From 9be0d4b4b08d5d6e8d8e102d9a2158e73b0378f7 Mon Sep 17 00:00:00 2001
From: Paul <Paul@Pauls-MacBook-Air.local>
Date: Mon, 11 May 2015 20:41:34 -0400
Subject: Minor updates

---
 final/main.tex | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

(limited to 'final/main.tex')

diff --git a/final/main.tex b/final/main.tex
index 3816d4a..066fabe 100644
--- a/final/main.tex
+++ b/final/main.tex
@@ -113,7 +113,7 @@ Problem~\ref{eq:opt}:
 which expresses that the ex-ante allocation probability is upper-bounded by
 $\hat{x}$.
 
-We use the notation from \cite{alaei} where $R(\hat{x})$ denotes the revenue
+We use the notation from \cite{alaei}, where $R(\hat{x})$ denotes the revenue
 obtained by the optimal mechanism solution to Problem~\ref{eq:opt} with ex-ante
 allocation constraint $\hat{x}$ given by \eqref{eq:ea}. The main question
 underlying this work can then be formulated as:
@@ -124,8 +124,8 @@ $\hat{x}$ and whose revenue is a constant approximation to $R(\hat{x})$?}
 
 \subsection{Examples and Motivations}
 
-\paragraph{}Note that when $\hat{x} = (1,\ldots,1)$, \eqref{eq:ea} does not further
-constrain Problem~\ref{eq:opt}, i.e. the constraint is trivially satisfied.  In
+\paragraph{}Note that when $\hat{x} = (1,\ldots,1)$, the ex-ante allocation constraint \eqref{eq:ea} does not further
+constrain the optimization problem given by Equation \ref{eq:opt}, i.e. the constraint is trivially satisfied.  In
 this case, as discussed above, the mechanism described in \cite{babaioff}
 provides a 6-approximation to $R((1,...,1))$.
 
-- 
cgit v1.2.3-70-g09d2

