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@@ -15,16 +15,16 @@ In our setup, experiments cannot be manipulated and hence measurements are consi is a cost $c_i$ associated with experimenting on subject $i$ which varies from subject to subject. This may be viewed as the cost subject $i$ incurs when tested, and hence $i$ needs to be reimbursed; or, it might be viewed as the incentive for $i$ -to participate in the experiment; or, it might be the inherent value of the data. When subjects are strategic, they may have an incentive to mis-report their cost. This economic aspect has always been inherent in experimental design: experimenters often work within strict budgets and design creative incentives. However, we are not aware of principled study of this setting from a strategic point of view. +to participate in the experiment; or, it might be the inherent value of the data. This economic aspect has always been inherent in experimental design: experimenters often work within strict budgets and design creative incentives. However, we are not aware of principled study of this setting from a strategic point of view. When subjects are strategic, they may have an incentive to mis-report their cost. Our contributions are as follows. \begin{itemize} \item -We formulate the problem of experimental design subject to a given budget, in presence of strategic agents who specify their costs. In particular, we focus on linear regression. This is naturally viewed as a budget feasible mechanism design problem with a sophisticated objective function that is related to the covariance of the $x_i$'s. In particular we formulate the {\em Experimental Design Problem} (\EDP) as follows: the experimenter \E\ wishes to maximize +We formulate the problem of experimental design subject to a given budget, in presence of strategic agents who specify their costs. In particular, we focus on linear regression. This is naturally viewed as a budget feasible mechanism design problem. We show that the objective function is sophisticated and related to the covariance of the $x_i$'s. In particular we formulate the {\em Experimental Design Problem} (\EDP) as follows: the experimenter \E\ wishes to find set $S$ of subjects to maximize \begin{align}V(S) = \log\det(I_d+\sum_{i\in S}x_i\T{x_i}) \label{obj}\end{align} -subject to the budget constraint $\sum_{i\in S}c_i\leq B$, where $B$ is \E's budget. %, and other {\em strategic constraints} we don't list here. - The objective function, which is the key, is motivated from the so-called $D$-objective criterion; in particular, it captures the reduction in the entropy of $\beta$ when the latter is learned through regression methods. +with a budget constraint $\sum_{i\in S}c_i\leq B$, where $B$ is \E's budget. %, and other {\em strategic constraints} we don't list here. + The objective function, which is the key, is motivated from the so-called $D$-objective criterion; in particular, it captures the reduction in the entropy of $\beta$ when the latter is learned through linear regression methods. \item The above objective is submodular. |