From 809c56b067c3728037df67f878249ac272d53f14 Mon Sep 17 00:00:00 2001
From: Paul <Paul@Pauls-MacBook-Air.local>
Date: Wed, 13 May 2015 11:48:07 -0400
Subject: Minor changes

---
 final/main.tex | 12 +++++++-----
 1 file changed, 7 insertions(+), 5 deletions(-)

(limited to 'final/main.tex')

diff --git a/final/main.tex b/final/main.tex
index 6da4e59..ab1afaf 100644
--- a/final/main.tex
+++ b/final/main.tex
@@ -52,7 +52,7 @@
 
 \author{Thibaut Horel \and Paul Tylkin}
 \title{The Additive Single Buyer Problem\\ with Ex-Ante Allocation Constraints \\ \vskip0.1in {\sc Economics 2099 Project}}
-\date{Fall 2014}
+\date{}
 \begin{document}
 
 \maketitle
@@ -262,7 +262,7 @@ The question we introduced in Section~\ref{sec:intro} can then be formulated for
 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
+The following Lemma shows that at least in the unconstrained case, the
 answer to the previous question is positive. In fact, the proof shows that the
 two-part tariff mechanism is rich enough to simulate both bundle pricing and
 separate posted pricing. Using the notation from \citep{babaioff}, let us write
@@ -345,7 +345,7 @@ Unfortunately, this approach breaks at step 1. To see why, consider for
 simplicity a discrete type space $T\subset \R_+^m$ and assume that the support
 of $F$ is exactly $T$. Writing $T = T_1\times\dots\times T_m$ and $F
 = F_1\times\dots\times F_m$, consider an ex-ante constraint $\hat{x}$ such that
-for all $i\in[m]$, $0<\hat{x}_i < \min_{t_i\in T_i} f_i(t)$, then this forces the
+for all $i\in[m]$, $0<\hat{x}_i < \min_{t_i\in T_i} f_i(t)$. Then, this forces the
 associated $\beta_i$ to be such that $\beta_i > \max_{t_i\in T_i} t_i$ for all
 $i\in [m]$. But a $\beta$-exclusive mechanism for such a $\beta$ will never
 allocate any item and hence will have a revenue of $0$. Hence the optimal
@@ -368,8 +368,8 @@ The goal of this section is to compare the two ways of constraining the single
 buyer problem that we discussed: namely $p$-exclusivity and ex-ante allocation
 constraints and draw a parallel in how these two notions are used to construct
 $n$-to-1 bidders reductions in \citep{yao} and \citep{alaei} respectively.
-
-
+\subsection{\citep{alaei}'s Approach}
+\subsection{\citep{yao}'s Approach}
 The notion of $p$-exclusivity introduced by \citep{yao} was crucial in his
 reduction from the $m$-item $n$-buyer setting to the $m$-item single buyer
 setting. He describes a mechanism known as \emph{Best-Guess Reduction}, which
@@ -382,6 +382,8 @@ 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.
 
+We will step through his results in greater detail, and also improve the approximation ratio in one of his lemmas, using the result from the published version of \citep{babaioff}.
+
 \bibliographystyle{abbrvnat}
 \bibliography{main}
 \end{document}
-- 
cgit v1.2.3-70-g09d2

