Bernoulli Factories and Bayesian Mechanism Design



Rad Niazadeh

Monday, February 6th, 2017
4:00pm 122 Gates Hall




In this talk, I am going to talk about a recent polynomial-time reduction from Bayesian incentive-compatible mechanism design to Bayesian algorithm design for welfare maximization problems. Unlike prior results, our reduction achieves exact incentive compatibility for problems with multi-dimensional and continuous type spaces. The key technical barrier preventing exact incentive compatibility in prior black-box reductions is that repairing violations of incentive constraints requires understanding the distribution of the mechanism’s output, which is typically #P-hard to compute. Reductions that instead estimate the output distribution by sampling inevitably suffer from sampling error, which typically precludes exact incentive compatibility. 

We overcome this barrier by employing and generalizing the computational model in the literature on Bernoulli Factories. In a Bernoulli factory problem, one is given a function mapping the bias of an “input coin” to that of an “output coin”, and the challenge is to efficiently simulate the output coin given only sample access to the input coin. Consider a generalization in which a problem instance is specified by a function mapping the expected values of a set of input distributions to a distribution over this set. The challenge is to give a polynomial time algorithm that exactly samples from the distribution over this set given only sample access to the input distributions. I will show how to incorporate Bernoulli factories for the function given by exponential weights: expected values of the input distributions correspond to the weights of an element in the set and we wish to select an element with probability proportional to an exponential function of its weight by efficient sampling. I then show how this algorithm solves a simple single agent truthful mechanism design problem. This truthful mechanism is the key ingredient that can be used to make the approximately incentive compatible reduction of Hartline et al. (2015) exactly incentive compatible.

This talk is based on a joint-work with Shaddin Dughmi, Jason Hartline, and Bobby Kleinberg.