This paper is concerned with a class of noncooperative games ofn players that are defined byn reward functions which depend continuously on the action variables of the players. This framework provides a realistic model of many interactive situations, including many common models in economics, sociology, engineering, and political science. The concept of Nash equilibrium is a suitable companion to such models.
A variety of different sufficient conditions for existence, uniqueness, and stability of a Nash equilibrium point have been previously proposed. By sharpening the noncooperative aspect of the framework (which is really only implicit in the original framework), this paper attempts to isolate one set of “natural” conditions that are sufficient for existence, uniqueness, and stability. It is argued thatl ∞ quasicontraction is such a natural condition. The concept of complete stability is introduced to reflect the full character of noncooperation. It is then shown that, in the linear case, the condition ofl ∞ quasicontraction is both necessary and sufficient for complete stability.
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This research was supported by the Air Force Office of Scientific Research under Grant No. AFOSR 77-3141 and by the National Science Foundation under Grant No. GK18748.
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Luenberger, D.G. Complete stability of noncooperative games. J Optim Theory Appl 25, 485–505 (1978). https://doi.org/10.1007/BF00933516
- Game theory
- contraction mappings