From cd1b1e9058966592eb80971d8e00fee9333e1c31 Mon Sep 17 00:00:00 2001
From: Paul <Paul@Pauls-MacBook-Air.local>
Date: Mon, 15 Dec 2014 19:03:58 -0500
Subject: Added stuff

---
 project2/main.tex | 11 ++++++-----
 1 file changed, 6 insertions(+), 5 deletions(-)

(limited to 'project2/main.tex')

diff --git a/project2/main.tex b/project2/main.tex
index 96a1d6b..8b29b5b 100644
--- a/project2/main.tex
+++ b/project2/main.tex
@@ -50,7 +50,7 @@
 
 
 
-\section{Introduction}
+\section{Introduction to the Problem}
 
 \blfootnote{We are deeply grateful to Jason Hartline who provided the original idea for this project.}We consider the problem of selling $m$ heterogeneous items to a single agent
 with additive utility. The type $t$ of the agent is drawn from a distribution
@@ -105,10 +105,6 @@ Problem~\ref{eq:opt}:
 which expresses the assumption that the ex-ante allocation rule is upper-bounded by
 $\hat{x}$.
 
-To provide some more intuition about this, we assume that the buyer is charged a price $p_0$ to participate in the mechanism, 
-and then is offered a menu of goods with prices $p_1,...,p_m$. The buyer's utility over the goods is additive, as above. However, there is an ex-ante constraint of being allocated a given good $i$, given by $\hat{x}_i$. For each good, if the buyer is allocated the good, which he is with probability $x_i \leq \hat{x}_i$, then he pays $p_i$; otherwise, he pays nothing. This is essentially the concept of a two-part tariff, as discussed in \cite{armstrong}.
-  
-
 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 $\hat{x}$.
 \eqref{eq:ea}.
@@ -117,6 +113,7 @@ solution to Problem~\ref{eq:opt} with ex-ante allocation constraint $\hat{x}$.
 there a simple mechanism whose ex-ante allocation rule is upper-bounded by
 $\hat{x}$ and whose revenue is a constant approximation to $R(\hat{x})$?}
 
+\section{Importance of the Problem and Possible Approach}
 \paragraph{} One reason why one might want to consider this problem is that it
 can be used as a building block for more general mechanisms. Indeed, in
 \cite{alaei}, Alaei shows that under fairly general assumptions, given an
@@ -126,6 +123,10 @@ for the multi-agent problem which is a $\gamma\cdot\alpha$ approximation to the
 revenue-optimal mechanism where $\gamma$ is a constant which is at least
 $\frac{1}{2}$.
 
+
+To provide some more intuition about this, we assume that the buyer is charged a price $p_0$ to participate in the mechanism, 
+and then is offered a menu of goods with prices $p_1,...,p_m$. The buyer's utility over the goods is additive, as above. However, there is an ex-ante constraint of being allocated a given good $i$, given by $\hat{x}_i$. For each good, if the buyer is allocated the good, which he is with probability $x_i \leq \hat{x}_i$, then he pays $p_i$; otherwise, he pays nothing. This is essentially the concept of a two-part tariff, as discussed in \cite{armstrong}.
+
 It is interesting to consider two specific cases for which we already have an
 answer to our problem:
 \begin{itemize}
-- 
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