From 0edaad75bb56814a2186fa488b540ede18581f6f Mon Sep 17 00:00:00 2001
From: Jeff Burdges <burdges@gnunet.org>
Date: Wed, 25 Apr 2018 02:07:10 +0200
Subject: sleepy time

---
 games/games.tex | 32 ++++++++++++++++++++++----------
 1 file changed, 22 insertions(+), 10 deletions(-)

(limited to 'games')

diff --git a/games/games.tex b/games/games.tex
index 402f4ad..e5bb2fc 100644
--- a/games/games.tex
+++ b/games/games.tex
@@ -493,13 +493,13 @@ Let \oraSet{Income} stand for access to the oracles
   \item $(C_1, \dots, C_\ell) \leftarrow \mathcal{A}^{\oraSet{Income}}(pkExchange)$
   \item Augment the wallets of all non-corrupted users with their
     transitive closure using the \algo{Link} protocol.
-    Spend all remaining value on coins in wallets of non-corrupted users with \algo{Deposit}.\footnote{If \algo{Deposit} can only be run once per coin, then run a similar alggorithm that ignores this check.} 
+    Spend all remaining value on coins in wallets of non-corrupted users with \algo{Deposit}.\footnote{If \algo{Deposit} can only be run once per coin, then run a similar algorithm that ignores this check.} 
   \item Let $L$ be the sum of unspent value for valid coins in $C_1, \dots\, C_\ell$, after
     accounting for the previous spending step.
   \item Let $w$ be the sum of coins withdrawn by non-corrupted users,
-    $w'$ be the sum of coins withdrawn by corruped users, and $s$ be the value marked as spent
+    $w'$ be the sum of coins withdrawn by corrupted users, and $s$ be the value marked as spent
     by non-corrupted users.
-  \item Return $p \over b + p$ where $b := w - s$ gives the lost coins and $p := L - w'$ gives the adversary's winnings.
+  \item If defined, we return $p \over b + p$ where $b := w - s$ gives the lost coins and $p := L - w'$ gives the adversary's winnings.
     Also, we note our adversary wins the strong income transparency game if $L - w' > 0$.
   \comment{$(L, w, w', s)$  Big stile break so split into two games.  Return ratio.  Two expectations is wrong. }
 \end{enumerate}
@@ -904,17 +904,29 @@ false planchet has only a $1\over\kappa$ chance of succeeding.
 If such a malicious refresh is detected, the coin becomes unspendable,
  and contributes to the lost coins $b := w - s$
  instead of the winnings $p := L - w'$ aka $|X|$.
-We observe $b + p$ gives the value of refreshes attempted with false planchets.
 
-We shall now prove $E({p \over b + p}) = {1\over\kappa}$ (\dag)
+We observe $b + p$ gives the value of refreshes attempted with
+false planchets.  Of course $p = 0$ and $b = 0$ if the adversary never
+uses false planchets, so no $R_C$ occur, leaving $p \over b + p$ undefined.
+
+We shall prove $E({p \over b + p}) = {1\over\kappa}$ (\dag)
+with the expectation taken over games with false planchets.
+We obtain a valid game by terminating a game early, so
+ we work by induction on the length of the game.
+
+% https://math.stackexchange.com/questions/852890/expectation-of-random-variables-ratio
+If we group together the games identical up until some $R_C$ with value $v$, 
+then we find $R_C$ 
+...
+contributes $v\over\kappa$ to $p$ and $\kappa v$ to $b + p$.
+
+
  by grouping games identical up to an $R_C$
  according to the simulator's random choice of $\gamma$. 
-We obtain a valid game by terminating a game early, so
- we construct the desire grouping by induction on the length of the game.
-Of course $b = 0$ if no $R_C$ occur.
+Of course 
 We defined $R_C$ to be an anti-chain in the refresh forest, but
-we emphasize that an adversary who wins an $R_C$ need not gamble again,
-while an an adversary who looses an $R_C$ looses $C$.
+emphasize that an adversary who wins an $R_C$ should not gamble again,
+while an adversary who looses an $R_C$ looses $C$.
 It follows that each $R_C$ contributes ${1\over\kappa}$ towards the total expectation.
 {RETHINK WHAT HAPPENS HERE AND ABOVE}
 We conclude that $E({p \over b + p}) = {1\over\kappa}$, as desired.
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