Nash, pareto, and stackelberg solutions in non-antagonistic two-person games

Abstract

Non-antagonistic differential games are formalized below on the basis of the formal theory of positional antagonistic differential games /1, 2/. There is a large literature (for part of it, see survey /3/) concerned with the types of solution indicated in the title. The Nash solution is best known in coalitionless games; Pareto optimality is a basic concept in cooperative games without collateral pay-offs; and finally, the Stackelberg solution /4, 5/ is typical for hierarchical games. All three types of solution are considered below in a unified approach. Our concept of a Pareto solution differs from the usual one in allowing for the individual scope of each player. A single structure of strategies for all types of solution is discovered for two-person games. Relations between the sets of solutions of different types are established. It is shown that the set of solutions of each type is characterized by the solutions of appropriate non-standard (optimal) control problems. The results are illustrated by the example of the plane motion of a material particle subject to the total action of control forces generated by the different players. The paper is related to /6–8/ and continues the studies of /9, 10/. (See also: A.F. Kleimenov. On the theory of hierarchical, two-person differential games, Preprint Inst, matematika i Mekhanika, UNTs AN SSSR, Sverdlovsk, 1985).