Stabilizing Cooperative Outcomes in Two-Person Games: Theory and Cases

We analyze the 78 2 x 2 distinct strict ordinal games, 57 of which are conflict games that contain no mutually best outcome. In 19 of the 57 games (33%), including Prisoners’ Dilemma and Chicken, a cooperative outcome—one that is at least next-best for each player—is not a Nash equilibrium (NE). But this outcome is a nonmyopic equilibrium (NME) in 16 of the 19 games (84%) when the players start at this outcome and make farsighted calculations, based on backward induction; in the other three games, credible threats can induce cooperation. In two of the latter games, the NMEs are “boomerang NMEs,” whereby players have an incentive to move back and forth between two diagonally opposite NMEs, one of which is cooperative. In Prisoners’ Dilemma, the NE and one NME are not Pareto-optimal, but we conjecture that in all two-person games with strict preferences, there is at least one Pareto-optimal NME. As examples of NMEs that are not NEs, we analyze two games that plausibly model the choices of players in international relations: (i) no first use of nuclear weapons, a policy that has been adopted by some nuclear powers; and (ii) the 2015 agreement between Iran, and a coalition of the United States and other countries, that has forestalled Iran’s possible development of nuclear weapons.