Existence of stable payoff configurations for cooperative games
1963; American Mathematical Society; Volume: 69; Issue: 1 Linguagem: Inglês
10.1090/s0002-9904-1963-10879-4
ISSN1088-9485
AutoresMorton D. Davis, Michael Maschler,
Tópico(s)Optimization and Variational Analysis
Resumo1. Let T be an w-person cooperative game with a (not necessarily superadditive) characteristic function v(B). We assume that T is normalized so that v(B)^0 for each coalition B, and v(i)~0 for i = l , 2, • • • , n. Let B = 5 i , J52, • • • , Bm be a coalition structure, i.e., a partition of the set N= {1, 2, • • • , into m nonempty coalitions. An outcome of the game with this coalition structure can be represented by a payoff configuration (x; B), where the payoff vector X2=(#i, #2, • • • , xn) represents the amount which the players receive. If we restrict ourselves to individually rational payoff configurations (i.r.p.c/s), i.e., to payoff configurations with x ^ O coordinate wise, then x must lie in the space Z ( B ) s 5 i X 5 , 2 X • • • XSm) where Sj = {x B i ^{x k }keBj:x k ^0 and ]£*€£,• xk = v(Bj)} are geometric simplices for j = l , • • • , m. Let (x; B) be an i.r.p.c. for a game I and let v and \x be two distinct players in a coalition Bj of B. An objection of v against fx in (x; B) is a vector y , where C is a coalition containing player v but not player jx, whose coordinates {yu}, kCiC, satisfy: yv>xv, yk^Xk a n d Hkecyk = v(C). A counter objection to this objection is a vector Z, where D is a coalition containing player ix but not player *>, whose coordinates {zk\, feG-D, satisfy: Zk^Xk for each fe in JD, z*E;y* for each k in CP\P, X^eD Zfc = z>(.D). DEFINITION. We shall say that player v is stronger than player /x (or, equivalently, tha t player \x is weaker than player J>), in (x; B), if v has an objection against fxt which cannot be countered. We denote this by v>ju. We shall say that both players are equal, and write v^p, if neither v> fx nor JX> v. REMARK. By definition, v~fx in (x; B) if v and jx belong to different coalitions of B. DEFINITION. A coalition Bj in B will be called stable in (x; B), if each two of its members are equal. DEFINITION. An i.r.p.c. (x; B) is called stable if each coalition in B is stable in (x; B). The set of all the stable i.r.p.c.'s is called the bargaining set M f of
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