Equilibrium computation of the Hart and Mas-Colell bargaining model

•Equilibria of the two-player Hart & Mas-Colell bargaining are efficiently computable.•Players in Prisoners’ Dilemma can cooperate in Hart & Mas-Colell bargaining model.•Quantitative efficiency, i.e. PoA, is analyzed.

The 8-th problem raised by [Hart, S., Mas-Colell, A., 2010. Bargaining and cooperation in strategic form games. Journal of the European Economics Association 8 (1), 7–33], is solved. To be specific, I show that the set of SP equilibria can be determined by a finite number of systems of linear inequalities, which are efficiently solvable when there are two players. This is more or less surprising because the Hart and Mas-Colell bargaining model and the SP equilibrium both seem to be rather complicated, and it is well known that an arbitrary Nash equilibrium is hard to compute, even when there are only two players. Using this algorithm, it is shown that players of Prisoners’ Dilemma can cooperate to some extent in the Hart and Mas-Colell bargaining, and full cooperation is attainable as ρρ, a parameter of this model, approaches to 1. Quantitative efficiency, i.e. price of anarchy, is also analyzed.