Mixture Selection, Mechanism Design, and Signaling

We pose and study a fundamental algorithmic problem which we term mixture selection, arising as a building block in a number of game-theoretic applications: Given a function g from the n-dimensional hypercube to the bounded interval [-1, 1], and an n × m matrix A with bounded entries, maximize g(Ax) over x in the m-dimensional simplex. This problem arises naturally when one seeks to design a lottery over items for sale in an auction, or craft the posterior beliefs for agents in a Bayesian game through the provision of information (a.k.a. Signaling). We present an approximation algorithm for this problem when g simultaneously satisfies two "smoothness" properties: Lipschitz continuity with respect to the L∞ norm, and noise stability. The latter notion, which we define and cater to our setting, controls the degree to which low-probability -- and possibly correlated -- errors in the inputs of g can impact its output. The approximation guarantee of our algorithm degrades gracefully as a function of the Lipschitz continuity and noise stability of g. In particular, when g is both O(1)-Lipschitz continuous and O(1)-stable, we obtain an (additive) polynomial-time approximation scheme (PTAS) for mixture selection. We also show that neither assumption suffices by itself for an additive PTAS, and both assumptions together do not suffice for an additive fully polynomial-time approximation scheme (FPTAS). We apply our algorithm for mixture selection to a number of different game-theoretic applications, focusing on problems from mechanism design and optimal signaling. In particular, we make progress on a number of open problems suggested in prior work by easily reducing them to mixture selection: we resolve an important special case of the small-menu lottery design problem posed by Dughmi, Han, and Nisan, we resolve the problem of revenue-maximizing signaling in Bayesian second-price auctions posed by Emek et al. And Miltersen and Sheffet, we design a quasipolynomial-time approximation scheme for the optimal signaling problem in normal form games suggested by Dughmi, and we design an approximation algorithm for the optimal signaling problem in the voting model of Alonso and Cámara.