Income transperency goes full optimality argument

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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.
+