From 55299b4183359a879d047544b659931786095b0f Mon Sep 17 00:00:00 2001
From: Paul <Paul@Pauls-MacBook-Air.local>
Date: Mon, 11 May 2015 21:28:34 -0400
Subject: Added natbib package and other changes

---
 final/main.tex | 33 ++++++++++++++++++---------------
 1 file changed, 18 insertions(+), 15 deletions(-)

(limited to 'final/main.tex')

diff --git a/final/main.tex b/final/main.tex
index 066fabe..1a101f5 100644
--- a/final/main.tex
+++ b/final/main.tex
@@ -3,8 +3,11 @@
 \usepackage[english]{babel}
 \usepackage{paralist} 
 \usepackage[utf8x]{inputenc}
-\usepackage[pagebackref=true,breaklinks=true,colorlinks=true]{hyperref}
+\usepackage[pagebackref=true,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*}%
@@ -58,7 +61,7 @@ for this project.}
 
 \subsection{Setup of the Problem}
 
-The exposition in this section draws from \cite{hartlinebayes}. We consider the
+The exposition in this section draws from \citep{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
 $\R_+^m$. We use the notation $[m]$ for $\{1,...,m\}$. For an allocation $x = (x_1,\ldots,x_m)$ where $x_i$, $i\in[m]$,
@@ -69,9 +72,9 @@ utility of an agent with type $t=(t_1,\ldots, t_m)$ is defined by:
 \end{displaymath}
 
 By the taxation principle, a mechanism for this setting can be described by
-a pair $x(\cdot), p(\cdot)$ where the allocation rule $x$ and the payment rule
+a pair $\left(x(\cdot), p(\cdot)\right)$ where the allocation rule $x$ and the payment rule
 $p$ are indexed by the agent's type. For an agent with type $t$, $\big(x(t),
-p(t)\big)$ is her preferred menu option.
+p(t)\big)$ is her preferred menu option (\citep{hartline}, Section 3.4).
 
 The revenue-optimal mechanism for this single-agent problem can be formally
 obtained as the solution to the following optimization problem:
@@ -86,17 +89,17 @@ obtained as the solution to the following optimization problem:
 \end{split}
 \end{equation}
 where the first constraint expresses incentive compatibility and the second
-constraint expresses individual rationality. As noted in \cite{hartlinebayes},
+constraint expresses individual rationality. As noted in \citep{hartlinebayes},
 this formulation can easily be extended to support additional constraints which
 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
+If the distribution $F$ is correlated, a result from \citep{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 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
+\citep{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.
@@ -113,7 +116,7 @@ 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 \citep{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:
@@ -126,7 +129,7 @@ $\hat{x}$ and whose revenue is a constant approximation to $R(\hat{x})$?}
 
 \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}
+this case, as discussed above, the mechanism described in \citep{babaioff}
 provides a 6-approximation to $R((1,...,1))$.
 
 \paragraph{Single-item case.} 
@@ -134,7 +137,7 @@ To make things more concrete, let us first look at the single-item case which
 is well understood. In this setting, $\hat{x}$ is a real number and the
 function $R(\hat{x})$ defined above is simply the revenue curve and turns out
 to be a very useful object to understand the revenue maximizing multiple-agent
-single-item auction (see \cite{hartline}, Chapter 3).
+single-item auction (see \citep{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
@@ -145,7 +148,7 @@ $p_{\hat{x}} = F^{-1}(1-\hat{x})$.
 \paragraph{Multiple-item case.} The multiple-agent multiple-item setting is not
 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
+For example, it is noted in \citep{alaei} that $R(\hat{x})$ is a concave
 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
@@ -166,7 +169,7 @@ $\gamma$ is a constant which is at least $\frac{1}{2}$.
 Given the simplicity of posted-price mechanisms and the fact that they are
 optimal in the single-agent single-item setting (for a regular distribution
 $F$), it would be desirable to obtain a mechanism as close as possible to posted-price.
-Unfortunately, Hart and Nisan \cite{hart-nisan} showed that even in the
+Unfortunately, Hart and Nisan \citep{hart-nisan} showed that even in the
 unconstrained, regular case, no posted-price mechanism for the single-agent
 problem has an approximation ratio better than $\Omega(\log n)$.
 
@@ -174,7 +177,7 @@ Instead, we consider the following candidate mechanism suggested by Jason
 Hartline: the buyer is charged a price $p_0$ to participate in the mechanism,
 and then is offered each item separately with posted prices $p_1,\ldots, p_m$.
 This is essentially the concept of a two-part tariff, as discussed in
-\cite{armstrong}.
+\citep{armstrong}.
 
 Note that as written above, the candidate mechanism is not individually
 rational because the agent gets charged $p_0$ regardless of her type. To
@@ -208,7 +211,7 @@ where we use the notation $\I[\text{expression}]$ to denote the indicator
 variable for the expression, which takes on value 1 when the expression is true and 0 otherwise.
 
 We note that this mechanism is exactly the $\beta$-bundle mechanism from
-\cite{yao}, even though the interpretation in terms of entrance fee plus posted
+\citep{yao}, even though the interpretation in terms of entrance fee plus posted
 prices is not explicit in this paper.
 
 \subsection{$p$-exclusivity}
@@ -236,6 +239,6 @@ Lemma.
 \end{lemma}
 
 
-\bibliographystyle{abbrv}
+\bibliographystyle{abbr}
 \bibliography{main}
 \end{document}
-- 
cgit v1.2.3-70-g09d2

