Competitive Many-to-One Matching: Sorting vs. Equality

Abstract: We study many-to-one matching with transfers and peer effects, such as matching workers to firms, students to schools, residents to neighborhoods, or consumers to status goods. With flexible prices (as in the labor market), competitive equilibrium exists and is efficient under general conditions. We characterize when workforces are segregated by skill and matched to firms in a positively assortative manner. In general, equilibrium features alternating intervals of workforce segregation and compression (mixing). Comparative statics characterize when workforces are more segregated or more compressed, and when profits and wages are more or less unequal. With uniform prices (as in school or neighborhood choice), the value generated by peer effects accrues to schools rather than students, and equilibrium can be excessively segregated. Our model generalizes both assignment models (optimal transport) and Bayesian persuasion.

• The paper introduces a unified framework for competitive many-to-one matching with endogenous peer effects, establishing existence and efficiency via strong duality.
• It delineates trade-offs between sorting and compression, showing that increasing differences drive full segregation while decreasing returns favor pooling.
• Comparative statics reveal that changes in firm productivity and worker skill distributions impact wage dispersion and market efficiency, guiding policy design.

Many-to-One Matching with Peer Effects: Sorting vs. Compression

Model and Equilibrium Foundations

The paper "Competitive Many-to-One Matching: Sorting vs. Equality" (2605.30879) establishes a unified framework to analyze many-to-one matching mechanisms in markets featuring both heterogeneous agents and endogenous peer effects. Examples include labor markets (workers to firms), educational assignment (students to schools), residential choice (residents to neighborhoods), and consumption of status goods. The model generalizes classical assignment models (optimal transport), and connects deeply with Bayesian persuasion literature.

Formally, the matching involves two sides: unit mass of firms with productivity $x$ (distribution $\mu$) and unit mass of workers with skill $y$ (distribution $\nu$). Each firm hires a measure-one workforce $\eta$ (a distribution over worker types). The value generated from matching is $V(x, \eta)$, with quasilinear utilities. A competitive equilibrium assigns a distribution over pairs $(x, \eta)$ (with appropriate marginals) and utility/pricing schedules $(p, q)$ satisfying core constraints: no coalition of firms and workers can profitably deviate.

The paper proves existence and efficiency of competitive equilibria via strong duality between a planner's maximization problem and its dual, using infinite-dimensional convexity and Lipschitz continuity. Distinctly, the equilibrium assignment—with peer effects—always yields the maximal total surplus, but personalized prices are essential, not always realistic. The structure of the equilibrium is tightly characterized, enabling explicit comparative statics.

Sorting vs. Compression: Tradeoff and Characterization

Central to the analysis is the tradeoff between sorting and compression. If value is $v(x, z)$ (mean-measurable: workforce affects output through mean skill $z$), two forces emerge:

• Sorting force (positive assortativity): If $\mu$0 (increasing differences), it is efficient to segregate workers by skill and match more productive firms with more skilled workforces.
• Pooling (compression) force: If $\mu$1 (decreasing returns), overly heterogeneous workforces decrease surplus; mixing workers can compress skill distribution and economize on diminishing marginal returns.

The equilibrium assignment is resolved via a concave maximization problem with a majorization constraint (distribution of workforce skill means is compressed relative to population skill distribution). The solution alternates between intervals of full segregation (PAS) and compression, precisely identified through first-order conditions.

Concrete characterization:

• PAS equilibrium (full sorting) arises when increasing differences dominate decreasing returns ($\mu$2, with $\mu$3 being the segregation mapping determined by cumulative distributions).
• Full compression surfaces when decreasing returns dominate; firms hire workforces with mixed or averaged skills, but sorting occurs between compressed groups.
• Hybrid patterns (upper/lower compression) are also characterized, with conditions giving intervals where firms either segregate or compress.

The paper formalizes these conditions via primal-dual optimality and envelope theorems, constructing explicit price and wage schedules consistent with the equilibrium assignment.

Comparative Statics and Inequality Implications

A major contribution is transparent comparative statics:

• Sorting force increases relative to pooling ($\mu$4), workforce segregation and inequality rise.
• Worker skill heterogeneity increases, workforce skill dispersion increases.
• Changing distributions: If firm productivity becomes more concentrated (or worker skill becomes more dispersed), payoffs and matching patterns adjust—the equilibrium becomes more (or less) sorted, affecting wage/profit dispersion.

Notably, the equilibrium wage schedules exhibit pronounced responsiveness to the underlying distributions. Unlike one-to-one matching, in many-to-one markets, changes affecting the upper tail of worker skill or productivity can reduce marginal returns for workers throughout the skill distribution—highlighting competition dynamics intensified by peer substitution. This result has direct implications for observed wage inequality trends due to sorting and labor market polarization.

Moment-Measurable Extensions: Heterogeneity in Workforce Composition

The analysis extends to non-mean-measurable cases (moment-measurable): firms may value different moments of their workforce skill distribution—e.g., CES aggregators, best-shot vs. weakest-link production, or learning functions depending on skill. The equilibrium assignment in these cases is unique and characterized via single-dipped or single-peakedness: more productive firms may employ more heterogeneous workforces if their technology rewards skill diversity.

Rigorous stability results show that, as production technologies perturb away from the mean-measurable case, the equilibrium assignment structure converges appropriately, resolving nonuniqueness in the matching between firms and individual workers.

Uniform Price Equilibrium and Over-Segregation

For cases where personalized pricing is infeasible (school admissions, neighborhoods), the paper introduces uniform price equilibrium (UPE), where all agents on one side pay a uniform price for a given match. Strikingly, under UPE, the value of peer effects accrues to the "clubs" (schools) rather than individual agents (students): regulating away personalized prices can systematically transfer surplus. Explicit comparative analysis shows that UPE can result in excessive segregation compared to the efficient competitive equilibrium, especially when peer effects are strong or decreasing returns are present.

Conditions are provided for PAS matching to be UPE, and for situations where excessive segregation can arise—clarifying theoretical foundations for policy debates around efficiency and equity in social assignment mechanisms (schools, neighborhoods, status goods). Regulatory implications are substantial: uniform pricing agreements among schools or goods providers should be scrutinized, as they may inefficiently shift surplus.

Practical and Theoretical Implications, Future Directions

Practically, the modeling framework enables granular prediction and analysis of matching patterns, wage/profit schedules, and the effect of market interventions (investment, scale, many-to-many setting). Theoretical implications are broad: the model unifies optimal transport and Bayesian persuasion, clarifies the interplay of sorting and peer effects, and yields tractable comparative static insights for markets where workforce composition matters.

Future extensions include endogenous investments with convex costs (leading to universal sorting), endogenous scale, and many-to-many matching. The model's tractability also invites further exploration of mechanism design with majorization constraints, and information design analogs.

Conclusion

This work rigorously elucidates the structure and implications of competitive equilibrium in many-to-one matching with peer effects. By unifying assignment and persuasion frameworks, characterizing key tradeoffs, and isolating comparative statics, the paper provides foundational tools for both empirical and theoretical investigation of sorting, equality, and efficiency in matching markets. The explicit conditions and structural results are critical for interpreting and guiding policy in labor, education, residential, and consumer settings where peer effects and workforce composition are salient.