A mathematical bridge between stable cooperation and asset pricing

What makes cooperation stable?

A cooperative game assigns a value to each admissible coalition of players: the amount that coalition can obtain together. With transferable utility, these values can be allocated among participants. The central question is whether the value produced by the entire group can be distributed without leaving any coalition less than it could obtain on its own.

The set of allocations satisfying those requirements is called the core. An allocation in the core distributes exactly the grand coalition’s value and gives every admissible coalition at least its own value. If the core is empty, no allocation meets all those requirements simultaneously.

In their preprint, Bernard Cornet and Miklós Pintér consider games whose player set may be infinite. Allocations are represented by bounded, finitely additive set functions: coalition-level assignments that add across disjoint coalitions. Finite additivity does not require the stronger property of additivity over countably infinite collections.

Their contribution is a new derivation of an existing characterization of when this core is nonempty, rather than a newly discovered criterion for cooperation.

Balancedness prevents conflicting demands

The relevant criterion is balancedness. Imagine assigning nonnegative weights to a finite collection of coalitions so that each player’s combined participation weight is exactly one. Balancedness requires the weighted sum of those coalitions’ values to be no greater than the value of the entire group.

This tests whether overlapping coalitions make collectively excessive demands. If an allocation satisfies every coalition, a weighted collection covering everyone exactly once cannot require more than the available total.

The infinite-player theorem recovered by the authors says that, for games bounded below, balancedness is both necessary and sufficient for a nonempty core. Bounded below means there is a single finite lower limit on the values of all admissible coalitions.

The authors also use a reversed construction, the anti-core, whose allocations give each coalition no more than its value while still matching the grand coalition’s total. Its corresponding condition, anti-balancedness, reverses the weighted inequality. For games bounded above, anti-balancedness characterizes a nonempty anti-core.

Turning coalitions into payoffs

The proof connects these game-theoretic objects to asset pricing through the Choquet extension. This construction turns a function defined on coalitions into a function defined on simple payoffs—payoffs taking only finitely many distinct values.

A coalition’s indicator payoff is one on that coalition and zero elsewhere. The extension assigns this payoff the coalition’s game value. More general simple payoffs are decomposed into a constant baseline and successive layers defined by their payoff thresholds. Coalition values then determine a price for each layer.

The resulting pricing rule need not be linear. The authors apply a theorem from Cornet’s earlier work that supplies a continuous linear valuation lying below such a rule, provided the rule satisfies a particular no-arbitrage condition and is bounded above near zero.

Here, no-arbitrage has a precise meaning: a finite portfolio assembled with nonnegative weights, whose combined future payoff is zero, cannot have a negative combined cost today. It is this stated condition—not an unrestricted claim about all financial markets—that drives the proof.

How the proof crosses the bridge

The authors establish three links. First, an upper bound on coalition values gives the Choquet extension the local upper bound required by the pricing theorem. Second, anti-balancedness implies the specified no-arbitrage condition. Third, the resulting continuous linear valuation defines a bounded, finitely additive allocation in the anti-core.

To obtain the usual core result, they apply this reasoning to the dual game. That game values a coalition by subtracting its complement’s value from the grand coalition’s value. Duality exchanges the core and anti-core and turns the original lower-bound assumption into an upper-bound assumption.

The technical choice of simple payoffs matters. An earlier asset-pricing derivation required bounded variation of the game and a sigma-algebra of coalitions. This proof removes those requirements: coalitions need only form a field, closed under complements and finite unions, rather than countable unions.

What the result does not establish

The remaining boundedness assumption cannot simply be discarded. The paper gives an example on the positive integers in which admissible coalitions are finite or have finite complements. Finite coalitions receive their size as their value; coalitions with finite complements receive minus the size of that complement. The grand coalition has value zero.

This game is additive and balanced but unbounded both above and below, and its core is empty under the paper’s bounded-allocation definition. Balancedness alone therefore does not guarantee the desired allocation in this setting.

The work is a mathematical preprint, not an empirical study or a causal analysis. It does not test actual bargaining or trading behavior. Its achievement is to show how an established asset-pricing existence theorem can recover a general cooperative-game existence theorem, making their shared mathematical structure explicit.