On the core,the Weber set and convexity in games with a priori unions

This paper deals with the concepts of core and Weber set with a priori unions à la Owen. As far as we know, the Owen approach to games with a priori unions has never been studied from the coalitional stability point of view. Thus we introduce the coalitional core and coalitional Weber set and characterize the class of convex games with a priori unions by means of the relationships between both solution concepts.     

The not-quite non-atomic game: Non-emptiness of the core in large production games

We show that, in cooperative production games, when the production functions are not concave, the core may well be empty. However, as the number of players increases (subject to some regularity conditions), the relative deficit obtained by using concavified functions decreases to zero. Furthermore, differentiability of the functions will cause the absolute deficit to go to zero.     

Values for strategic games in which players cooperate

In this paper we propose a new method to associate a coalitional game with each strategic game. The method is based on the lower value of finite two-player zero-sum games. We axiomatically characterize this new method, as well as the method that was described in Von Neumann and Morgenstern (1944). As an intermediate step, we provide axiomatic characterizations of the value and the lower value of matrix games and finite two-player zero-sum games, respectively.The authors acknowledge the financial support of Ministerio de Ciencia y Tecnologia, FEDER andXunta de Galicia through projects BEC2002-04102-C02-02 and PGIDIT03PXIC20701PN.We wish to thank Professor William Thomson as well as an anonymous referee for useful comments.     

Mixing weighted values of non-atomic games

Weighted values of non-atomic games were introduced by Hart and Monderer (1997). They study these values by using two approaches: the potential approach and the asymptotic approach. In this study we develop the random order approach (the mixing value, Aumann and Shapley, 1974) to weighted values and prove that these values coincide with the asymptotic weighted values of Hart and Monderer in pNA. Received April 1995/Revised version May 1996/ Final version October 1997     

Assignment games with stable core

We prove that the core of an assignment game (a two-sided matching game with transferable utility as introduced by Shapley and Shubik, 1972) is stable (i.e., it is the unique von Neumann-Morgenstern solution) if and only if there is a matching between the two types of players such that the corresponding entries in the underlying matrix are all row and column maximums. We identify other easily verifiable matrix properties and show their equivalence to various known sufficient conditions for core-stability. By these matrix characterizations we found that on the class of assignment games, largeness of the core, extendability and exactness of the game are all equivalent conditions, and strictly imply the stability of the core. In turn, convexity and subconvexity are equivalent, and strictly imply all aformentioned conditions. Final version: April 1, 2001     

On the core of network synthesis games

We use polynomial formulations to show that several rational and discrete network synthesis games, including the minimum cost spanning tree game, satisfy the assumptions of Owen's linear production game model. We also discuss computational issues related to finding and recognizing core points for these classes of games.     

The class of microarray games and the relevance index for genes

Nowadays, microarray technology is available to generate a huge amount of information on gene expression. This information must be statistically processed and analyzed, in particular, to identify those genes which are useful for the diagnosis and prognosis of specific diseases. We discuss the possibility of applying game-theoretical tools, like the Shapley value, to the analysis of gene expression data. Via a “truncation” technique, we build a coalitional game whose aim is to stress the relevance (“sufficiency”) of groups of genes for the specific disease we are interested in. The Shapley value of this game is used to select those genes which deserve further investigation. To justify the use of the Shapley value in this context, we axiomatically characterize it using properties with a genetic interpretation. The authors are grateful to two anonymous referees for their extremely helpful comments. An earlier version of this paper was presented at the VI Spanish Meeting on Game Theory and Practice, July 12–14, 2004, Elche, Spain. S. Moretti gratefully acknowledges the financial support of the EU project NewGeneris, European Union 6th FP (FOOD-CT-2005-016320).     

