When cooperation can survive a local setback

A merger may cut costs, yet complicate the division of profits. A production team may benefit from expanding, yet gain less from one additional worker than from the previous recruit. These possibilities challenge a convenient assumption in cooperative game theory: that a participant’s marginal contribution never falls as the group receiving it expands.

In Cores of 3-distant convex games, Keisuke Bando, Toshiyuki Hirai, Kenzo Imamura and Ryuhei Mizutani show how much of the standard theory survives a particular relaxation of that assumption. Their arXiv preprint, submitted on November 5, 2026, offers mathematical proofs and stylized applications, not evidence from observed firms or workers.

What counts as stable cooperation?

The paper studies transferable-utility games: each possible coalition has a specified worth, and its members can redistribute that value through payments. A division belongs to the core if it distributes the entire grand coalition’s worth while giving every subgroup at least what that subgroup could obtain independently. No coalition can then improve its members’ position by breaking away and redistributing its own worth.

For conventional convex games, the core always exists. One can find an allocation simply by ordering the players and paying each the value added upon joining earlier arrivals. But fixed communication costs or merger-related cost savings can disrupt convexity.

The authors instead examine 3-distant convexity. The usual convexity inequality must hold whenever two coalitions differ in the membership of at least three players. Violations can occur only between coalitions obtained from each other by swapping one member. The “3” measures membership differences, not geographic distance or coalition size.

One inequality replaces a family of checks

Under this restriction, and the paper’s standing assumption that everyone together is worth at least the sum of their standalone worths, the authors establish an exact condition for a nonempty core. Add up each player’s contribution to the grand coalition—the total worth minus the worth remaining without that player. A stable allocation exists if and only if this sum is at least the grand coalition’s worth.

Equivalently, the combined worth of all coalitions missing exactly one player cannot exceed n−1 times the full coalition’s worth, where n is the number of players. This is not a criterion for arbitrary cooperative games: its sufficiency depends on 3-distant convexity.

The comparison is with the classical Bondareva–Shapley characterization, which requires inequalities for every balanced collection of coalitions—weighted collections covering each player equally. Here, just one such collection suffices. The proof uses linear-programming duality and an “uncrossing” argument that restricts which coalitions need positive weight in a suitably chosen optimal dual solution.

Existence does not rescue the ordinary allocation shortcut. In the paper’s three-player example, individuals are worth zero, every pair is worth two, and all three together are worth three. The unique core allocation gives each player one. Every sequential marginal-contribution allocation instead assigns some permutation of zero, two and one, leaving a pair underpaid.

The authors therefore construct a modified greedy algorithm. It builds allocations through successively larger groups, adjusts payments when necessary, and explicitly handles intermediate groups whose cores are empty. It uses O(n²) coalition-worth evaluations and arithmetic operations. Earlier work already supplied a polynomial-time route through the ellipsoid method; the advance is a strongly polynomial combinatorial construction, not the first efficient solvability result.

Applications, with boundaries

One application models differentiated-product firms competing on prices, with identical initial marginal costs and mergers generating coalition-specific cost reductions that do not decrease as coalitions expand. A coalition’s worth is its maximum guaranteed profit against outsiders choosing prices to minimize that profit—not its profit against ordinary equilibrium responses.

With identical cost reductions across merging coalitions, the authors derive a higher threshold for retaining 3-distant convexity than for retaining full convexity. With heterogeneous reductions, their sufficient conditions require additional restrictions on how synergy increments vary. Importantly, equal profit-sharing already lies in this oligopoly model’s core. The algorithm’s added role is to construct allocations responsive to coalition-specific synergies.

The second application assigns each worker’s indivisible resource to one of two complementary tasks, with output (x₁x₂)^r and 1/2 < r < 1. The general 3-distant-convexity result requires r sufficiently close to one and resources sufficiently similar: any two workers’ resources together must exceed any third worker’s.

These qualifications matter. The theory neither guarantees stability for unrestricted production technologies nor predicts actual bargaining outcomes. Its computational bound counts access to coalition values; it does not make those values cheap to calculate. The contribution is narrower and precise: within a specified class, local failures of convexity leave both stability testing and stable allocation tractable.