Mechanism Design for Stochastic Optimization Problems

We study the problem of welfare maximization in a novel setting motivated by the standard stochastic two-stage optimization with recourse model. We identify and address algorithmic and game-theoretic challenges that arise from this framework. In contrast, prior work in algorithmic mechanism design has focused almost exclusively on optimization problems without uncertainty. We make two kinds of contributions.

First, we introduce a family of mechanisms that induce truth-telling in general two-stage stochastic settings. These mechanisms are not simple extensions of VCG mechanisms, as the latter do not readily address incentive issues in multi-stage settings. Our mechanisms implement the welfare maximizer in sequential ex post equilibrium for risk-neutral agents. We provide formal evidence that this is the strongest implementation one can expect.

Next, we investigate algorithmic issues by studying a novel combinatorial optimization problem called the Coverage Cost problem, which includes the well-studied Fixed-Tree Multicast problem as a special case. We note that even simple instances of the stochastic variant of this problem are #P-Hard. We propose an algorithm that approximates optimal welfare with high probability, using a combination of sampling and supermodular set function maximization|the techniques may be of independent interest. To the best of our knowledge, our work is the first to address both game-theoretic and algorithmic challenges of mechanism design in multi-stage settings with data uncertainty.