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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...

Two solution concepts for cooperative games in characteristic-function form, the kernel and the nucleolus, are studied in their relationship to a number of other concepts, most notably the core. The unifying technical idea is to analyze the behavior of the strong ϵ-core as ϵ varies. One of the central results is that the portion of the prekernel that falls within the core, or any other strong ϵ-core, depends only on the latter's geometrical shape. The prekernel is closely related to the kernel and often coincides with it, but has a simpler definition and simpler analytic properties. A notion of “quasi-zero-monotonicity” is developed to aid in enlarging the class of games in which kernel considerations can be replaced by prekernel considerations. The nucleolus is approached through a new, geometrical definition, equivalent to Schmeidler's original definition but providing very elementary proofs of existence, unicity, and other properties. Finally, the intuitive interpretations of the two solution concepts are clarified: the kernel as a kind of multi-bilateral bargaining equilibrium without interpersonal utility comparisons, in which each pair of players bisects an interval which is either the battleground over which they can push each other aided by their best allies (if they are strong) or the no-man's-land that lies between them (if they are weak); the nucleolus as the result of an arbitrator's desire to minimize the dissatisfaction of the most dissatisfied coalition.

Covering both noncooperative and cooperative games, this comprehensive introduction to game theory also includes some advanced chapters on auctions, games with incomplete information, games with vector payoffs, stable matchings and the bargaining set. Mathematically oriented, the book presents every theorem alongside a proof. The material is presented clearly and every concept is illustrated with concrete examples from a broad range of disciplines. With numerous exercises the book is a thorough and extensive guide to game theory from undergraduate through graduate courses in economics, mathematics, computer science, engineering and life sciences to being an authoritative reference for researchers.

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