Bernoulli factories and black-box reductions in mechanism design

Abstract

We provide a 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 which we call the expectations from samples computational model, in which a problem instance is specified by a function mapping the expected values of a set of input distributions to a distribution over outcomes. The challenge is to give a polynomial time algorithm that exactly samples from the distribution over outcomes given only sample access to the input distributions.In this model, we give a polynomial time algorithm for the function given by exponential weights: expected values of the input distributions correspond to the weights of alternatives and we wish to select an alternative with probability proportional to an exponential function of its weight. This algorithm is the key ingredient in designing an incentive compatible mechanism for bipartite matching, which can be used to make the approximately incentive compatible reduction of exactly incentive compatible.

Algorithmic Mechanism Design With Investment

“…Nisan and Ronen (2007) showed that VCG‐based mechanisms with nearly‐optimal allocation algorithms are generically non‐truthful, while Lehmann, Oćallaghan, and Shoham (2002) introduced a truthful mechanism for the knapsack problem in which the allocation is determined by a greedy algorithm. In addition, Hartline and Lucier (2015) developed a method for converting a (non‐optimal) algorithm for optimization into a Bayesian incentive compatible mechanism with weakly higher social welfare or revenue; Dughmi, Hartline, Kleinberg, and Niazadeh (2017) generalized this result to multidimensional types. For a more comprehensive review of results on approximation in mechanism design, see Hartline (2016).…”

Auction learning as a two-player game

“…On the conceptual front, this approach does not leverage state-of-the-art work in Auction Theory for the multi-bidder setting. Our work, on the other hand, leverages an exciting line of recent works (Hartline and Lucier, 2010;Hartline et al, 2011;Bei and Huang, 2011;Daskalakis and Weinberg, 2012;Rubinstein and Weinberg, 2018;Dughmi et al, 2017;Cai et al, 2019) on ε-truthful-to-truthful reductions. 1 On the technical front, we identify three areas for improvement.…”

“…To address this, Hartline et al (2015); Bei and Huang (2011); Dughmi et al (2017) develop a sophisticated reduction for turning algorithms into truthful mechanisms, which was later extended in Daskalakis and Weinberg (2012); Rubinstein and Weinberg (2018); Cai et al (2019) to reduce ε-truthful mechanisms into truthful mechanisms with small loss in revenue. For example, Lemma 2 follows immediately from Theorem 5.2 in Rubinstein and Weinberg (2018).…”