Welfare Undominated Groves Mechanisms

Abstract: A common objective in mechanism design is to choose the outcome (for example, allocation of resources) that maximizes the sum of the agents' valuations, without introducing incentives for agents to misreport their preferences. The class of Groves mechanisms achieves this; however, these mechanisms require the agents to make payments, thereby reducing the agents' total welfare. In this paper we introduce a measure for comparing two mechanisms with respect to the final welfare they generate. This measure induces a partial order on mechanisms and we study the question of finding minimal elements with respect to this partial order. In particular, we say a non-deficit Groves mechanism is welfare undominated if there exists no other non-deficit Groves mechanism that always has a smaller or equal sum of payments. We focus on two domains: (i) auctions with multiple identical units and unit-demand bidders, and (ii) mechanisms for public project problems. In the first domain we analytically characterize all welfare undominated Groves mechanisms that are anonymous and have linear payment functions, by showing that the family of optimal-in-expectation linear redistribution mechanisms, which were introduced in [6] and include the Bailey-Cavallo mechanism [1,2], coincides with the family of welfare undominated Groves mechanisms that are anonymous and linear in the setting we study. In the second domain we show that the classic VCG (Clarke) mechanism is welfare undominated for the class of public project problems with equal participation costs, but is not undominated for a more general class.

• The paper introduces a welfare dominance ordering of Groves mechanisms that minimizes total agent payments while maintaining efficiency and strategy-proofness.
• It demonstrates that, in multi-unit auctions with unit demand, optimal-in-expectation linear (OEL) mechanisms, including the Bailey–Cavallo variant, are uniquely undominated under anonymity and linearity.
• In public project settings, the analysis shows that VCG is welfare undominated for symmetric cost shares, while highlighting potential improvements in asymmetric environments.

Welfare Undominated Groves Mechanisms: A Technical Synthesis

Introduction and Conceptual Framework

The paper presents a rigorous analysis of welfare comparison among Groves mechanisms, focusing on mechanisms' impact on agents' total welfare, specifically through the sum of payments required by participants. The main contribution is introducing a partial order over Groves mechanisms based on welfare dominance: a Groves mechanism is said to "welfare dominate" another if the total sum of payments it collects from agents is always weakly smaller and strictly smaller for at least one type profile. The minimal elements in this ordering, termed "welfare undominated," are examined with the aim of characterizing such mechanisms in canonical domains—multi-unit auctions with unit demand and public project settings.

The authors emphasize the importance of minimizing total payments in settings without a profit-seeking central authority, contrasting with much of the traditional mechanism design literature, which is revenue-oriented. They distinguish their notion of "welfare dominance" from prior concepts such as individual dominance, and argue for its strategic and operational relevance in welfare-centric applications.

Formal Model and Welfare Dominance Notion

The paper adopts the standard framework of tax-based mechanisms $(f,t)$ with quasilinear utilities, focusing exclusively on Groves mechanisms due to their efficiency and strategy-proofness (by Holmström's theorem). The core property under study is welfare dominance: for mechanisms $M, M'$, $M$ welfare dominates $M'$ if for all type vectors $\theta$, the sum of agents' payments under $M$ never exceeds that under $M'$, and is strictly smaller for some $\theta$.

The authors also distinguish between budget balance, non-deficit (feasibility), and pay-only mechanisms, and clarify that full budget balance is unsatisfiable together with efficiency and strategy-proofness in general domains, motivating the search for mechanisms that are optimal in terms of minimizing the payment burden while retaining feasibility, efficiency, and strategy-proofness.

Multi-Unit Auctions with Unit Demand

Characterization of Welfare Undominated Mechansisms

For the domain of multi-unit auctions where each of $n$ agents demands at most one of $m$ identical goods, the authors prove that the optimal-in-expectation linear redistribution (OEL) mechanisms coincide precisely with the class of welfare undominated, anonymous, linear Groves mechanisms. This class includes the Bailey-Cavallo (BC) mechanism and extends earlier known results in distributed surplus redistribution.

The construction of these mechanisms is based on parameterizations indexed by $k=0,\ldots,n$, which induce different linear coefficients for redistribution as a function of the order statistics of the other agents' bids. For instance, the well-known BC mechanism corresponds to $k=m+1$, yielding an explicit formula where each agent receives a multiple of the $(m+1)$-th highest bid among their competitors in addition to the VCG payment. The technical proof leverages properties of linear symmetric functions and the restriction of feasibility to show that any anonymous, linear, undominated Groves mechanism must have the structure of an OEL mechanism, and conversely, all OEL mechanisms are welfare undominated.

