From fae05f1565ab895abe78a2daa93ab93b4c3d97d9 Mon Sep 17 00:00:00 2001
From: Paul <Paul@Pauls-MacBook-Air.local>
Date: Mon, 11 May 2015 18:45:27 -0400
Subject: Some updates.

---
 final/main.tex | 33 +++++++++++++++++----------------
 1 file changed, 17 insertions(+), 16 deletions(-)

(limited to 'final/main.tex')

diff --git a/final/main.tex b/final/main.tex
index ac8f87d..198b8ab 100644
--- a/final/main.tex
+++ b/final/main.tex
@@ -44,6 +44,7 @@
 
 \author{Thibaut Horel \and Paul Tylkin}
 \title{Economics 2099 Project}
+\date{Fall 2014}
 \begin{document}
 
 \maketitle
@@ -54,12 +55,12 @@ for this project.}
 \section{Introduction}
 \label{sec:intro}
 
-\subsection{Problem}
+\subsection{Setup of the Problem}
 
 The exposition in this section draws from \cite{hartlinebayes}. 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 $F$ over
-$\R_+^m$. For an allocation $x = (x_1,\ldots,x_m)$ where $x_i$, $i\in[m]$,
+$\R_+^m$. We use the notation $[m]$ for $\{1,...,m\}$. For an allocation $x = (x_1,\ldots,x_m)$ where $x_i$, $i\in[m]$,
 denotes the probability of being allocated item $i$ and payment $p$, the
 utility of an agent with type $t=(t_1,\ldots, t_m)$ is defined by:
 \begin{displaymath}
@@ -86,17 +87,17 @@ obtained as the solution to the following optimization problem:
 where the first constraint expresses incentive compatibility and the second
 constraint expresses individual rationality. As noted in \cite{hartlinebayes},
 this formulation can easily be extended to support additional constraints which
-arise in natural scenarios: budget constraints or matroid constraints for
+arise in natural scenarios: budget constraints or matroid constraints, for
 example.
 
 If the distribution $F$ is correlated, a result from \cite{hart} shows that the
 gap between the revenue of auctions with finite menu size and
 a revenue-maximizing auction can be arbitrarily large. When $F$ is a product
-distribution however, there is a hope to obtain simple auctions with constant
-approximation ratios to the optimal revenue. In fact, a result from
-\cite{babaioff} shows that the maximum of \emph{item pricing} where each item
-is priced separately at its monopoly reserve price and \emph{bundle pricing}
-where the grand bundle of all items is priced and sold as a single item
+distribution, however, there is a possibility of obtaining a simple auction with a constant
+approximation ratio to the optimal revenue. In fact, a result from
+\cite{babaioff} shows that the maximum of \emph{item pricing}, where each item
+is priced separately at its monopoly reserve price, and \emph{bundle pricing},
+where the grand bundle of all items is priced and sold as a single item,
 achieves a 6-approximation to the optimal revenue.
 
 We propose to extend this result by considering a more general version of
@@ -136,7 +137,7 @@ single-item auction (see \cite{hartline}, Chapter 3).
 
 In particular, if the type of the agent (her value) is drawn from a regular
 distribution, the posted price revenue curve is concave and equal to the
-revenue curve. In this case, the optimal mechanism which serves with ex-ante
+revenue curve. In this case, the optimal mechanism which serves the agent with ex-ante
 allocation probability $\hat{x}$ is a simple posted-price mechanism with price
 $p_{\hat{x}} = F^{-1}(1-\hat{x})$.
 
@@ -144,20 +145,20 @@ $p_{\hat{x}} = F^{-1}(1-\hat{x})$.
 fully understood yet, but the revenue function $R(\hat{x})$ defined above still
 seems to play an important role in understanding the revenue optimal auction.
 For example, it is noted in \cite{alaei} that $R(\hat{x})$ is a concave
-function of $\hat{x}$. This allows to define a convex optimization program
-(where the objective function is the sum of the revenue curve for all agents)
+function of $\hat{x}$. This allows us to define a convex optimization program
+(where the objective function is the sum of the revenue curves for all agents)
 whose optimal value is an upper bound on the expected value of the
-multiple-agent revenue optimal mechanism.
+multiple-agent revenue-optimal mechanism.
 
 Furthermore, in the same paper, Alaei shows that under fairly general
 assumptions, given an $\alpha$-approximate mechanism for the single-agent,
-ex-ante allocation constrained problem it is possible to construct in
+ex-ante allocation constrained problem, it is possible to construct in
 polynomial time a mechanism 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 $\frac{1}{2}$.
 
 
-\section{A two-part tariff mechanism}
+\section{A Two-Part Tariff Mechanism}
 
 \subsection{Definition of the Mechanism}
 
@@ -180,8 +181,8 @@ restore individual rationality, we need to have the agent pay only when:
 \begin{displaymath}
     \sum_{i=1}^m (t_i - p_i)^+ \geq p_0
 \end{displaymath}
-where we used the notation $(x)^+\eqdef\max\{x, 0\}$, so that the sum in the
-above inequality could also be written $\sum_{i: t_i\geq p_i} (t_i- p_i)$.
+where we used the notation $(x)^+\eqdef\max\{x, 0\}$, so that the
+above inequality could also be written as $$\sum_{i: t_i\geq p_i} (t_i- p_i)\geq p_0.$$
 
 We can now formally describe the candidate mechanism.
 
-- 
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