From 461d4189cca0dd5ce9ed4dcc5da783d1c0a32a1f Mon Sep 17 00:00:00 2001
From: Thibaut Horel <thibaut.horel@gmail.com>
Date: Tue, 12 May 2015 17:44:46 -0400
Subject: Fix 1.2

---
 final/main.tex | 27 +++++++++++++++++----------
 1 file changed, 17 insertions(+), 10 deletions(-)

(limited to 'final/main.tex')

diff --git a/final/main.tex b/final/main.tex
index 395f7ce..1ecfbca 100644
--- a/final/main.tex
+++ b/final/main.tex
@@ -147,19 +147,26 @@ $\hat{x}$ and whose revenue is a constant approximation to $\Rev(\hat{x},F)$?}
 
 \paragraph{Single-item case.} 
 To make things more concrete, let us first look at the single-item case, which
-has already been studied extensively and is well-understood. In this setting, $\hat{x}$ is a real number and the
-function $\Rev(\hat{x},F)$ defined above is obtained by maximizing the revenue curve $R(\hat{x})$ subject to the allocation constraint, and turns out
-to be a very useful object to understand the revenue maximizing multiple-agent
-single-item auction (see \citep{hartline}, Chapter 3).
+has already been studied extensively and is well understood. In this setting,
+$\hat{x}$ is a real number and the function $\Rev(\hat{x},F)$ defined above is
+obtained by maximizing the revenue curve $R(x)$ subject to the allocation
+constraint $x\leq \hat{x}$, and turns out to be a very useful object to
+understand the revenue maximizing multiple-agent 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 optimal mechanism which serves the agent with ex-ante
-allocation probability $\hat{x}$ has revenue $\Rev(\hat{x},F)$, given by solving \begin{align*}
-\max_{p} p(1-F(p)) \\
-\text{subject to }& 1 - F(p) \leq \hat{x} 
+distribution, the revenue curve $R(x)$ is equal to the posted price revenue
+curve $P(x) = xF^{-1}(1-x)$ and the optimal mechanism which serves the agent
+with ex-ante allocation probability at most $\hat{x}$ has revenue $\Rev(\hat{x},F)$,
+given by solving
+\begin{align*}
+    \begin{split}
+        \max_{x} & \;xF^{-1}(1-x) \\
+\text{subject to }& \; x \leq \hat{x} 
+    \end{split}
 \end{align*}
-
-which is a concave function.
+which is a convex optimization program since $P(x) = xF^{-1}(1-x)$ is concave
+for regular distributions.
 
 \paragraph{Multiple-item case.} The multiple-agent multiple-item setting is not
 fully understood yet, but the revenue function $\Rev(\hat{x},F)$ defined above still
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