Added stuff

1 files changed, 16 insertions, 7 deletions

diff --git a/project2/main.tex b/project2/main.tex
index dc863df..96a1d6b 100644
--- a/project2/main.tex
+++ b/project2/main.tex
@@ -22,6 +22,13 @@
      \setlength{\itemsep}{-1ex} \setlength{\parsep}{0pt}}%
   {\end{description}}
 
+\newcommand\blfootnote[1]{%
+  \begingroup
+  \renewcommand\thefootnote{}\footnote{#1}%
+  \addtocounter{footnote}{-1}%
+  \endgroup
+}
+
 \DeclareMathOperator*{\E}{\mathbb{E}}
 \let\Pr\relax
 \DeclareMathOperator*{\Pr}{\mathbb{P}}
@@ -37,14 +44,15 @@
 
 \author{Thibaut Horel \and Paul Tylkin}
 \title{Economics 2099 Project}
-
 \begin{document}
 
 \maketitle
 
+
+
 \section{Introduction}
 
-We consider the problem of selling $m$ heterogeneous items to a single agent
+\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
 $F$ over $\R_+^m$. 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 $p$,
@@ -99,9 +107,10 @@ $\hat{x}$.
 
 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}.
+  
 
-Let us denote by $R(\hat{x})$ the revenue obtained by the optimal mechanism
-solution to Problem~\ref{eq:opt} with ex-ante allocation constraint
+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}.
 
 \paragraph{Problem.} \emph{Given an ex-ante allocation constraint $\hat{x}$, is
@@ -123,12 +132,12 @@ answer to our problem:
     \item when $\hat{x} = (1,\ldots,1)$, \eqref{eq:ea} does not further
         constrain Problem~\ref{eq:opt}, i.e. the constraint is trivially satisfied. 
         In this case, as discussed above, the
-        mechanism described in \cite{babaioff} provides a 6-approximation.
-    \item when $\hat{x} = (\frac{1}{m},\ldots,\frac{1}{m})$, by summing the
+        mechanism described in \cite{babaioff} provides a 6-approximation to $R((1,...,1))$.
+    \item when $\hat{x} = \left(\frac{1}{m},\ldots,\frac{1}{m}\right)$, by summing the
         constraint \eqref{eq:ea} for all $i\in[m]$, we see that the ex-ante
         allocation constraint implies that in expectation, no more than
         one-item is sold to the agent. This is exactly the unit-demand case for
-        which TODO:cite provides a 2 approximation.
+        which TODO:cite provides a 2 approximation to $R\left( \left(\frac{1}{m},\ldots,\frac{1}{m}\right)\right)$.
 \end{itemize}
 
 \section{Related Work}