From 705f11ce5e6cf6dbd41e8a055743d711b8f8fbdb Mon Sep 17 00:00:00 2001
From: Thibaut Horel <thibaut.horel@gmail.com>
Date: Mon, 5 Nov 2012 16:27:15 +0100
Subject: More fixes, reapply some lost changes overwritten by previous merge

---
 problem.tex | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

(limited to 'problem.tex')

diff --git a/problem.tex b/problem.tex
index f0eeb43..b07cd42 100644
--- a/problem.tex
+++ b/problem.tex
@@ -134,7 +134,7 @@ incentivize her participation in the study. Note that, in this setup, the featur
 Ideally, motivated by the $D$-optimality criterion, we would like to design a mechanism that maximizes or approximates \eqref{dcrit} . Since \eqref{dcrit} may take arbitrarily small negative values, to define a meaningful approximation one would consider  the (equivalent) maximization of $V(S) = \det\T{X_S}X_S$. %, for some strictly increasing, on-to function $f:\reals_+\to\reals_+$. 
 However, the following lower bound implies that such an optimization goal cannot be attained under the constraints of  truthfulness, budget feasibility, and individual rationality.
 \begin{lemma}
-For any $M>1$, there is no $M$-approximate, truthful, budget feasible, individionally rational mechanism for a budget feasible reverse auction with $V(S) =  \det{\T{X_S}X_S}$.
+For any $M>1$, there is no $M$-approximate, truthful, budget feasible, individually rational mechanism for a budget feasible reverse auction with $V(S) =  \det{\T{X_S}X_S}$.
 \end{lemma}
 \begin{proof}
 \input{proof_of_lower_bound1}
@@ -151,7 +151,7 @@ This negative result motivates us to study a problem with a modified objective:
 \end{center} where $I_d\in \reals^{d\times d}$ is the identity matrix.
 One possible interpretation of \eqref{modified} is that, prior to the auction, the experimenter has free access to $d$ experiments whose features form an orthonormal basis in $\reals^d$. However, \eqref{modified} can also be motivated in the context of \emph{Bayesian experimental design} \cite{chaloner1995bayesian}. In short, the objective \eqref{modified} arises naturally when the experimenter retrieves the model $\beta$ through \emph{ridge regression}, rather than the linear regression \eqref{leastsquares} over the observed data; we explore this connection in Section~\ref{sec:bed}. 
 
-Note that maximizing \eqref{modified} is equivalent to maximizing \eqref{dcrit} in the full-information case. In particular, $\det(\T{X_S}X_S)> \det(\T{X_{S'}}X_{S'})$ iff $\det(I_d+\T{X_S}X_S)>\det(I_d+\T{X_{S'}}X_{S'})$. In addition, \eqref{edp}---and the equivalent problem with objective \eqref{dcrit}---are NP-hard; to see this, note that \textsc{Knapsack} reduces to EDP with dimension $d=1$ by mapping the weight of each item $w_i$ to an experiment with $x_i=w_i$. Nevertheless, \eqref{modified} is submodular, monotone and satifies $V(\emptyset) = 0$.
+Note that maximizing \eqref{modified} is equivalent to maximizing \eqref{dcrit} in the full-information case. In particular, $\det(\T{X_S}X_S)> \det(\T{X_{S'}}X_{S'})$ iff $\det(I_d+\T{X_S}X_S)>\det(I_d+\T{X_{S'}}X_{S'})$. In addition, \eqref{edp}---and the equivalent problem with objective \eqref{dcrit}---are NP-hard; to see this, note that \textsc{Knapsack} reduces to EDP with dimension $d=1$ by mapping the weight of each item $w_i$ to an experiment with $x_i=w_i$. Nevertheless, \eqref{modified} is submodular, monotone and satisfies $V(\emptyset) = 0$.
 %, allowing us to use the extensive machinery for the optimization of submodular functions, as well as recent results in the 
 % context of budget feasible auctions \cite{chen,singer-mechanisms}. 
 
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