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Solutions to Cooperative Games Without Side Payments

1963; American Mathematical Society; Volume: 106; Issue: 2 Linguagem: Inglês

10.2307/1993770

1088-6850

Bezalel Peleg,

Digital Platforms and Economics

An extension of Von Neumann Morgenstern solution theory to cooperative games without side payments has been outlined in [1].In this paper we revise some of the definitions given in [1] and prove that in the new theory every threeperson constant sum game is solvable (see [1, Theorem 1]).Other results that were formulated in [1] had already been proved in [2].[1 ; 2] are also necessary for a full understanding of the basic definitions of this paper.1. Basic definitions.If N is a set with n members, we denote by F^the n-dimensional euclidean space the coordinates of whose points are indexed by the members of N. Subsets of N will be denoted by S. If x e EN and i e N, x' will denote the coordinate of x corresponding to i; xs will denote the set {xl: ieS}.The superscript JV will be omitted, thus we write x instead of xN.We write x s ^ y s if x ' 2: y' for all ieS; similarly for > and = .0 denotes the empty set.Definition 1.1.An n-person characteristic function is a pair iN,v) where N is a set with n members, and v is a function that carries each S <= N into a set v (S) <= EN so that(1) viS) is closed,(2) v (S) is convex,(3) vi0) = E\ (4) if xeviS) and xs ^ y s then ye viS).Definition 1.2.An n-person game is a triad iN,v,H), where iN,v) is an nperson characteristic function and H is a convex compact subset of t>(iV).We notice that this definition is not identical with that given in [1 ; 2].In the first place v is not assumed to be superadditive, i.e., the condition: viSy US2) => viSy) C\viS2) for every pair of disjoint coalitions Sy and S2 is dropped.Secondly H need not be a polyhedron.2. Solutions.Let G = (N, v, H) be an n-person game.Definition 2.1.Let x,yeEN, S ^ 0. x dominates y via S, written x^-sy, if x e d(S) and xs > ys.Definition 2.2.x dominates y, written x ^y, if there is an S such that x J=~s y.For xeF^the following sets are defined: domsx = {y : x^sy} and dorn x = {y: x^-y}.LetK <=EN.We define domsK = |JX6KdomsxanddomK = (JxeKdomx.