Predicting Behavior in Unstructured Bargaining with a Probability Distribution

25 Pages Posted: 16 Nov 2010 Last revised: 3 Apr 2013

See all articles by David Wolpert

David Wolpert

Santa Fe Institute

James Bono

Microsoft Corporation

Date Written: November 15, 2010


In experimental tests of human behavior in unstructured bargaining games, typically many joint utility outcomes are found to occur, not just one. This suggests we predict the outcome of such a game as a probability distribution. This is in contrast to what is conventionally done (e.g, in the Nash bargaining solution), which is predict a single outcome. We show how to translate Nash’s bargaining axioms to provide a distribution over outcomes rather than a single outcome. We then prove that a subset of those axioms forces the distribution over utility outcomes to be a power-law distribution. 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 the experimentalist’s Bayes-optimal predictions for the joint utility. Nor are the bargains corresponding to the modes of those joint utility distributions the modes of the distribution over bargains in general, since more than one bargain may result in the same joint utility. After introducing distributional bargaining solution concepts, we show how an external regulator can use them to optimally design an unstructured bargaining scenario. Throughout we demonstrate our analysis in computational experiments involving flight rerouting negotiations in the National Airspace System. We emphasize that while our results are formulated for unstructured bargaining, they can also be used to make predictions for noncooperative games where the modeler knows the utility functions of the players over possible outcomes of the game, but does not know the move spaces the players use to determine those outcomes.

Keywords: Nash, Bargaining, PGT, Distribution-valued solution concept, Bargaining management

JEL Classification: C7

Suggested Citation

Wolpert, David and Bono, James, Predicting Behavior in Unstructured Bargaining with a Probability Distribution (November 15, 2010). Available at SSRN: or

David Wolpert (Contact Author)

Santa Fe Institute ( email )

1399 Hyde Park Road
Santa Fe, NM 897501
United States

James Bono

Microsoft Corporation ( email )

One Microsoft Way
Redmond, WA 98052
United States

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