Income transperency goes full optimality argument

2 files changed, 41 insertions, 21 deletions

diff --git a/games/games.tex b/games/games.tex
index 72c1793..365013f 100644
--- a/games/games.tex
+++ b/games/games.tex
@@ -901,11 +901,11 @@ We may assume the first case that $\cal A$ provides a false planchet
 outright because we produce a key exchange failure in the second case.
 
 We shall prove $E({p \over b + p} | X \neq \emptyset) \le {1\over\kappa}$ (\dag)
-with the expectation taken over games with false planchets, meaning
- any probability space used by the adversary,
+meaning  any probability space used by the adversary,
  times the uniform distribution on exchange keys $\V{skE}$,
  times the space of $\gamma$ choices by the exchange,
  i.e. the uniform distribution on $\kappa^X$.
+with the expectation taken over games with false planchets
 
 We shall optimizes our adversary in ways that improve its $p \over b + p$.
 We cannot actually produce this optimized adversary ourselves,
@@ -930,30 +930,31 @@ for at most one choice of $\gamma$ in $R_C$,
 
 We defined $R_C$ to be an anti-chain in the refresh forest, but
 emphasize that an adversary who looses an $R_C$ looses $C$ completely,
-while an optimal adversary who wins an $R_C$ should not gamble again.
-There is no way to influence $p$ or $b$ through withdrawals or spends
-by corrupted users of course.  In principle, one could decrease $b$ by
- sharing from a corrupted user to a non-corrupted users,
-but we may assume this does not occur either, again by optimality.
-
+while an optimal adversary who wins an $R_C$ should not gamble again. 
+Indeed, an adversary has no reason to touch its winnings from an $R_C$.
 
+% There is no way to influence $p$ or $b$ through withdrawals or spends
+% by corrupted users of course.  In principle, one could decrease $b$ by
+%  sharing from a corrupted user to a non-corrupted users,
+% but we may assume this does not occur either, again by optimality.
 
-We obtain a valid game run by terminating a game early, or even by
-blocking oracle calls on decedents of a particular coin $C$,
- which we refer to as a strictly simpler game run.
-We shall prove $E({p \over b + p}) = {1\over\kappa}$ (\dag)
-with the expectation taken over games with false planchets
- in which the adversary plays optimally
-  in that no strictly simpler game run increases $p \over b + p$.
-
-For any $R_C$, 
+An adversary may respond adaptively to $\gamma$ of course, but,
+for any $R_C$, 
 there are $\kappa$ game runs identical up through the commitment phase
-of $R_C$ that diverge in simulator's random choice of $\gamma$,
- and blocks $C$ afterwards. 
+of $R_C$ that diverge in simulator's random choice of $\gamma$.
+% and suppress $C$'s decendents by optimality. 
 If $v$ is the value of $R_C$ then one run adds $v$ to $p$, while
- the remaining $\kappa-1$ runs add $v$ to $b$.
+ the remaining $\kappa-1$ runs add $v$ to $b$. 
+
+We define $p_C$ and $b_C$ to be these contributions summed
+ over the $\kappa$ runs, so $\kappa p_C = b_C + p_C$. 
+Now $p = {1\over\kappa |X|} \sum_{C \in X} p_C$
+ and $b = {1\over\kappa |X|} \sum_{C \in X} b_C$
+so $E({p \over b + p} | X \neq \emptyset) = {1\over\kappa}$,
+ which yields the inequality (\dag).
+
+
 
-An adversary may respond adaptively to $\gamma$ of course, but
 induction on the length of the game now still produces disjoint groupings
 $\mathbb{G}$ of optimal games in which
   $\sum_{\mathbb{G}} {p \over b + p} = {1\over\kappa} |\mathbb{G}|$.
diff --git a/games/rm.tex b/games/rm.tex
new file mode 100644
index 0000000..61e1691
--- /dev/null
+++ b/games/rm.tex
@@ -0,0 +1,19 @@
+
+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.
+