Some revisions

1 files changed, 11 insertions, 12 deletions

diff --git a/final/main.tex b/final/main.tex
index a79a0bb..4eccb83 100644
--- a/final/main.tex
+++ b/final/main.tex
@@ -124,10 +124,11 @@ given by Equation \ref{eq:opt}, i.e. the constraint is trivially satisfied.  In
 this case, as discussed above, the mechanism described in \citep{babaioff}
 provides a 6-approximation to $R((1,...,1))$.
 
-We use the notation from \citep{alaei}, where $R(\hat{x})$ denotes the revenue
+Following \citep{hartline}, we use the notation $R(x)$ to denote the revenue obtained by the optimal mechanism as a function of equality constraint $x$. 
+We slightly modify the notation from \citep{alaei} by replacing his $R$ with $\hat{R}$, and let $\hat{R}(\hat{x})$ denote 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}. When we want to make
-the type distribution $F$ explicit we will write $R(\hat{x}, F)$. Finally, when
+the type distribution $F$ explicit we will write $\hat{R}(\hat{x}, F)$. Finally, when
 the ex-ante allocation constraint \eqref{eq:ea} is not present, we use $\Rev(F)
 = R((1,\dots, 1), F)$ to denote the revenue of the revenue optimal IR-IC
 mechanism.
@@ -135,14 +136,12 @@ mechanism.
 The main question
 underlying this work can then be formulated as:
 
-\paragraph{Problem.} \emph{Given an ex-ante allocation constraint $\hat{x}$, is
+\paragraph{Question.} \emph{Given an ex-ante allocation constraint $\hat{x}$, is
 there a simple mechanism whose ex-ante allocation rule is upper-bounded by
-$\hat{x}$ and whose revenue is a constant approximation to $R(\hat{x})$?}
+$\hat{x}$ and whose revenue is a constant approximation to $\hat{R}(\hat{x})$?}
 
 \subsection{Examples and Motivations}
 
-TODO: this section is mostly wrong, to be discussed.
-
 \paragraph{Single-item case.} 
 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
@@ -236,7 +235,7 @@ two-part tariff mechanism for an optimal choice of $\hat{v}$:
 \begin{displaymath}
     \TPRev(F) = \sup_{\hat{v}\in\R_+^{m+1}} \E_{t\sim F}[p(t, \hat{v})]
 \end{displaymath}
-and by $\TPRev(\hat{x}, F)$ the best revenue which can be obtained by the
+and by $\TPRev(\hat{x}, F)$ the best revenue which can be obtained by the
 two-part tariff mechanism with ex-ante allocation constraint $\hat{x}$:
 \begin{displaymath}
     \begin{split}
@@ -245,7 +244,7 @@ two-part tariff mechanism with ex-ante allocation constraint $\hat{x}$:
         i\in[m]\\
 \end{split}
 \end{displaymath}
-The problem we introduced in Section~\ref{sec:intro} can then be formulated for
+The question we introduced in Section~\ref{sec:intro} can then be formulated for
 the two-part tariff mechanism: \emph{is $\TPRev(\hat{x}, F)$ a constant
 approximation to $\Rev(\hat{x}, F)$?}
 
@@ -253,9 +252,9 @@ The following simple Lemma shows that at least in the unconstrained case, the
 answer to the previous question is positive. In fact, the proof shows that the
 two-part tariff mechanism is rich enough to simulate both bundle pricing and
 separate posted pricing. Using the notation from \citep{babaioff}, let us write
-$\BRev(F)$ the revenue obtained by the optimal grand bundle pricing and by
-$\SRev(F)$ the revenue obtained by selling each item at its monopoly reserve
-price, then we have:
+$\BRev(F)$ for the revenue obtained by the optimal grand bundle pricing and 
+$\SRev(F)$ for the revenue obtained by selling each item at its monopoly reserve
+price. Then we have:
 
 \begin{lemma} For any product distribution $F$,
     \begin{displaymath}
@@ -264,7 +263,7 @@ price, then we have:
 \end{lemma}
 
 \begin{proof}
-    The second inequality is the main result from \citep{babaioff}, we will
+    The second inequality is the main result from \citep{babaioff}; we will
     thus only prove the first inequality. 
 
     In the specific case where $\hat{v} = (0,\hat{v}_1,\dots,\hat{v}_m)$,