ with false planchets.
Also ${\cal A}_n$ inherits optimality from $\cal A$.
We shall now prove $E({p \over b + p} | X \neq \emptyset) = {1\over\kappa}$ for ${\cal A}_n$.
We shall prove $E({p \over b + p} | X \neq \emptyset) = {1\over\kappa}$ (\dag)
with the expectation taken over games with false planchets
in which the adversary plays optimally
in that no strictly simpler game run increases $p \over b + p$.
induction on the length of the game now still produces disjoint groupings
$\mathbb{G}$ of optimal games in which
$\sum_{\mathbb{G}} {p \over b + p} = {1\over\kappa} |\mathbb{G}|$.
We conclude that $E({p \over b + p}) = {1\over\kappa}$, as desired.
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