Computing Bayes-Nash Equilibria in Combinatorial Auctions with Continuous Value and Action Spaces

Abstract

Combinatorial auctions (CAs) are widely used in practice, which is why understanding their incentive properties is an important problem. However, finding Bayes-Nash equilibria (BNEs) of CAs analytically is tedious, and prior algorithmic work has only considered limited solution concepts (e.g. restricted action spaces). In this paper, we present a fast, general algorithm for computing symmetric pure ε-BNEs in CAs with continuous values and actions. In contrast to prior work, we separate the search phase (for finding the BNE) from the verification step (for estimating the ε), and always consider the full (continuous) action space in the best response computation. We evaluate our method in the well-studied LLG domain, against a benchmark of 16 CAs for which analytical BNEs are known. In all cases, our algorithm converges quickly, matching the known results with high precision. Furthermore, for CAs with quasi-linear utility functions and independently distributed valuations, we derive a theoretical bound on ε. Finally, we introduce the new Multi-Minded LLLLGG domain with eight goods and six bidders, and apply our algorithm to finding an equilibrium in this domain. Our algorithm is the first to find an accurate BNE in a CA of this size.

Computing Bayes-Nash Equilibria in Combinatorial Auctions with Verification

“…19. Note that Theorems 1 and 2 are significantly stronger versions of the theorem found in our earlier work(Bosshard et al, 2017), producing much tighter bounds on ε. The general idea of using a finite subset of the value space to obtain a bound on the whole value space was also used more recently byBalcan et al (2019) who employ learning theory to estimate approximate incentive compatibility of non-truthful mechanisms.…”

“…This paper is a significantly extended version of our earlier work published in the conference proceedings of IJCAI (Bosshard et al, 2017). We thank the editor and the anonymous reviewers of JAIR for their very helpful comments.…”

Equilibrium Learning in Combinatorial Auctions: Computing Approximate Bayesian Nash Equilibria via Pseudogradient Dynamics

“…Rabinovich et al [28] study best-response dynamics on mixed strategies in auctions with finite action spaces. Most recently, Bosshard et al [9,10] proposed a method to find BNE in combinatorial auctions that relies on smoothed best-response dynamics and is applicable to any Bayesian game. The method explicitly computes point-wise best-responses in a fine-grained linearization of the strategy space via sophisticated Monte-Carlo integration.…”

“…The goal of an individual update step is thus to marginally improve the expected utility of player 𝑖 across all possible joint valuations 𝑣 ∼ 𝐹 . This perspective ultimately considers low-probability events less important than high-probability events, which is in contrast to some other methods, which explicitly aim to optimize all ex-interim states [9]. Second, to compute the gradient oracle ∇ 𝛽 ũ in self-play, we rely on access to other players strategies, but evaluating each player's policy relies only on their own valuation.…”

A megújuló áramtermelés támogatására vonatkozó aukciók elemzése

“…A folyamatos költségcsökkenés miatt azonban a megújuló támogatások esetén egyre inkább általánossá válhat majd, hiszen az ajánlatadók egyre nagyobb részének nem lesz már szüksége támogatásra (vagyis a modellem terminológiáját követve: egyre többek értékelése lesz éppen 0). Bosshard et al (2017) összefoglalja, hogy a korábbi irodalomban milyen különböző algoritmusok születtek a Nash-egyensúly illetve a Bayes-i Nash egyensúly (BNE) keresés esetén. Munkájukban olyan aukciókra fókuszálnak, ahol a javak egy kombinációjára adható ajánlat.…”