In a combinatorial auction with item bidding, agents participate in multiple single-item second-price auctions at once. As some items might be substitutes, agents need to strategize in order to maximize their utilities. A number of results indicate that high welfare can be achieved this way, giving bounds on the welfare at equilibrium. Recently, however, criticism has been raised that equilibria are hard to compute and therefore unlikely to be attained. In this paper, we take a different perspective. We study simple best-response dynamics. That is, agents are activated one after the other and each activated agent updates his strategy myopically to a best response against the other agents' current strategies. Often these dynamics may take exponentially long before they converge or they may not converge at all. However, as we show, convergence is not even necessary for good welfare guarantees. Given that agents' bid updates are aggressive enough but not too aggressive, the game will remain in states of good welfare after each agent has updated his bid at least once. In more detail, we show that if agents have fractionally subadditive valuations, natural dynamics reach and remain in a state that provides a @math approximation to the optimal welfare after each agent has updated his bid at least once. For subadditive valuations, we can guarantee an @math approximation in case of @math items that applies after each agent has updated his bid at least once and at any point after that. The latter bound is complemented by a negative result, showing that no kind of best-response dynamics can guarantee more than an @math fraction of the optimal social welfare.
A final point of reference are truthful mechanisms for combinatorial auctions. While no mechanism can achieve a better than @math approximation for submodular valuations with valuation queries alone @cite_5 , Dobzinski @cite_22 recently managed to improve a long-standing approximation guarantee of @math for submodular valuations to @math for fractionally subadditive valuations, requiring access to both value and demand oracles.
@cite_5 @cite_22
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A long-standing open question in algorithmic mechanism design is whether there exist computationally efficient truthful mechanisms for combinatorial auctions, with performance guarantees close to those possible without considerations of truthfulness. In this article, we answer this question negatively: the requirement of truthfulness can impact dramatically the ability of a mechanism to achieve a good approximation ratio for combinatorial auctions. More precisely, we show that every universally truthful randomized mechanism for combinatorial auctions with submodular valuations that approximates optimal social welfare within a factor of m1 2−e must use exponentially many value queries, where m is the number of items. Furthermore, we show that there exists a class of succinctly represented submodular valuation functions, for which the existence of a universally truthful polynomial-time mechanism that provides an m1 2−e-approximation would imply NP e RP. In contrast, ignoring truthfulness, there exist constant-factor approximation algorithms for this problem, and ignoring computational efficiency, the VCG mechanism is truthful and provides optimal social welfare. These are the first hardness results for truthful polynomial-time mechanisms for any type of combinatorial auctions, even for deterministic mechanisms. Our approach is based on a novel direct hardness technique that completely skips the notoriously hard step of characterizing truthful mechanisms. The characterization step was the main obstacle for proving impossibility results in algorithmic mechanism design so far. We study a central problem in Algorithmic Mechanism Design: constructing truthful mechanisms for welfare maximization in combinatorial auctions with submodular bidders. Dobzinski, Nisan, and Schapira provided the first mechanism that guarantees a non-trivial approximation ratio of @math [STOC'06], where @math is the number of items. This was subsequently improved to @math [Dobzinski, APPROX'07] and then to @math [Krysta and Vocking, ICALP'12]. In this paper we develop the first mechanism that breaks the logarithmic barrier. Specifically, the mechanism provides an approximation ratio of @math . Similarly to previous constructions, our mechanism uses polynomially many value and demand queries, and in fact provides the same approximation ratio for the larger class of XOS (a.k.a. fractionally subadditive) valuations. We also develop a computationally efficient implementation of the mechanism for combinatorial auctions with budget additive bidders. Although in general computing a demand query is NP-hard for budget additive valuations, we observe that the specific form of demand queries that our mechanism uses can be efficiently computed when bidders are budget additive.