Judicial precedent as a dynamic rationale for axiomatic bargaining theory

Axiomatic bargaining theory (e.g., Nash's theorem) is static. We attempt to provide a dynamic justification for the theory. Suppose a judge or arbitrator must allocate utility in an (infinite) sequence of two-person problems; at each date, the judge is presented with a utility possibility set in ℝ₊². He/she must choose an allocation in the set, constrained only by Nash's axioms, in the sense that a penalty is paid if and only if a utility allocation is chosen at date T that is inconsistent, according to one of the axioms, with a utility allocation chosen at some earlier date. Penalties are discounted with t and the judge chooses any allocation, at a given date, that minimizes the penalty he/she pays at that date. Under what conditions will the judge's chosen allocations converge to the Nash allocation over time? We answer this question for three canonical axiomatic bargaining solutions-Nash, Kalai-Smorodinsky, and "egalitarian"-and generalize the analysis to a broad class of axiomatic models.