When coalition bargaining has one payoff prediction—and when it has several

Coalition bargaining models describe situations in which participants must choose both their partners and how to share the gains. If a model permits several equilibrium payoff vectors, the same underlying circumstances can support different predicted distributions. Comparing outcomes after a change in productivity or bargaining rules then requires a way to select among equilibria.

In Multiplicity and Uniqueness of Stationary Expected Payoffs in Coalitional Bargaining, Vanderbilt University economist Kirill S. Evdokimov identifies conditions that remove this ambiguity. The theoretical preprint establishes a uniqueness result, but also supplies counterexamples showing that agreement without delay is not enough. These are mathematical findings, not empirical estimates of business negotiations or political bargaining.

How the bargaining works

In the model, one player is randomly selected to make a proposal each period. Players can have different selection probabilities and different discount factors, which determine how they value future consumption. The proposer can pass or suggest a feasible coalition containing herself, together with an allocation of its surplus.

Coalition members vote simultaneously. Agreement requires all of them to accept; otherwise bargaining continues. Once a coalition forms, the game ends, rather than allowing additional coalitions to form afterward.

The surplus is transferable: it can be divided among participants. A coalition's total surplus does not depend on which member proposes it. However, adding a member can reduce that surplus or make a coalition infeasible.

The analysis focuses on stationary equilibria, whose strategies do not change with bargaining history. A prospective partner's discounted continuation payoff—the value of moving to another bargaining round—acts as the price of recruiting that partner. Proposers choose coalitions that maximize the surplus left after accounting for members' continuation values.

Immediate agreement can still leave several answers

One three-player example produces three equilibrium payoff vectors through feedback involving delay. A player whose participation reduces coalition surplus may pass. Passing preserves the other players' continuation opportunities, potentially increasing their recruitment prices and reinforcing the incentive to pass.

A second three-player example also has exactly three equilibrium payoff vectors, despite every equilibrium featuring immediate agreement. Here, adding members strictly increases coalition surplus, and every nonempty coalition is feasible.

In that example, one player is selected to propose much more frequently than the others. Changes in coalition choices alter recruitment prices, which in turn reinforce those choices. The paper shows that multiplicity persists under sufficiently small parameter changes, including changes that break symmetry between two players. Thus it is not confined to one precisely balanced numerical specification.

The condition that delivers uniqueness

The main theorem combines two requirements. First, every equilibrium must make passing strictly suboptimal for every proposer. Second, coalition surplus must satisfy a mathematical exchange condition called M-natural-concavity.

Roughly, this condition says that a member can move between two feasible coalitions, possibly in exchange for another member, without reducing their combined surplus. It restricts both coalition feasibility and coalition values. It does not require surplus to increase whenever someone joins.

The proof represents equilibrium payoffs as fixed points: continuation values must reproduce themselves through optimal proposals and recruitment probabilities. Fixed-point index theory supplies a global accounting constraint. Evdokimov shows that each qualifying equilibrium payoff vector is isolated and contributes a local index of positive one; because the total index is one, only one payoff vector can remain.

The difficult cases involve proposers indifferent between several coalitions. M-natural-concavity supplies an “equitable grouping” structure that controls how tied coalition choices differ. This is a technical property of recruitment patterns, not a claim that players receive equal payoffs.

The unique payoff vector also varies continuously with model parameters within the theorem's domain. Unique payoffs do not, however, imply unique proposal strategies or a uniquely determined coalition.

Applications and boundaries

One application models joint ventures whose productive contributions are offset by coordination costs across nested groups of firms. Another values governing coalitions through the best assignment of parties to cabinet positions, given their expertise. These specifications illustrate where the theorem applies; they are not tests against observed ventures or governments.

The paper also proves conditional comparative results. Uniformly increasing a player's contribution to every coalition containing her strictly raises her continuation payoff, by no more than the surplus increase. Increasing her discount factor weakly raises that payoff. Both results require the parameter changes to preserve immediate agreement in every equilibrium.

Importantly, the patience result concerns discounted continuation payoffs. The paper's separately defined ex ante payoff can fall as patience increases. More broadly, the uniqueness theorem does not cover arbitrary coalition values, history-dependent strategies or general delay equilibria. As a preprint, its contribution remains a theoretical result awaiting further scrutiny, rather than a universal prediction about who wins negotiations.