From fca9d9fca8141e104b792ce0b27346d83af71b82 Mon Sep 17 00:00:00 2001
From: Stratis Ioannidis <stratis@stratis-Latitude-E6320.(none)>
Date: Mon, 11 Feb 2013 19:29:32 -0800
Subject: small changes

---
 problem.tex | 24 ++++++++++++------------
 1 file changed, 12 insertions(+), 12 deletions(-)

(limited to 'problem.tex')

diff --git a/problem.tex b/problem.tex
index 3bdbb94..347bce7 100644
--- a/problem.tex
+++ b/problem.tex
@@ -138,13 +138,13 @@ We study the strategic case, in wich the costs $c_i$ are {\em not} common knowle
 
 
 
-When the experiment subjects are strategic, the experimental design problem becomes a special case of  a \emph{budget feasible reverse auction}, as introduced by \citeN{singer-mechanisms}. Formally, given a budget $B$ and a value function $V:2^{\mathcal{N}}\to\reals_+$, a \emph{mechanism} $\mathcal{M} = (S,p)$ comprises (a) an \emph{allocation function}
+When the experiment subjects are strategic, the experimental design problem becomes a special case of  a \emph{budget feasible reverse auction}, as introduced by \citeN{singer-mechanisms}. Formally, given a budget $B$ and a value function $V:2^{\mathcal{N}}\to\reals_+$, a \emph{mechanism} $\mathcal{M} = (S,p)$ comprises (a) an \emph{allocation function}\footnote{Note that $S$ would be more aptly termed as a \emph{selection} function, as this is a reverse auction, but we retain the term ``allocation'' to align with the familiar term from standard auctions.}
 $S:\reals_+^n \to 2^\mathcal{N}$ and (b) a \emph{payment function}
 $p:\reals_+^n\to \reals_+^n$. Given the
 vector of costs $c=[c_i]_{i\in\mathcal{N}}$, the allocation function $S$ determines the set in
 $ \mathcal{N}$ of experiments to be purchased, while the payment function
 returns a vector of payments $[p_i(c)]_{i\in \mathcal{N}}$.
- Let $s_i(c) = \id_{i\in S(c)}$ be the binary indicator of  $i\in S(c)$. Mechanism design in budget feasible reverse auctions seeks mechanisms that have the following properties \cite{singer-mechanisms,chen}:
+ Let $s_i(c) = \id_{i\in S(c)}$ be the binary indicator of  $i\in S(c)$. As usual, we seek mechanisms that have the following properties \cite{singer-mechanisms}:
 \begin{itemize}
 \item \emph{Normalization.} Unallocated experiments receive zero payments, \emph{i.e.},
  \begin{align}s_i(c)=0\text{ implies }p_i(c)=0.\label{normal}\end{align}
@@ -162,16 +162,16 @@ returns a vector of payments $[p_i(c)]_{i\in \mathcal{N}}$.
      \begin{align}   \sum_{i\in\mathcal{N}} p_i(c) \leq B.\label{budget}\end{align}
     %\end{displaymath}
 \end{itemize}
-We define the \emph{Strategic} version of \EDP{} as
-\begin{center}
-\textsc{StrategicExperimentalDesignProblem} (SEDP)
-%\begin{subequations}
-\begin{align*}
-\text{Maximize:}&\quad V(S) = \log\det(I_d+\T{X_{S}}X_{S}) \\ %\label{modified} \\
-\text{subject to:}&\quad \mathcal{M}=(S,p)\text{ satisfies }\eqref{normal}-\eqref{budget}   
-\end{align*}
-%\end{subequations}
-\end{center}
+%We define the \emph{Strategic} version of \EDP{} as
+%\begin{center}
+%\textsc{StrategicExperimentalDesignProblem} (SEDP)
+%%\begin{subequations}
+%\begin{align*}
+%\text{Maximize:}&\quad V(S) = \log\det(I_d+\T{X_{S}}X_{S}) \\ %\label{modified} \\
+%\text{subject to:}&\quad \mathcal{M}=(S,p)\text{ satisfies }\eqref{normal}-\eqref{budget}   
+%\end{align*}
+%%\end{subequations}
+%\end{center}
 Given that the full information problem \EDP{} is NP-hard, we further seek mechanisms that have the following two additional properties:
 \begin{itemize}
     \item \emph{Approximation Ratio.} The value of the allocated set  should not
-- 
cgit v1.2.3-70-g09d2