Equity in auction design with unit-demand agents and non-quasilinear preferences

We study a model of auction design where a seller is selling a set of objects to a set of agents who can be assigned no more than one object. Each agent’s preference over (object, payment) pair need not be quasilinear. If the domain contains all classical preferences, we show that there is a unique mechanism, the minimum Walrasian equilibrium price (MWEP) mechanism, which is strategy-proof, individually rational, and satisfies equal treatment of equals, no-wastage (every object is allocated to some agent), and no-subsidy (no agent is subsidized). This provides an equity-based characterization of the MEWP mechanism, and complements the efficiency-based characterization of the MWEP mechanism known in the literature.

💡 Research Summary

The paper investigates auction design for a seller who must allocate a set of heterogeneous objects to a set of agents, each of which can receive at most one object (unit‑demand). Unlike most of the auction literature, the authors do not assume quasilinear utilities; instead, agents may exhibit income effects, leading to non‑quasilinear (classical) preferences. A “classical” preference is defined by four axioms: monotonicity in money, strict preference for any real object over the null object, continuity, and the ability to be compensated (i.e., for any bundle there exists a transfer that makes the agent indifferent). The domain considered is the Cartesian product of all classical preferences, which the authors call a “sufficiently rich” domain because it contains every possible classical preference.

The authors impose five desiderata on any deterministic allocation mechanism:

1. Strategy‑proofness (SP) – truthful reporting is a dominant strategy for every agent.
2. Individual Rationality (IR) – each agent receives at least as much utility as from the null bundle (0, 0).
3. Equal Treatment of Equals (ETE) – agents with identical preferences receive bundles that are indifferent to each other.
4. No‑Wastage (NW) – every real object must be allocated to some agent (no object is left unsold).
5. No‑Subsidy (NS) – payments are non‑negative; the mechanism never pays agents.

The central contribution is a uniqueness theorem: when the preference domain contains all classical preferences, the only mechanism satisfying SP, IR, ETE, NW, and NS is the Minimum Walrasian Equilibrium Price (MWEP) mechanism. The MWEP mechanism selects the Walrasian equilibrium with the lowest possible price vector: each object receives the smallest price that still supports a Walrasian equilibrium, and agents are allocated their most‑preferred affordable object at those prices. Because the price vector is minimal, NS is automatically satisfied; the Walrasian equilibrium guarantees that every object is demanded by at least one agent, ensuring NW; and the construction yields a deterministic, strategy‑proof allocation.

A key corollary follows: any mechanism meeting the five axioms must be Pareto efficient. In other words, the combination of the weak fairness condition (ETE) and the weak efficiency condition (NW) forces efficiency when agents may have arbitrary income effects. This strengthens earlier results that required the stronger “no‑envy” condition together with NW to guarantee efficiency (Svensson, 1983). The paper thus shows that equity, even in its weakest form, can substitute for explicit efficiency objectives in auction design.

The authors discuss the necessity of the richness assumption. If the domain is restricted (e.g., only quasilinear preferences, or only “normal goods” preferences), other mechanisms can satisfy the five axioms. Likewise, if objects are identical rather than heterogeneous, the MWEP mechanism is no longer unique. Hence, heterogeneity of objects and the presence of non‑quasilinear preferences are essential for the characterization.

The paper situates its contribution within a broader literature on fair mechanisms. Prior work often used stronger fairness notions such as anonymity or no‑envy, which either limited the set of admissible mechanisms or required additional continuity or revenue‑maximization axioms. By focusing on ETE, the authors obtain a clean, prior‑free characterization that does not rely on any revenue or continuity assumptions beyond those already embedded in the definition of classical preferences.

Practical implications are highlighted. Many public auctions (e.g., spectrum licenses, public housing, sports franchise sales) involve heterogeneous items and large monetary transfers, making income effects realistic. Policymakers concerned with legal challenges based on discrimination can adopt the MWEP mechanism, confident that it simultaneously respects equal treatment, avoids waste, and delivers efficient outcomes without needing to compute complex optimal auctions.

The paper concludes with suggestions for future research: extending the analysis to multi‑unit demand agents, exploring randomized mechanisms, and incorporating prior information (Bayesian settings). These extensions would address settings where Walrasian equilibria may not exist or where strategy‑proofness is harder to achieve, but the current work provides a solid baseline for understanding the interplay between equity and efficiency in non‑quasilinear auction environments.