From 665c37fb26906eef4adbb94e8bea801e40fa4cb4 Mon Sep 17 00:00:00 2001
From: Thibaut Horel <thibaut.horel@gmail.com>
Date: Tue, 12 May 2015 17:53:33 -0400
Subject: Move footnote on notation up

---
 final/main.tex | 6 ++++--
 1 file changed, 4 insertions(+), 2 deletions(-)

(limited to 'final/main.tex')

diff --git a/final/main.tex b/final/main.tex
index 5ebba6e..adde64f 100644
--- a/final/main.tex
+++ b/final/main.tex
@@ -299,11 +299,13 @@ price. Then we have:
 \subsection{$p$-exclusivity}
 
 As noted by \citep{yao}, the above mechanism $\M$ has the additional property of being
-$p$-exclusive, where $p$-exclusivity is defined as follows: for a vector $p = (p_1,\dots,p_m)$
+$p$-exclusive, where $p$-exclusivity\footnote{\citep{yao} actually uses the
+notation $\beta$-exclusive for the same thing, but use $p$ here to have
+consistent notations.} is defined as follows: for a vector $p = (p_1,\dots,p_m)$
 a mechanism is said to be $p$-exclusive if $x_i = 0$ whenever $p_i > t_i$. This
 is essentially saying that there is a reserve price for each item.
 
-The notion of $p$-exclusivity introduced\footnote{\citep{yao} actually uses the notation $\beta$-exclusive for the same thing, but we thought that $p$ was a more natural choice.} by \citep{yao} was crucial in his reduction
+The notion of $p$-exclusivity introduced by \citep{yao} was crucial in his reduction
 from the $k$-item $m$-buyer setting to the $k$-item single buyer setting. He describes a mechanism known as \emph{Best-Guess Reduction}, which conducts $m$ single-buyer $k$-item auctions, using an IR-IC $p$-exclusive mechanism, for a particular value of $p$ drawn from the joint bid distribution over all buyers conditioned on the bids of all other buyers, and then combines this with the Vickrey second-price auction, showing that this mechanism has revenue that is a constant approximation to the optimal $k$-item, $m$-buyer mechanism. He then defines another mechanism, \emph{Second-Price Bundling}, which is meant to heuristically approximate this combined mechanism, and shows that its revenue is also a constant approximation to the optimal mechanism.
 
 $p$-exclusivity
-- 
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