From daf111ce8d9703bfe9599a93515b0b7a473d6757 Mon Sep 17 00:00:00 2001
From: Jeff Burdges <burdges@gnunet.org>
Date: Wed, 19 Sep 2018 18:23:17 -0400
Subject: Try to fix expectations

---
 taler-fc19/paper.tex | 14 ++++++--------
 1 file changed, 6 insertions(+), 8 deletions(-)

diff --git a/taler-fc19/paper.tex b/taler-fc19/paper.tex
index c76b15d..2b9f1bd 100644
--- a/taler-fc19/paper.tex
+++ b/taler-fc19/paper.tex
@@ -1293,8 +1293,8 @@ Our instantiation satisfies {weak income transparency}.
   where the expectation runs over
   any probability space used by the adversary and challenger.
 
-  We shall optimiz our adversary in ways that maximize $p \over b + p$.
-  %TODO: Explain
+  We shall optimize our adversary in ways that maximize $p \over b + p$. 
+  %TODO: Explain.   % We cannot actually produce this optimize adversary ourselves, but its existence suffices to prove the inequality, and restrict our analysis to them.  This is not a reduction
 
   As a reminder, if a refresh operation is initiated using a false commitment
   that is detected by the exchange, then the new coin cannot be obtained, and
@@ -1335,13 +1335,11 @@ Our instantiation satisfies {weak income transparency}.
     b_i &= (\kappa - 1)v
   \end{align*}
   and thus $\kappa p_i = b_i + p_i$.  Now
-  \begin{align*}
-    \Exp{p} &= {1\over|F|} \sum_{R_i \in F} p_i\\
-    \Exp{b} &= {1\over|F|} \sum_{R_i \in F} b_i,
-  \end{align*}
-  so
   \begin{equation*}
-    \Exp{{p \over b + p} \middle| F \neq \emptyset} = {1\over\kappa},
+    \Exp{{p \over b + p} \middle| F \neq \emptyset}
+	= |F| \sum_{R_i\in F} {p_i \over b_i + p_i} 
+    = |F| \sum_{R_i\in F} {p_i \ovver \kappa p_i}
+	= {1\over\kappa},
   \end{equation*}
   which yields the equality (\ref{eq:income-transparency-proof}).
 
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