Contingent Transfers as an Incentive for Cooperation in Noncooperative Games

Consider a noncooperative game at whose outcomes commodities accrue to players, which are both valued by them and susceptible to transfer between them. In this situation, outcome-contingent transfers of commodities form a natural schema for incentivizing cooperation, in the following sense. Transfers agreed upon by the players give rise to a new noncooperative game whose Nash Equilibria (NE) may engender a Pareto-improvement over some designated status quo NE of the original game. However, as in the folk theorem for repeated games, an embarrassingly large set of NE may be sustainable via transfers. The main source for this pathology is the possibility of “threats†, which can be understood as “off-shell†transfers, i.e. transfers at outcomes that are not actually being reached with positive probability at the NE under consideration. Our endeavor is to restore the discriminatory nature of NE by means of two simple ideas. We say that an NE of the post-transfers game is transparent if there are no off-shell transfers. This can also be viewed via the lens of “credibility†, of something being seen in order to be believed. The other condition is budget balance, i.e., at the NE, there should be no need for a net injection of commodities from the outside in order to sustain transfers. An NE is considered eligible by us if it satisfies these two conditions, in addition to engendering Pareto-improvement. If a social welfare function is specified, then one may seek to determine eligible NE that maximize welfare on the domain of feasible transfers. We carry out this analysis for three classical noncooperative games. These are the Centipede Game, Contest, and the Prisoners’ Dilemma; the last of which we discuss in some detail.