Added abstract

1 files changed, 8 insertions, 4 deletions

diff --git a/final/main.tex b/final/main.tex
index bf8e6c8..5508b59 100644
--- a/final/main.tex
+++ b/final/main.tex
@@ -60,6 +60,10 @@
 \blfootnote{We are grateful to Jason Hartline who provided the original idea
 for this project.}
 
+\begin{abstract}
+In this paper, given the problem of selling $m$ heterogeneous items, with ex-ante allocation constraint $\hat{x}$, to a single buyer with additive utility, having type drawn from the distribution $F$, we focus on finding a simple mechanism obeying the allocation constraint whose revenue is a constant approximation to the revenue of the optimal mechanism with this constraint. In particular, following a suggestion by J.D. Hartline, we define a mechanism based on a two-part tariff approach (in which the price charged to the buyer consists of an entrance fee plus posted prices for the items) and then draw a connection with a technique from \citep{yao}, in which we relate this mechanism to a $p$-exclusive mechanism. This mechanism can be viewed as a generalization of the work of \citep{babaioff}, where we have introduced allocation constraints.
+\end{abstract}
+
 \section{Introduction}
 \label{sec:intro}
 
@@ -228,7 +232,7 @@ and 0 otherwise.
 
 We note that this mechanism is exactly the $\beta$-bundle mechanism from
 \citep{yao}, even though the interpretation as a two-part tariff is
-not explicit in this paper.
+not explicit in his paper.
 
 For a choice $\hat{v} = (\hat{v}_0,\hat{v}_1,\dots,\hat{v}_m)$ of the entrance
 fee and the posted prices, let us denote by $x(t, \hat{v})$ and $p(t, \hat{v})$
@@ -277,7 +281,7 @@ price. Then we have:
     $\TPRev(F)$ optimizes (in particular) over all $\hat{v}$, with $\hat{v}_0
     = 0$, we have that $\TPRev(F)\geq \SRev(F)$.
 
-    When $\hat{v}_1=\dots=\hat{v}_m = 0$, the two part tariff mechanism then
+    When $\hat{v}_1=\dots=\hat{v}_m = 0$, the two-part tariff mechanism then
     simply sells all the items at price $\hat{v}_0$ whenever $\sum_{i=1}^m
     t_i\geq \hat{v}_0$, \emph{i.e} is a grand bundle pricing mechanism. For the
     optimal choice of $\hat{v}_0$, the revenue reaches $\BRev(F)$. As in the
@@ -288,13 +292,13 @@ price. Then we have:
 
 \subsection{$p$-exclusivity}
 
-As noted by \citep{yao}, the above mechanism has the additional property of being
+As noted by \citep{yao}, the above mechanism $\M$ has the additional property of being
 $p$-exclusive, where $p$-exclusivity is defined as follows: for a vector $p = (p_1,\dots,p_m)$
 a mechanism is said to be $p$-exclusive if $x_i = 0$ whenever $p_i > t_i$. This
 is essentially saying that there is a reserve price for each item.
 
 The notion of $p$-exclusivity introduced\footnote{\citep{yao} actually uses the notation $\beta$-exclusive for the same thing, but we thought that $p$ was a more natural choice.} by \citep{yao} was crucial in his reduction
-from the $k$-item n-buyer setting to the $k$-item single buyer setting. $p$-exclusivity
+from the $k$-item $m$-buyer setting to the $k$-item single buyer setting. $p$-exclusivity
 can easily be enforced in the optimization we formulated in Section~\ref{sec:intro},
 by adding the following non-linear constraints:
 \begin{displaymath}