Paths leading to the Nash set for nonsmooth games

Maschler, Owen and Peleg (1988) constructed a dynamic system for modelling a possible negotiation process for players facing a smooth n-person pure bargaining game, and showed that all paths of this system lead to the Nash point. They also considered the non-convex case, and found in this case that the limiting points of solutions of the dynamic system belong to the Nash set. Here we extend the model to i) general convex pure bargaining games, and to ii) games generated by "divide the cake" problems. In each of these cases we construct a dynamic system consisting of a differential inclusion (generalizing the Maschler-Owen-Peleg system of differential equations), prove existence of solutions, and show that the solutions converge to the Nash point (or Nash set). The main technical point is proving existence, as the system is neither convex valued nor continuous. The intuition underlying the dynamics is the same as (in the convex case) or analogous to (in the division game) that of Maschler, Owen, and Peleg.