From 767e7ba86a7e916680faef79f885f1a5ce8c6b2b Mon Sep 17 00:00:00 2001
From: Stratis Ioannidis <stratis@stratis-Latitude-E6320.(none)>
Date: Sat, 6 Jul 2013 22:27:45 -0700
Subject: moved algo to appendix

---
 myerson.tex | 18 ++++++++++--------
 1 file changed, 10 insertions(+), 8 deletions(-)

(limited to 'myerson.tex')

diff --git a/myerson.tex b/myerson.tex
index bcfedc4..8c80cf6 100644
--- a/myerson.tex
+++ b/myerson.tex
@@ -10,20 +10,22 @@ other users:
 We distinguish four cases depending on the value of $s_i(c_i, c_{-i})$ and
 $s_i'(c_i', c_{-i})$.
 
-Since the mechanism is normalized, if $s_i(c_i, c_{-i})= s_i(c_i', c_{-i})=0$,
+\begin{enumerate}
+\item $s_i(c_i, c_{-i})= s_i(c_i', c_{-i})=0$.
+Since the mechanism is normalized 
 we have $p_i(c_i, c_{-i}) = p_i(c_i', c_{-i})= 0$ and \eqref{eq:local-foobar}
 is true.
-
+\item $s_i(c_i', c_{-i}) = s_i(c_i, c_{-i}) = 1$.
 Note that $i$ is paid her threshold payment when allocated, and since this
 payment does not depend on $i$'s reported cost, \eqref{eq:local-foobar} is true
-(and is in fact an equality) when $s_i(c_i', c_{-i}) = s_i(c_i, c_{-i}) = 1$.
-
-If $s_i(c_i', c_{-i}) = 0$ and $s_i(c_i, c_{-i}) = 1$, we have $p_i(c_i',
+(and is in fact an equality).
+\item $s_i(c_i', c_{-i}) = 0$ and $s_i(c_i, c_{-i}) = 1$.
+ We then have $p_i(c_i',
 c_{-i}) = 0$ by normalization and \eqref{eq:local-foobar} follows from
 individual rationality.
-
-Finally, let us assume that $s_i(c_i', c_{-i}) = 1$ and $s_i(c_i, c_{-i}) = 0$.
+\item $s_i(c_i', c_{-i}) = 1$ and $s_i(c_i, c_{-i}) = 0$.
 By $\delta$-decreasingness of $s_i$, $c_i \geq c_i'+\delta$, and $s_i(c_i,
 c_{-i}) = 0$ implies that $i$'s threshold payment is less than $c_i$,
 \emph{i.e.} $p_i(c_i', c_{-i}) \leq c_i$. This last inequality is equivalent to
-\eqref{eq:local-foobar} in this final case.
+\eqref{eq:local-foobar} in this final case. \qed
+\end{enumerate}
-- 
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