An Efficient $\varepsilon$-BIC to BIC Transformation and Its Application to Black-Box Reduction in Revenue Maximization

Abstract

We consider the black-box reduction from multi-dimensional revenue maximization to virtual welfare maximization. Cai et al. [12,13,14,15] show a polynomial-time approximation-preserving reduction, however, the mechanism produced by their reduction is only approximately Bayesian incentive compatible (ε-BIC). We provide a new polynomial time transformation that converts any ε-BIC mechanism to an exactly BIC mechanism with only a negligible revenue loss. Our transformation applies to a very general mechanism design setting and only requires sample access to the agents' type distributions and query access to the original ε-BIC mechanism. Other ε-BIC to BIC transformations exist in the literature [23,35,18] but all require exponential time to run. As an application of our transformation, we improve the reduction by Cai et al. [12,13,14,15] to generate an exactly BIC mechanism.Our transformation builds on a novel idea developed in a recent paper by Dughmi et al. [24]: finding the maximum entropy regularized matching using Bernoulli factories. The original algorithm by Dughmi et al. [24] can only handle graphs with nonnegative edge weights, while our transformation requires finding the maximum entropy regularized matching on graphs with both positive and negative edge weights. The main technical contribution of this paper is a new algorithm that can accommodate arbitrary edge weights.

From the Bernoulli Factory to a Dice Enterprise via Perfect Sampling of Markov Chains

“…Assume that we are given a p-coin, and write p 0 = 1 − p and p 1 = p, to consider a fine and connected ladder of the form π : (0, 1) → ∆ k as in equation (5). In this case the definition of neighbourhoods simplifies and so does the Markov chain defined in Proposition 3.4.…”

“…Corollary 3.8. Let π : (0, 1) → ∆ k be a fine and connected ladder as in equation (5). The transition matrix of the Markov chain (X t ) t∈N defined in Proposition 3.4 can be rewritten as…”

“…Proposition 3.15. Let π : (0, 1) → ∆ k be a univariate fine and connected ladder as in (5). Assume further that π is strictly log-concave and that the Markov chain and update function defined in Corollary 3.8 are used.…”

“…Relations between the Bernnoulli Factory and other fundamental simulation questions in statistics and computer science have been explored in [21] and [9], respectively. More recently, Bernoulli Factory techniques have been successfully applied to perform exact simulations of diffusions [3,24], develop perfect simulation algorithms [2,10,25], design MCMC algorithms that can tackle intractable likelihood models and perform Bayesian inference [11,12,15,38], design particle filters in scenarios where weights are not available analytically [37], and also to reductions in mechanism design [5,7,30].…”

Bernoulli Factories for Flow-Based Polytopes

“…A Bernoulli factory is a technique from applied probability that allows one to exactly sample certain random variables having only sample access to its parameters. Its original goal (Asmussen et al (1992); Keane and O'Brien (1994)) was to perform exact simulation of certain stochastic processes, but it has since found applications in various fields such as mechanism design (Dughmi et al (2017); Cai et al (2019)), Bayesian inference (Gonçalves et al (2017b); Herbei and Berliner (2014)) and quantum physics (Dale et al (2015); Yuan et al (2016)).…”