Does the grand coalition form? Persistence, arrival, and the role of the sharing rule in a dynamic process of nested binding agreements

Abstract

We study a dynamic coalition-formation process in the tradition of Konishi and Ray (2003): players repeatedly form and dissolve binding agreements, evaluate states by discounted long-term expected payoffs, and hold self-confirming beliefs about the process. States and payoff sharing follow Heitzig and Kornek (2018): a state is a hierarchy of nested agreements, and the members of a new agreement share the surplus it generates, measured against the state without that agreement. All payoff assumptions are structural. We prove that every grand state ever reached is absorbing, and that every absorbing state is grand, for every discount factor. A grand state is actually reached, almost surely, for small discount factors, for three players, and, at every discount factor, whenever distributional stakes are smaller than each player s share of the efficiency gain. Otherwise the process can fail only by cycling for ever among non-grand states. We give exact necessary conditions on such a cycle, decidable for a fixed candidate cycle by linear programming, and exhibit, under an earlier and weaker notion of profitability, a four-player payoff structure whose only closed class is a cycle of two pairs forming and dissolving alternately. Under the present definition, and under either termination rule, no equilibrium traverses a cycle on a fixed schedule: somebody always reaches a state they would rather not leave, and the axioms give them the floor. Whether arrival can fail by cycling at random is open. The axioms are not merely postulated: we exhibit a bargaining game proposal, amendment by substitutes voted on by their own signatories, final unanimity, and an arbitrarily small delay on failure whose equilibria satisfy them as the delay vanishes, and which settles each period in its first round.