From 8dad587f8867494ad0e24a2033145893845e4ad6 Mon Sep 17 00:00:00 2001
From: Paul <Paul@Pauls-MacBook-Air.local>
Date: Tue, 12 May 2015 15:17:28 -0400
Subject: Small changes

---
 final/main.tex | 14 +++++++-------
 1 file changed, 7 insertions(+), 7 deletions(-)

(limited to 'final/main.tex')

diff --git a/final/main.tex b/final/main.tex
index 3345ee6..bf8e6c8 100644
--- a/final/main.tex
+++ b/final/main.tex
@@ -3,12 +3,11 @@
 \usepackage[english]{babel}
 \usepackage{paralist} 
 \usepackage[utf8x]{inputenc}
-\usepackage[pagebackref=true,breaklinks=true,colorlinks=true,citecolor=blue]{hyperref}
+\usepackage[pagebackref=false,breaklinks=true,colorlinks=true,citecolor=blue]{hyperref}
 \usepackage[capitalize, noabbrev]{cleveref}
 \usepackage[square,sort]{natbib}
 
 
-
 % these are compressed lists to help fit into a 1 page limit
 \newenvironment{enumerate*}%
   {\vspace{-2ex} \begin{enumerate} %
@@ -43,6 +42,7 @@
 
 \newcommand{\inprod}[1]{\left\langle #1 \right\rangle}
 \newcommand{\R}{\mathbb{R}}
+\newcommand{\M}{\mathfrak{M}}
 \newcommand{\eqdef}{\mathbin{\stackrel{\rm def}{=}}}
 \newcommand{\llbracket}{[\![}
 
@@ -201,7 +201,7 @@ where we used the notation $(x)^+\eqdef\max(x, 0)$, so that the
 above inequality could also be written as $$\sum_{i: t_i\geq \hat{v}_i} (t_i-
 \hat{v}_i)\geq \hat{v}_0.$$
 
-We can now formally describe the candidate mechanism.
+We can now formally describe the candidate mechanism $\M$.
 
 \begin{center}
 \fbox{
@@ -248,8 +248,8 @@ two-part tariff mechanism with ex-ante allocation constraint $\hat{x}$:
 \end{split}
 \end{displaymath}
 The question we introduced in Section~\ref{sec:intro} can then be formulated for
-the two-part tariff mechanism: \emph{is $\TPRev(\hat{x}, F)$ a constant
-approximation to $\Rev(\hat{x}, F)$?}
+the two-part tariff mechanism: $$\text{\emph{is $\TPRev(\hat{x}, F)$ a constant
+approximation to $\Rev(\hat{x}, F)$?}}$$
 
 The following simple Lemma shows that at least in the unconstrained case, the
 answer to the previous question is positive. In fact, the proof shows that the
@@ -288,12 +288,12 @@ price. Then we have:
 
 \subsection{$p$-exclusivity}
 
-As noted by Yao, the above mechanism has the additional property of being
+As noted by \citep{yao}, the above mechanism has the additional property of being
 $p$-exclusive, where $p$-exclusivity 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 by Yao was crucial in his reduction
+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
 from the $k$-item n-buyer setting to the $k$-item single buyer setting. $p$-exclusivity
 can easily be enforced in the optimization we formulated in Section~\ref{sec:intro},
 by adding the following non-linear constraints:
-- 
cgit v1.2.3-70-g09d2

