Worst-case optimal redistribution of VCG payments in heterogeneous-item auctions with unit demand

Many important problems in multiagent systems involve the allocation of multiple resources among the agents. For resource allocation problems, the well-known VCG mechanism satisfies a list of desired properties, including efficiency, strategy-proofness, individual rationality, and the non-deficit property. However, VCG is generally not budget-balanced. Under VCG, agents pay the VCG payments, which reduces social welfare. To offset the loss of social welfare due to the VCG payments, VCG redistribution mechanisms were introduced. These mechanisms aim to redistribute as much VCG payments back to the agents as possible, while maintaining the aforementioned desired properties of the VCG mechanism. We continue the search for worst-case optimal VCG redistribution mechanisms -- mechanisms that maximize the fraction of total VCG payment redistributed in the worst case. Previously, a worst-case optimal VCG redistribution mechanism (denoted by WCO) was characterized for multi-unit auctions with nonincreasing marginal values [7]. Later, WCO was generalized to settings involving heterogeneous items [4], resulting in the HETERO mechanism. [4] conjectured that HETERO is feasible and worst-case optimal for heterogeneous-item auctions with unit demand. In this paper, we propose a more natural way to generalize the WCO mechanism. We prove that our generalized mechanism, though represented differently, actually coincides with HETERO. Based on this new representation of HETERO, we prove that HETERO is indeed feasible and worst-case optimal in heterogeneous-item auctions with unit demand. Finally, we conjecture that HETERO remains feasible and worst-case optimal in the even more general setting of combinatorial auctions with gross substitutes.Mingyu Gu

Undominated Groves Mechanisms

The family of Groves mechanisms, which includes the well-known VCG mechanism (also known as the Clarke mechanism), is a family of efficient and strategy-proof mechanisms. Unfortunately, the Groves mechanisms are generally not budget balanced. That is, under such mechanisms, payments may flow into or out of the system of the agents, resulting in deficits or reduced utilities for the agents. We consider the following problem: within the family of Groves mechanisms, we want to identify mechanisms that give the agents the highest utilities, under the constraint that these mechanisms must never incur deficits. We adopt a prior-free approach. We introduce two general measures for comparing mechanisms in prior-free settings. We say that a non-deficit Groves mechanism M in- dividually dominates another non-deficit Groves mechanism M′ if for every type profile, every agent’s utility under M is no less than that under M′, and this holds with strict inequality for at least one type profile and one agent. We say that a non-deficit Groves mechanism M collectively dominates another non-deficit Groves mechanism M′ if for every type profile, the agents’ total utility under M is no less than that under M′, and this holds with strict inequality for at least one type profile. The above definitions induce two partial orders on non-deficit Groves mechanisms. We study the maximal elements corresponding to these two partial orders, which we call the individually undominated mechanisms and the collectively undominated mechanisms, respectively

Stochastic Mechanisms for Truthfulness and Budget Balance in Computational Social Choice

In this thesis, we examine stochastic techniques for overcoming game theoretic and computational issues in the collective decision making process of self-interested individuals. In particular, we examine truthful, stochastic mechanisms, for settings with a strong budget balance constraint (i.e. there is no net flow of money into or away from the agents). Building on past results in AI and computational social choice, we characterise affine-maximising social choice functions that are implementable in truthful mechanisms for the setting of heterogeneous item allocation with unit demand agents. We further provide a characterisation of affine maximisers with the strong budget balance constraint. These mechanisms reveal impossibility results and poor worst-case performance that motivates us to examine stochastic solutions. To adequately compare stochastic mechanisms, we introduce and discuss measures that capture the behaviour of stochastic mechanisms, based on techniques used in stochastic algorithm design. When applied to deterministic mechanisms, these measures correspond directly to existing deterministic measures. While these approaches have more general applicability, in this work we assess mechanisms based on overall agent utility (efficiency and social surplus ratio) as well as fairness (envy and envy-freeness). We observe that mechanisms can (and typically must) achieve truthfulness and strong budget balance using one of two techniques: labelling a subset of agents as ``auctioneers'' who cannot affect the outcome, but collect any surplus; and partitioning agents into disjoint groups, such that each partition solves a subproblem of the overall decision making process. Worst-case analysis of random-auctioneer and random-partition stochastic mechanisms show large improvements over deterministic mechanisms for heterogeneous item allocation. In addition to this allocation problem, we apply our techniques to envy-freeness in the room assignment-rent division problem, for which no truthful deterministic mechanism is possible. We show how stochastic mechanisms give an improved probability of envy-freeness and low expected level of envy for a truthful mechanism. The random-auctioneer technique also improves the worst-case performance of the public good (or public project) problem. Communication and computational complexity are two other important concerns of computational social choice. Both the random-auctioneer and random-partition approaches offer a flexible trade-off between low complexity of the mechanism, and high overall outcome quality measured, for example, by total agent utility. They enable truthful and feasible solutions to be incrementally improved on as the mechanism receives more information and is allowed more processing time. The majority of our results are based on optimising worst-case performance, since this provides guarantees on how a mechanism will perform, regardless of the agents that use it. To complement these results, we perform empirical, average-case analyses on our mechanisms. Finally, while strong budget balance is a fixed constraint in our particular social choice problems, we show empirically that this can improve the overall utility of agents compared to a utility-maximising assignment that requires a budget imbalanced mechanism