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diff --git a/problem.tex b/problem.tex index 03ce49b..0ba9db9 100644 --- a/problem.tex +++ b/problem.tex @@ -34,7 +34,21 @@ Value function \eqref{dcrit} has several appealing properties. To begin with, it \subsection{Budget Feasible Mechanism Design} -In this paper, we approach the problem of optimal experimental design from the perspective of \emph{a budget feasible reverse auction} \cite{singer-mechanisms}. In particular, we assume that each experiment $i\in \mathcal{N}$ is associated with a cost $c_i$, that the experimenter needs to pay in order to conduct the experiment. The experimenter has a budget $B\in\reals_+$. In the \emph{full information case}, the costs are common knowledge; optimal design in this context amounts to selecting a set $S$ maximizing the value $V(S)$ subject to the constraint $\sum_{i\in S} c_i\leq B$. +In this paper, we approach the problem of optimal experimental design from the +perspective of \emph{a budget feasible reverse auction} +\cite{singer-mechanisms}. In particular, we assume that each experiment $i\in +\mathcal{N}$ is associated with a cost $c_i$, that the experimenter needs to +pay in order to conduct the experiment. The experimenter has a budget +$B\in\reals_+$. In the \emph{full information case}, the costs are common +knowledge; optimal design in this context amounts to selecting a set $S$ +maximizing the value $V(S)$ subject to the constraint $\sum_{i\in S} c_i\leq +B$. We write: +\begin{equation}\label{eq:non-strategic} + OPT(V,\mathcal{N}, B) = \max_{S\subseteq\mathcal{N}} \left\{V(S) \mid + \sum_{i\in S}\leq B\right\} +\end{equation} +the best value we can reach under the budget constraint $B$ when selecting +experiments from the set $\mathcal{N}$. As in \cite{singer-mechanisms,chen}, we focus in a \emph{strategic scenario}: experiment $i$ corresponds to a \emph{strategic agent}, whose cost $c_i$ is @@ -55,7 +69,7 @@ returns a vector of payments $[p_i]_{i\in \mathcal{N}}$. As in ($i\notin S$ implies $p_i=0$), individually rational ($p_i\geq c_i$) and have no positive transfers $p_i\geq 0$. -Furthermore, we want to design a mechanism which is: +Moreover, we want to design a mechanism which has the following properties: \begin{itemize} \item \emph{Truthful.} An agent has no incentive to report a cost $c_i'$ different from his true cost $c_i$. Formally, let us write $(c_i', c_{-i})$ @@ -93,16 +107,20 @@ a characterization of truthful mechanisms. \begin{theorem}[Myerson \cite{myerson}]\label{thm:myerson} A normalized mechanism $\mathcal{M} = (f,p)$ for a single parameter auction is truthful iff: -\begin{itemize} +\begin{enumerate} \item $f$ is \emph{monotone}: for any agent $i$ and $c_i' \leq c_i$, for any fixed costs $c_{-i}$ of agents in $\mathcal{N}\setminus\{i\}$, $i\in f(c_i, c_{-i})\implies i\in f(c_i', c_{-i})$. \item agents are paid their \emph{threshold payments}: agent $i$ receives $\inf\{c_i: i\in f(c_i, c_{-i})\}$. -\end{itemize} +\end{enumerate} \end{theorem} -\stratis{Explain why this is important and what it implies about the things we need to prove. Also, don't overuse bullets.} +This theorem is particularly useful when designing a truthful mechanism: we +can focus on designing a monotone allocation function. Then the mechanism will +be truthful as long as we give each agent her threshold payment. Finally, it +suffices to prove that the sum of threshold payments does not exceed the budget +to ensure budget feasibility. \begin{comment} \subsection{Experimental Design} |