Equivalence of Welfare Undominance and Undominance

A notable assertion is that, within the class of anonymous linear Groves mechanisms in the unit demand setting, welfare undominance and individual undominance collapse—these two optimality notions lead to the same set (the OEL mechanisms). This is significant because, in general, welfare undominance is strictly stronger and can lead to a strict subset of undominated mechanisms. The result is substantiated via careful algebraic induction on the structure of the redistribution function.

Formal Results

The main theorems are:

• No feasible (i.e., non-deficit) anonymous linear Groves mechanism welfare dominates any OEL mechanism.
• The only undominated feasible anonymous linear Groves mechanisms are the OEL mechanisms.

These results provide a complete characterization, resolving a key open question in the design of redistribution mechanisms for multi-unit auctions.

Public Project Problems

Classical Symmetric Cost Shares

In the standard public project environment, where $n$ agents must decide whether to finance a project of cost $c$ with cost shares $c_i = c/n$, the authors show that VCG (the Clarke mechanism) is welfare undominated among all Groves mechanisms. They rigorously prove that neither redistribution (as in the BC mechanism) nor alternative anonymous constructions can reduce the sum of payments below that induced by VCG, given the constraints of strategy-proofness, efficiency, and non-deficit.

Significantly, they prove that in this domain, the BC transformation applied to VCG yields VCG itself, i.e., no surplus can be redistributed. This establishes VCG's optimality in terms of minimizing agents' net welfare loss from taxes within the Groves framework.

Generalized Asymmetric Cost Shares

When the model is generalized so that agents may have heterogeneous cost shares (i.e., $c_i$ arbitrary, $\sum_i c_i = c$), the optimality of VCG is shown to be more nuanced:

• It is proved that VCG is undominated among pay-only mechanisms, i.e., no other pay-only Groves mechanism can dominate it.
• However, there exist feasible Groves mechanisms (that can involve rebates) which can welfare dominate VCG for certain instances. The authors supply explicit constructions and parameter settings illustrating this separation, e.g., for $n=3$ agents with particular cost shares.

Thus, symmetric cost shares are a sharp threshold for the welfare undominance of VCG; in the absence of symmetry, further improvements in agent welfare (via lowered total payments) are theoretically possible.

Technical Contributions and Proof Approach

Central technical tools involve:

• The use of anonymous permutations to reduce non-anonymous mechanisms to anonymous ones, ensuring that any welfare-dominating mechanism can be symmetrized without loss.
• Inductive arguments on the linear structure of payments for multi-unit auctions, which yield uniqueness results for linear coefficients in feasible Groves mechanisms.
• Construction and examination of the Bailey–Cavallo–Guo–Conitzer (BCGC) transformation, which operationalizes the maximally redistributive mechanism within Groves.
• Explicit counterexamples and instances demonstrating the strictness between welfare dominance and individual dominance, and the failure of welfare optimality under certain constraints.

Implications and Directions

The results have substantial implications for the design of mechanisms in settings where maximizing agents' net welfare is paramount and where there is no designer with a direct interest in tax revenue. The complete characterization guides both mechanism designers and theorists in identifying the limits of payment minimization within the Groves universe, highlighting precise structural and domain-dependent distinctions.

Of particular note is that, under linearity and anonymity, there is no scope for further welfare-improving mechanism innovation for multi-unit auctions with unit demand beyond OEL (and BC) mechanisms. Conversely, in public project domains, symmetry (equal cost shares) is critical; any departure allows in principle for mechanisms that further lower agents' payments, but at the potential cost of introducing non-pay-only features.

Future work is suggested in several areas: complete characterization of welfare undominated Groves mechanisms in broader or more complex domains (e.g., non-linear, non-anonymous settings), computational consideration for practical mechanism synthesis under non-deficit constraints, and exploration of more refined solution concepts relating to agent risk attitudes or dynamic preferences. Another open direction raised is extension of welfare dominance results to social choice environments beyond quasi-linear utility settings.

Conclusion

This work gives a definitive treatment of welfare undominated Groves mechanisms in two classical domains, rigorously identifying the unique position of OEL mechanisms and VCG in settings with strong symmetry, and delineating the limitations and possibilities that arise in asymmetric environments. These results refine the understanding of feasible, efficient, strategy-proof mechanism design when aggregate agent welfare is paramount, and will be foundational in both theoretical exploration and practical design of non-deficit allocation protocols.