We study the design of mechanisms in combinatorial auction domains. We focus on settings where the auction is repeated, motivated by auctions for licenses or advertising space. We consider models of agent behaviour in which they either apply common learning techniques to minimize the regret of their bidding strategies, or apply short-sighted best-response strategies. We ask: when can a black-box approximation algorithm for the base auction problem be converted into a mechanism that approximately preserves the original algorithm's approximation factor on average over many iterations? We present a general reduction for a broad class of algorithms when agents minimize external regret. We also present a new mechanism for the combinatorial auction problem that attains an @math approximation on average when agents apply best-response dynamics.
The problem of designing combinatorial auction mechanisms that implement approximations at equilibria (and, in particular, Bayes-Nash equilibria for partial information settings) was considered in @cite_9 for submodular CAs, and in @cite_16 for general CA problems. Implementation at equilibrium, especially for the alternative goal of profit maximization, has a rich history in the economics literature; see, for example, Jackson @cite_11 for a survey.
@cite_9 @cite_16 @cite_11
2077121235 2404718321 2022890659 1846856248
We study mechanisms for utilitarian combinatorial allocation problems, where agents are not assumed to be single-minded. This class of problems includes combinatorial auctions, multi-unit auctions, unsplittable flow problems, and others. We focus on the problem of designing mechanisms that approximately optimize social welfare at every Bayes-Nash equilibrium (BNE), which is the standard notion of equilibrium in settings of incomplete information. For a broad class of greedy approximation algorithms, we give a general black-box reduction to deterministic mechanisms with almost no loss to the approximation ratio at any BNE. We also consider the special case of Nash equilibria in full-information games, where we obtain tightened results. This solution concept is closely related to the well-studied price of anarchy. Furthermore, for a rich subclass of allocation problems, pure Nash equilibria are guaranteed to exist for our mechanisms. For many problems, the approximation factors we obtain at equilibrium improve upon the best known results for deterministic truthful mechanisms. In particular, we exhibit a simple deterministic mechanism for general combinatorial auctions that obtains an O(√m) approximation at every BNE. We consider the problem of maximizing revenue in prior-free auctions for general single parameter settings. The setting is modeled by an arbitrary downward-closed set system, which captures many special cases such as single item, digital goods and single-minded combinatorial auctions. We relax the truthfulness requirement by the solution concept of Nash equilibria. Implementation by Nash equilibria is a natural and relevant framework in many applications of computer science, where auctions are run repeatedly and bidders can observe others’ strategies, but the auctioneer needs to design a mechanism in advance and cannot use any information on the bidders’ private valuations. We introduce a worst-case revenue benchmark which generalizes the second price of single item auction and the F2 benchmark, introduced by , for digital goods. We design a mechanism whose Nash equilibria obtains at least a constant factor of this benchmark and prove that no truthful mechanisms can achieve a constant approximation. The Generalized Second Price Auction has been the main mechanism used by search companies to auction positions for advertisements on search pages. In this paper we study the social welfare of the Nash equilibria of this game in various models. In the full information setting, socially optimal Nash equilibria are known to exist (i.e., the Price of Stability is 1). This paper is the first to prove bounds on the price of anarchy, and to give any bounds in the Bayesian setting. Our main result is to show that the price of anarchy is small assuming that all bidders play un-dominated strategies. In the full information setting we prove a bound of 1.618 for the price of anarchy for pure Nash equilibria, and a bound of 4 for mixed Nash equilibria. We also prove a bound of 8 for the price of anarchy in the Bayesian setting, when valuations are drawn independently, and the valuation is known only to the bidder and only the distributions used are common knowledge. Our proof exhibits a combinatorial structure of Nash equilibria and uses this structure to bound the price of anarchy. While establishing the structure is simple in the case of pure and mixed Nash equilibria, the extension to the Bayesian setting requires the use of novel combinatorial techniques that can be of independent interest. We study anonymous posted price mechanisms for combinatorial auctions in a Bayesian framework. In a posted price mechanism, item prices are posted, then the consumers approach the seller sequentially in an arbitrary order, each purchasing her favorite bundle from among the unsold items at the posted prices. These mechanisms are simple, transparent and trivially dominant strategy incentive compatible (DSIC). We show that when agent preferences are fractionally subadditive (which includes all submodular functions), there always exist prices that, in expectation, obtain at least half of the optimal welfare. Our result is constructive: given black-box access to a combinatorial auction algorithm A, sample access to the prior distribution, and appropriate query access to the sampled valuations, one can compute, in polytime, prices that guarantee at least half of the expected welfare of A. As a corollary, we obtain the first polytime (in n and m) constant-factor DSIC mechanism for Bayesian submodular combinatorial auctions, given access to demand query oracles. Our results also extend to valuations with complements, where the approximation factor degrades linearly with the level of complementarity.