An approach to solution uniqueness in game problems
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AN APPROACH TO SOLUTION UNIQUENESS IN GAME PROBLEMS1
UDC 517.9
E. R. Smol’yakov
A new concept of strong conflict equilibrium is proposed that supplements the well-known fundamental system of conflict equilibria and considerably increases the possibility of finding a unique strongest equilibrium (solution) in any game problem. The efficiency of this new equilibrium is illustrated by static and dynamic game problems. Keywords: conflict equilibrium, games and the solution uniqueness problem. INTRODUCTION In this paper, a new concept of equilibrium is proposed for any game problems (in particular, for any conflict problems such as antagonistic, noncooperative, cooperative, static, and dynamic ones) that considerably extends the possibilities of finding a unique strongest equilibrium (solution). The place of this equilibrium among the hierarchical series of well-known equilibria is determined and its high efficiency is illustrated by static and dynamic model problems from the viewpoint of resolution of the solution uniqueness problem. It is proved in the theory of conflict equilibria [1] containing an extensive basic system of concepts of equilibrium (in contrast to the classical game theory in which problems of neither solution existence nor solution uniqueness of any game problems were not solved as a result of restricted capabilities of classical concepts of equilibrium and insufficiency of their number [2–4]) that any conflict problem always have a solution and, in most cases (except for cases of some symmetry in problems), this solution must be unique. This theory provided a mechanism of finding solutions to any problem. Nevertheless, the uniqueness problem (when any symmetry in a problem is absent) remains uncompletely solved, possibly, since a complete system of basic equilibria is not found till now. In this work, a new equilibrium concept is proposed that supplements already well-known concepts, that is illustrated by examples, and that turns out to be very useful in searching for a unique solution. However, its determination in problems is more labor-consuming than the determination of any well-known equilibrium. In this connection, it is recommended to use this equilibrium in practice only when all well-known types of equilibria are found and a unique strongest equilibrium is not found with their help. The possibilities of finding a unique strongest equilibrium are illustrated by examples of static game problems and dynamic game problems (differential games). PROBLEM STATEMENT AND A BASIC SYSTEM OF EQUILIBRIA We formulate some equilibria from the basic system [1] in a general form. Without them, it is impossible to understand the meaning and role of the proposed new concept of D ¢-equilibrium. Without loss of generality, we restrict ourselves to the following assumption.
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This work was carried out under the program “Basic foundations of information technologies and systems” of the Russian Academy of Science (Project No. 1–3). M. V. Lomonosov Moscow State University, Moscow, Russia, [email protected].
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