5 research outputs found
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 from the -dimensional hypercube to the
bounded interval , and an matrix with bounded entries,
maximize over in the -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
simultaneously satisfies two smoothness properties: Lipschitz continuity with
respect to the norm, and noise stability. The latter notion, which
we define and cater to our setting, controls the degree to which
low-probability errors in the inputs of can impact its output. When is
both -Lipschitz continuous and -stable, we obtain an (additive)
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 FPTAS.
We apply our algorithm to different game-theoretic applications from
mechanism design and optimal signaling. 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 C\^{a}mara