Fix income transperency computations

We need more work here because 1/2 is maybe not ideal, maybe we want 1/kappa.

1 files changed, 4 insertions, 5 deletions

diff --git a/games/games.tex b/games/games.tex
index ce63274..1b5c7a0 100644
--- a/games/games.tex
+++ b/games/games.tex
@@ -327,13 +327,12 @@ Let \oraSet{Income} stand for access to the oracles \ora{AddClient}, \ora{Withdr
     corrupted and non-corrupted users, respectively.
     Also let $b$ denote the number of coins lost in refresh operations
     with false planchets.  Our adversary wins if both $\ell > w$ and 
-    ${b \over w'} \le 1-{1\over\kappa}$ with $C_1, \dots, C_\ell$
+    ${b \over l-w} \le 1-{1\over\kappa}$ with $C_1, \dots, C_\ell$
      all being distinct valid unspent coins.
 \end{enumerate}
 
-
 Note that we want to show in the end that the probability of winning one Income Transparency game is at most $1/\kappa$.
-This is why $\kappa$ does not appear directly in the proof.
+FALSE: This is why $\kappa$ does not appear directly in the game.
 
 \subsection{Others?}
 Let adversary distinguish between freshly withdrawn coin and coin obtained via refresh protocol.  Why?
@@ -706,9 +705,9 @@ as otherwise the refresh always fails.
 As our $\gamma$ are chosen randomly, any given refresh with one
 false planchet has a $1-{1\over\kappa}$ chance of contributing to
 $b$ instead of $|X|$.  So $E[{b \over f}] = 1-{1\over\kappa}$ where
-$f \le w'$ denotes the number of refreshes attempted with false planchets.
+$l-w \le f \le w'$ denotes the number of refreshes attempted with false planchets.
 It follows that 
- $P[{b \over w'} \ge (1-{1\over\kappa})] = 1/2$.
+ $P[{b \over l-w} \ge 1-{1\over\kappa}] \ge 1/2$.
 \end{proof}
 
 \begin{corollary}