From 767e7ba86a7e916680faef79f885f1a5ce8c6b2b Mon Sep 17 00:00:00 2001 From: Stratis Ioannidis 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} -- cgit v1.2.3-70-g09d2