The kernel of a cooperative game

Abstract

M o r t o n D a v i s T h e C i t y U n i v e r s i t y of N e w Y o r k and M i c h a e l M a s c h l e r T h e H e b r e w U n i v e r s i t y J e r u s a l e m , I s r a e l ABST RAG T The kernel of a cooperative n -p e r s o n game i s defined. It i s a subs e t of the bargaining s e t m:i). Its existence and some of i t s p r o p e r t i e sa r e studied. We apply it to the 3-person games, to the 4-person constants u m games, to the s y m m e t r i c and n-quota g a m e s and to games in which only the n and the (n-1)-person coalitions a r e allowed to be non-flat.In o r d e r to illustrate i t s m e r i t s and d e m e r i t s a s a predictor of an actual outcome in a real-life situation, we exhibit an example in which the k e r n e l prediction s e e m s frustrating. The opinions of other authors a r e quoted in o r d e r to throw some light on this interesting example.authors were staying at the Econometric R e s e a r c h P r o g r a m , Princeton University, Princeton, New J e r s e y .tA coalition is called flat if its value is equal to the sum of the values of its m e m b e r s -considered a s 1-person coalitions. 223x. 2 0, i = 1,2,. . . , n (individual rationality)The symbol (x; 8) will be called an individually rational payoff configuration (i.r.p.c. o r P.c.).If we fix the coalition structure 8, then the set of all the payoffs X, satisfying (2.5) and (2.6), is a Cartesian product of m simplices:(2.7) 1 2 r 1 j = 1 , 2 ,..., m . DEFINITION 2.1: Let (x;%) be an i.r.p.c. for a game r, and let D be an arbitrary coalition. The excess of D with respect to (x;$) is (2.9) e(D) = v(D)xi. ie D The excess of D therefore represents the total amount that the members of D gain (or lose, i f e(D) < O), i f they withdraw from (x;B) and form the coalition D. Clearly, (2.10) e(Bj) = 0, j = 1,2,. . . , m .Let k and P be two distinct players in a coalition B. of 8; we denote by 3 the set J k, P of all the coalitions which contain player k but do not contain player 1 ; i.e., (2.11) 5 {DID CN, keD,P(D}. DEFINITION 2.2: Let (x;%) be an i.r.p.c. for a game r, and let k and P be two distinct players in a coalition B. of 53. The maximum surplus of k over 1 with respect to (x;8) is J (2.12) % Max e(D). DE ' k,P 9The maximum surplus, therefore, represents the maximal amount player k can gain (or the minimal amount that he must lose), by withdrawing from (x;%) and joining a coalition D which does not require the consent of P (since PbD), with the understanding that the other members of D will be satisfied with getting the same amount they had in (x;53). DEFINITION 2.3: JLet (x; 8) be an i.r.p.c. for a game r, and let k,P be two distinct players in a coalition B. of $3. Player k is said to outweigh player P with respect to (x; 53), 226 M. DAVIS AND M. MASCHLER and this is denoted by k >> Q, or, equivalently, by Q << k, if (2.13)If neither k >> P nor Q >> k, we say that k and Q are in equilibrium. For the sake of completeness we define each player to be in equilibrium with himself. Similarly, we also regard any two players, who...

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