Econometrica: Jul, 1965, Volume 33, Issue 3

Existence and Uniqueness of Equilibrium Points for Concave N-Person Games

https://doi.org/0012-9682(196507)33:3<520:EAUOEP>2.0.CO;2-2
p. 520-534

J. B. Rosen

A constrained n-person game is considered in which the constraints for each player, as well as his payoff function, may depend on the strategy of every player. The existence of an equilibrium point for such a game is shown. By requiring appropriate concavity in the payoff functions a concave game is defined. It is proved that there is a unique equilibrium point for every strictly concave game. A dynamic model for nonequilibrium situations is proposed. This model consists of a system of differential equations which specify the rate of change of each player's strategy. It is shown that for a strictly concave game the system is globally asymptotically stable with respect to the unique equilibrium point of the game. Finally, it is shown how a gradient method suitable for a concave mathematical programming problem can be used to find the equilibrium point for a concave game.