Efficient and Envy-free Random Assignment Beyond Expected Utility

Abstract: We consider the random assignment problem with abstract continuous and convex preferences. In particular, we admit preference relations that are not constrained by independence or transitivity. By extending the Hylland--Zeckhauser pseudo-market mechanism, we show that weakly efficient and envy-free random assignments always exist. For preferences that can be represented via skew-symmetric bilinear (SSB) utility functions -- which generalize linear expected utility functions -- we prove the existence of efficient and approximately envy-free random assignments. Efficient and envy-free random assignments exist under a mild additional assumption on preferences. These findings have notable implications for ordinal random assignment, where ordinal preferences are extended to preferences over lotteries via the pairwise comparison (PC) extension. While the probabilistic serial rule and popular random assignments frequently and significantly violate PC-efficiency and PC-envy-freeness, respectively, random assignments that satisfy both conditions do exist.

• The paper introduces a generalized pseudo-market mechanism that ensures efficiency and envy-freeness under continuous and convex lottery preferences beyond standard expected utility.
• The paper provides a counterexample in the SSB domain and establishes strict maximality conditions or approximate envy allowances to reconcile efficiency with fairness.
• It applies ordinal analyses and pairwise comparisons to reveal computational and strategic challenges in traditional assignment rules, offering improved fairness guarantees.

Efficient and Envy-Free Random Assignment Beyond Expected Utility

Motivation and Model

The paper "Efficient and Envy-free Random Assignment Beyond Expected Utility" (2606.13730) addresses the assignment of indivisible objects to agents under general lottery preferences, relaxing the usual independence and transitivity requirements typical of expected utility (vNM) theory. The authors formalize the assignment problem in the setting of continuous and convex preferences over lotteries, generalizing assignment rules to accommodate agents whose behavior may violate standard rational choice axioms—a feature motivated by empirical observations such as Allais's paradox and recognized departures from transitivity. This model encompasses SSB (skew-symmetric bilinear) utility representations, strictly generalizing the space of linear expected utility functions.

Existence Results: Generalized Pseudo-Market Equilibria

The central mechanism extended is the Hylland-Zeckhauser pseudo-market scheme, which relies on equal artificial budgets and market-clearing prices to decentralize the allocation process without payments. The authors prove via Kakutani's fixed-point theorem that, under minimal assumptions (continuity and convexity), equilibrium random assignments exist and are both weakly efficient and envy-free. For budgets equalized across agents, this mechanism is shown to preserve envy-freeness, even absent independence or transitivity, by leveraging the convexity/continuity structure in the agents' preference graphs and upper hemicontinuity of demand correspondences.

Incompatibility and Domain Restrictions

A constructive counterexample demonstrates the generic incompatibility of efficiency and envy-freeness in the SSB domain. Specifically, for a three-agent preference profile represented by SSB utility matrices, no random assignment simultaneously satisfies efficiency and envy-freeness. The source of incompatibility lies in intransitive regions of indifference, which can be arbitrarily large within SSB utility models.

To circumvent this, the paper identifies a strict maximality domain restriction within SSB preferences. If the strict preference part "separates" the set of maximal lotteries from all others—an assumption guaranteed when preferences are transitive over degenerate lotteries—the pseudo-market equilibrium with appropriate cost-minimization tie-breaking yields both efficiency and envy-freeness. This domain includes significant generalizations beyond vNM, while also supporting computational tractability within the refined mechanism.

Approximate Envy-Freeness and Efficiency for SSB Preferences

For the broader SSB domain without restriction, the authors establish the existence of efficient and approximately envy-free random assignments for any $\alpha > 0$ (where $\alpha$ quantifies tolerated envy). The result exploits an SSB efficiency-welfare theorem connecting efficiency to affine welfare maximization under positive weights [ABB14b], coupled with a fixed-point argument. Iterative adjustment of agent welfare weights dampens maximal envy in the system, converging to an $\alpha$-envy-free assignment. The limiting assignment for $\alpha \to 0$ is always weakly efficient and exactly envy-free, but the efficient assignment set is not generally closed under SSB preferences (the paper provides an explicit example).

Ordinal Random Assignment and Pairwise Comparison

The analysis has significant ordinal implications. The pairwise comparison relation, a canonical SSB representation of ordinal agent preferences, refines stochastic dominance and is cyclic for four or more objects (Steinhaus-Trybula paradox). The probabilistic serial rule (PS) and popular random assignments fail to meet both strong efficiency and envy-freeness in this space. Numerical evidence (including explicit examples) shows PS frequently violates weak efficiency, especially as $n$ grows, with large probability margins. Similarly, popular assignments can breach envy-freeness with increasing prevalence. By contrast, the generalized pseudo-market equilibrium with pairwise comparison utility guarantees both efficiency and pairwise comparison envy-freeness—a property unique among well-known ordinal assignment rules.

Computational and Strategic Implications

The mechanism inherits non-strategyproof properties from its vNM antecedents, verifying the impossibility of combining efficiency, equal treatment of equals, and strategyproofness (per Zhou's negative results). Approximate strategyproofness in large markets holds, but strict strategyproofness remains unattainable. The computational complexity of finding equilibria, even in the vNM or ordinal domains, is PPAD-hard in general; more efficient computation is possible only in restricted bi-valued settings.

Conclusion

This work establishes that efficient and envy-free random assignments are guaranteed under generalized pseudo-market mechanisms for continuous and convex (and SSB) preferences, albeit with necessary restrictions or by tolerating approximate envy. It pinpoints structural incompatibilities in unrestricted preference domains and refines conditions where both efficiency and envy-freeness can coexist. These findings are consequential for ordinal assignment problems, enabling mechanisms that surpass the widely adopted probabilistic serial and popular assignment rules in efficiency and fairness. The results imply new avenues for fair division in domains where agent decision behaviour cannot be reliably modeled by expected utility, and for algorithmic market design in assignment settings subject to computational and strategic constraints.

Future research directions include tightening the computational tractability of generalized equilibria, exploring extensions to assignment problems with priorities or richer combinatorial constraints, and further examining incentive compatibility in highly non-linear preference domains.