A VALUE FOR n-PERSON GAMFB

L. S. Shapley

INTROOOCTION

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At the foundation of the theory of games is the assumption that

the players f a game can evaluate, in their utility scales, every "pros-pect" that ght arise as a result of a play. In attempting to apply the

theory to an fieid, on~ would n,ormally expect to be permitted to include,in the class of "prospects," the prospect of having to playa game. Thepossibility f evaluating games is therefore of critical importance. Solong as the heory is unable to assign values to the games typically foundin applicati n, only relatively simple situatlons --where games do not

t d~end on ot er games --will be susceptible to analysis and solution.2

In the fln1te theory of von Neumann and Morgenstern difficultyin evaluatio perslsts for the "essential" games, and for only those. Inthis note we deduce a value for the "essential" case and examine a number

of its eleme tary properties. We proceed from a set of three axioms,i,J having simpl intuitive interpretations, which suffice to determine the!t,1

value unique y.~J present work, though mathematlcally self-contained, is founded'1Ii

conceptual on the von Neumann-Morgenstern theory up to their introductlon~ of charact stic functlons. We thereby iriheri t certain important under-:

lylilg assump ions: (a) that utllity is objective and transfer~ble;j (b) that g es are cooper~ti~e affairs; (c) that games, ~anting (a) and

~ (b), are ade uately represented by their characteristic functions. However,~ we are not c mmitted to the assumptlons rega~ing rational behavior embodied" 'j in the von N umann-Morgenstern notion of "solut~on."j -~ W shall think of a "game" as a set of rules with specified1

players in t e playing positions. The rules alone describe what we shall

1 tThe prepar tion of thi~ paper was sponsored (in part) by the RANDCorporation.

2Reference [3] at the ~nd of ,this paper. Examples of infinite games with-out values y be found in [2), pp. 58-59,. and in [1], p. 110. See alsoKarlin [2]; p. 152-r~~. Jj'

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317A VALUE FOR n-PERSON GAMES-'

BIBLIOGRAPKi

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BOREL, E. and VllJ..E, J., ..Applications aux j eux de hasard," Trai te duCalcul des Probabilit@s et de ses Applications, vol. 4, part 2 (paris,

Gauthier-Villars, 1938).KUHN, H. W. and TUCKER, A. W., eds., Contribut,ions to the Theory of

Games (Annals of Mathematics Study No. 24), Princeton, 1950.

von NEUMANN, J. and MdRGENSTERN, 0., Theory of Games and Economic

Behavior, Princeton 1944, 2nd ed. 1947.

SHAPLEY, L. S., "Quota solutions of n-person games," this Study.[4]

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