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| -rw-r--r-- | problem.tex | 9 |
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@@ -24,13 +24,7 @@ Our contributions are as follows. 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. 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} (\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} 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. -<<<<<<< HEAD The objective function, which is the key, is obtained by optimizing the information gain in $\beta$ when it is learned through linear regression methods, and is the so-called $D$-objective criterion in the literature. - -======= - The objective function, which is the key, is motivated from the so-called $D$-optimality criterion; in particular, it captures the reduction in the entropy of $\beta$ when the latter is learned through linear regression methods. - ->>>>>>> c29302b25adf190f98019eb8ce5f79b10b66d54d \item The above objective is submodular. There are several recent results in budget feasible diff --git a/problem.tex b/problem.tex index a0a4419..8b1cb2b 100644 --- a/problem.tex +++ b/problem.tex @@ -108,19 +108,10 @@ c_{-i})$ implies $i\in f(c_i', c_{-i})$, and (b) %\end{enumerate} \end{lemma} \fussy -<<<<<<< HEAD Myerson's Theorem % is particularly useful when designing a budget feasible truthful mechanism, as it allows us to focus on designing a monotone allocation function. Then, the mechanism will be truthful as long as we give each agent her threshold payment---the caveat being that the latter need to sum to a value below $B$. -======= -Myerson's Theorem is particularly useful when designing a budget feasible truthful -mechanism. One can focus on designing a monotone allocation function, and the -resulting mechanism will be truthful as long as each agent is given her -threshold payment---the caveat being that the latter need to sum to a value -below $B$. ->>>>>>> c29302b25adf190f98019eb8ce5f79b10b66d54d - \subsection{Budget Feasible Experimental Design} We approach the problem of optimal experimental design from the |