A stable menu of job contracts can exist when no single deal is stable

Labor-market stability becomes difficult when workers complement one another: a team may generate more value together than its members would separately. In matching theory, such complementarities can leave every hiring arrangement vulnerable to a firm and workers proposing a better deal. A preprint by Madhuparna Karmokar and Debasis Mishra explores an alternative: making a collection of outcomes stable, rather than requiring any single outcome to withstand every challenge.

Their central result is conditional but broad within the model. If exactly one assignment of workers to firms maximizes total surplus, then all individually rational ways of dividing that assignment’s surplus together form a von Neumann–Morgenstern, or vNM, stable set. This can hold even when the conventional core—the collection of individually stable outcomes—is empty.

What makes a menu stable?

The study uses a many-to-one matching model: a firm can hire several workers, but each worker is assigned to at most one firm. Each possible firm–team combination produces a surplus, which can reflect the firm’s value from the team minus workers’ costs of taking those jobs. Surplus can be negative, and adding workers need not increase it.

An outcome specifies both the assignment and its payoff division. Salaries allow surplus to be transferred between a firm and its workers, but the model prohibits transfers across firms. Individual rationality means that every firm and worker receives a nonnegative payoff, with unmatched workers receiving zero. Efficiency means maximizing the sum of firms’ team surpluses, not selecting any particular distribution of that total.

The stability test compares outcomes through a domination relation. One outcome dominates another if a firm and the workers assigned to it in the proposed outcome all receive at least as much as before, with at least one receiving strictly more. People outside that group need not benefit.

A vNM stable set must satisfy two conditions. Internally, none of its outcomes may dominate another. Externally, every individually rational outcome outside the set must be dominated by an outcome inside it. Thus, membership does not mean an individual contract is immune to challenges. Stability belongs to the menu as a whole.

A proof-based result, not a market experiment

The authors establish their findings through mathematical proofs and constructed examples, rather than employment data or field experiments.

For the main existence theorem, internal stability follows because outcomes sharing the same assignment cannot dominate one another: with the same team surplus to divide, the firm and its workers cannot all gain. For external stability, the authors prove that every inefficient, individually rational outcome can be dominated by an efficient one. When the efficient assignment is unique, that dominating outcome necessarily belongs to the proposed menu.

A further proof uses a network-flow construction. The authors express payoff requirements as linear inequalities, translate them into a directed network, and apply David Gale’s demand theorem to establish feasibility. This supports their characterization of a more restrictive class of stable sets.

The paper calls existence “generic” because, holding workers’ job costs fixed, a unique efficient assignment occurs for a set of firm-value specifications with full Lebesgue measure. This is a mathematical statement about the model’s parameter space—not evidence about how frequently real labor markets meet the condition.

Ties and distribution change the picture

Multiple equally efficient assignments complicate matters. In one example, three firms each generate positive surplus only with a particular pair drawn from three workers. The overlapping pairs create three tied efficient assignments, and the authors show that no vNM stable set exists. Other examples do admit stable sets despite ties, so multiplicity alone does not imply failure.

Nor does collective stability by itself guarantee efficiency. The paper constructs a stable set containing only inefficient outcomes, even in an economy with a unique efficient assignment. The main existence theorem therefore identifies a stable menu; it does not establish that every stable menu has that form.

Stronger conclusions follow from an additional condition called richness. For every assignment represented, a rich menu must include the division giving firms all the surplus and, separately for each worker, the division giving that worker their matched firm’s entire team surplus while other workers receive zero.

A rich stable set exists exactly when efficient assignments are “essentially unique”: any firm whose team differs across them must generate zero surplus with both teams. In that case, all efficient, individually rational outcomes form the unique rich stable set.

These conclusions concern a complete-information model and immediate improvements, not strategic sequences of future moves. Richness also requires extreme payoff divisions that the authors acknowledge may be questionable as standards of social behavior. Reported as a preprint, the study offers a theoretical account of collective stability—not a tested hiring procedure, a prediction of wages, or a guarantee of fair pay.