up to myerson

1 files changed, 3 insertions, 3 deletions

diff --git a/intro.tex b/intro.tex
index 0712540..85f4f3f 100644
--- a/intro.tex
+++ b/intro.tex
@@ -21,7 +21,7 @@ In our setting, experiments cannot be manipulated and hence measurements are rel
  \E{} has a total budget of $B$ to conduct all the experiments. 
 There is a cost $c_i$ associated with experimenting on
 subject $i$ which varies from subject to subject.  This cost $c_i$ is determined by the subject $i$: it may be viewed as the  
-cost $i$ incurs when tested and for which she needs to be reimbursed; or, it might be viewed as the incentive for $i$ to participate in the experiment; or, it might be the intrinsic worth of the data to the user. The economic aspect of paying subjects has always been inherent in experimental design: experimenters often work within strict budgets and design creative incentives. Subjects often negotiate better incentives or higher payments. 
+cost $i$ incurs when tested and for which she needs to be reimbursed; or, it might be viewed as the incentive for $i$ to participate in the experiment; or, it might be the intrinsic worth of the data to the subject. The economic aspect of paying subjects has always been inherent in experimental design: experimenters often work within strict budgets and design creative incentives. Subjects often negotiate better incentives or higher payments. 
 However, we are not aware of a principled study of this setting from a strategic point of view, when subjects declare their costs and therefore determine their payment.  Such a setting is increasingly realistic, given the growth of these experiments over the Internet and associated data markets. 
 
 % When subjects are strategic, they may have an incentive to misreport their cost, leading to the need for a sophisticated choice of experiments and payments. Arguably, user incentiviation is of particular pertinence due to the extent of statistical analysis over user data on the Internet. %, which has led to the rise of several different research efforts in studying data markets \cite{...}.
@@ -29,7 +29,7 @@ However, we are not aware of a principled study of this setting from a strategic
 Our contributions are as follows.
 \begin{itemize}
 \item
-We initiate the study of experimental design problem in presence of a budget and strategic subjects. 
+We initiate the study of experimental design in the presence of a budget and strategic subjects. 
 %formulate the problem of experimental design subject to a given budget, in the presence of strategic agents who may lie about their costs. %In particular, we focus on linear regression. This is naturally viewed  as a budget feasible mechanism design problem, in which the objective function %is sophisticated and 
 %is  related to the covariance of the $x_i$'s.
  In particular, we formulate the  {\em  Experimental Design Problem} (\SEDP) as
@@ -41,7 +41,7 @@ subject to a budget constraint $\sum_{i\in S}c_i\leq B$, where $B$ is \E's budge
 \smallskip
 The objective function, which is the key, is formally obtained by optimizing  the information gain in  $\beta$ when the latter is learned  through ridge regression, and is related to  the so-called $D$-optimality criterion~\cite{pukelsheim2006optimal,atkinson2007optimum}. 
 \item
-We present a polynomial time, truthful mechanism for \SEDP{}, yielding a constant factor ($\approx 12.98$) approximation to the optimal value of \eqref{obj}. In contrast to this, we show that no truthful, budget-feasible mechanisms are possible for \SEDP{}  within a factor 2 approximation. 
+We present a polynomial time, $\epsilon$-truthful mechanism for \SEDP{}, yielding a constant factor ($\approx 12.98$) approximation to the optimal value of \eqref{obj}. In contrast to this, we show that no truthful, budget-feasible mechanisms are possible for \SEDP{}  within a factor 2 approximation. 
 
 \smallskip
 We note that the objective \eqref{obj} is submodular. Using this fact, applying previous results on budget feasible mechanism design under general submodular objectives~\cite{singer-mechanisms,chen} would yield either a deterministic, truthful, constant-approximation mechanism that requires exponential time,  or a non-deterministic, (universally) truthful, poly-time mechanism that yields a constant approximation ratio only \emph{in expectation} (\emph{i.e.}, its approximation guarantee for a given instance may in fact be unbounded).