A theory of unstructured bargaining using distribution-valued solution concepts

In experiments it is typically found that many joint utility outcomes arise in any given unstructured bargaining game. This suggests using a positive unstructured bargaining concept that maps a bargaining game to a probability distribution over outcomes rather than to a single outcome. We show how to "translate" Nash's bargaining axioms to apply to such distributional bargaining concepts. We then prove that a subset of those axioms forces the distribution over outcomes to be a power-law. Unlike Nash's original result, our result holds even if the feasible set is finite. When the feasible set is convex and comprehensive, the mode of the power law distribution is the Harsanyi bargaining solution, and if we require symmetry it is the Nash bargaining solution. However in general these modes of the joint utility distribution are not Bayes-optimal predictions for the joint uitlity, nor are the bargains corresponding to those outcomes the most likely bargains. We then show how an external regulator can use distributional solution concepts to optimally design an unstructured bargaining scenario. Throughout we demonstrate our analysis in computational experiments involving flight rerouting negotiations in the National Airspace System.