From 342435c8e708ebcb0ec442960f1ebd6a64c9711a Mon Sep 17 00:00:00 2001
From: Paul <Paul@Pauls-MacBook-Air.local>
Date: Mon, 15 Dec 2014 18:36:44 -0500
Subject: Added stuff

---
 project2/main.tex | 12 ++++++++----
 1 file changed, 8 insertions(+), 4 deletions(-)

(limited to 'project2/main.tex')

diff --git a/project2/main.tex b/project2/main.tex
index cfc3831..dc863df 100644
--- a/project2/main.tex
+++ b/project2/main.tex
@@ -55,7 +55,7 @@ the utility of an agent with type $t=(t_1,\ldots, t_m)$ is defined by:
 
 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
-$p$ are indexed by the agent's type. For an agent with type $t$, $\big(x(t),
+$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.
 
 The revenue-optimal mechanism for this single-agent problem can be formally
@@ -94,9 +94,12 @@ Problem~\ref{eq:opt}:
     \label{eq:ea}
     \E_{t\sim F}\big[x_i(t)] \leq \hat{x}_i,\quad\forall i\in[m]
 \end{equation}
-which simply expresses that the ex-ante allocation rule is upper-bounded by
+which expresses the assumption that the ex-ante allocation rule is upper-bounded by
 $\hat{x}$.
 
+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
 \eqref{eq:ea}.
@@ -114,11 +117,12 @@ for the multi-agent problem which is a $\gamma\cdot\alpha$ approximation to the
 revenue-optimal mechanism where $\gamma$ is a constant which is at least
 $\frac{1}{2}$.
 
-It is interesting to consider two specific case for which we already have an
+It is interesting to consider two specific cases for which we already have an
 answer to our problem:
 \begin{itemize}
     \item when $\hat{x} = (1,\ldots,1)$, \eqref{eq:ea} does not further
-        constrain Problem~\ref{eq:opt}. In this case, as discussed above, the
+        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
         constraint \eqref{eq:ea} for all $i\in[m]$, we see that the ex-ante
-- 
cgit v1.2.3-70-g09d2

