, Volume 45, Issue 3, pp 201240
First online:
LongTerm Behavior in the Theory of Moves
 Stephen J. Willson
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This paper proposes a revised Theory of Moves (TOM) to analyze matrix games between two players when payoffs are given as ordinals. The games are analyzed when a given player i must make the first move, when there is a finite limit n on the total number of moves, and when the game starts at a given initial state S. Games end when either both players pass in succession or else a total of n moves have been made. Studies are made of the influence of i, n, and S on the outcomes. It is proved that these outcomes ultimately become periodic in n and therefore exhibit longterm predictable behavior. Efficient algorithms are given to calculate these ultimate outcomes by computer. Frequently the ultimate outcomes are independent of i, S, and n when n is sufficiently large; in this situation this common ultimate outcome is proved to be Paretooptimal. The use of ultimate outcomes gives rise to a concept of stable points, which are analogous to Nash equilibria but consider longterm effects. If the initial state is a stable point, then no player has an incentive to move from that state, under the assumption that any initial move could be followed by a long series of moves and countermoves. The concept may be broadened to that of a stable set. It is proved that every game has a minimal stable set, and any two distinct minimal stable sets are disjoint. Comparisons are made with the results of standard TOM.
 Title
 LongTerm Behavior in the Theory of Moves
 Journal

Theory and Decision
Volume 45, Issue 3 , pp 201240
 Cover Date
 199812
 DOI
 10.1023/A:1004946714084
 Print ISSN
 00405833
 Online ISSN
 15737187
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 Game theory
 Noncooperative games
 Theory of Moves (TOM)
 Prisoner's Dilemma
 Stable set
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