On the bargaining set, kernel and core of superadditive games

We prove that for superadditive games a necessary and sufficient condition for the bargaining set to coincide with the core is that the monotonic cover of the excess game induced by a payoff be balanced for each imputation in the bargaining set. We present some new results obtained by verifying this condition for specific classes of games. For N-zero-monotonic games we show that the same condition required at each kernel element is also necessary and sufficient for the kernel to be contained in the core. We also give examples showing that to maintain these characterizations, the respective assumptions on the games cannot be lifted. Received: March 1998/Revised version: December 1998     

Stability and largeness of core for symmetric games

Largeness of the core is sufficient for stability of the core. In general the necessity is not known. In this paper we answer affirmatively the necessity for symmetric games. We also prove its equivalence to n specified vectors being imputations and also to the convexity of the lower boundary of the set of all acceptable pay-off vectors of the game. In this paper we establish the equivalence of a condition given by Shapley to the newly evolved condition, thereby give an alternate proof to Shapley’s condition. Received: January 1997/final version: March 1999     

Weighted multiple majority games with unions: Generating functions and applications to the European Union

An a priori system of unions or coalition structure is a partition of a finite set of players into disjoint coalitions which have made a prior commitment to cooperate in playing a game. This paper provides “ready-to-apply” procedures based on generating functions that are easily implementable to compute coalitional power indices in weighted multiple majority games. As an application of the proposed procedures, we calculate and compare coalitional power indices under the decision rules prescribed by the Treaty of Nice and the new rules proposed by the Council of the European Union.     

Additive stable solutions on perfect cones of cooperative games

Closed kernel systems of the coalition matrix turn out to correspond to cones of games on which the core correspondence is additive and on which the related barycentric solution is additive, stable and continuous. Different perfect cones corresponding to closed kernel systems are described. Received: December 2001/Revised: July 2002 RID="*" ID="*"  This note contains the new results, which were presented by the first author in an invited lecture at the XIV Italian Meeting on Game Theory and Applications in Ischia, July 2001. The lecture was dedicated to Irinel Dragan on the occasion of his seventieth birthday.     

A test of the core solution in finite strategy non-sidepayment games

This paper reports a test of the core solution in cooperative non-sidepayment games where players have finite strategy sets. Two laboratory experiments were conducted with three-person and four-person games; in both experiments, the core solution was tested competitively against the von Neumann-Morgenstern stable set and the imputation set. Predictions from these solution concepts were computed under parameters of α-effectiveness and strict preference. Results show that the frequency of outcomes falling in core is substantially higher than that observed in previous experiments (most of which involve sidepayment games). In addition, goodness-of-fit tests show that the core solution predicts the observed outcomes more accurately than do the stable set or the imputation set.     

ON THE NON-COMPACTNESS OF THE CORE OF NONATOMIC CONVEX σ-CONTINUOUS MEASURE GAMES

This paper concerns with the core of nonatomic gaxaes of foma f(μ), where μ is a nonatomic nonnegative measure and f is a continuous convex function on the domaln of μ. The main result of this paper is that the core of the game is not compact under the norm topology unless the game itself is a measure. This shows the largeness of the core in a sense other than that defined by Sharky for finite cases.     

The selectope for cooperative games

The selectope of a cooperative transferable utility game is the convex hull of the payoff vectors obtained by assigning the Harsanyi dividends of the coalitions to members determined by so-called selectors. The selectope is studied from a set-theoretic point of view, as superset of the core and of the Weber set; and from a value-theoretic point of view, as containing weighted Shapley values, random order values, and sharing values. Received May 1997/Revised version September 1999     

Approximating the least core value and least core of cooperative games with supermodular costs

We study the approximation of the least core value and the least core of supermodular cost cooperative games. We provide a framework for approximation based on oracles that approximately determine maximally violated constraints. This framework yields a 3-approximation algorithm for computing the least core value of supermodular cost cooperative games, and a polynomial-time algorithm for computing a cost allocation in the 2-approximate least core of these games. This approximation framework extends naturally to submodular profit cooperative games. For scheduling games, a special class of supermodular cost cooperative games, we give a fully polynomial-time approximation scheme for computing the least core value. For matroid profit games, a special class of submodular profit cooperative games, we give exact polynomial-time algorithms for computing the least core value as well as a least core cost allocation.     

The bargaining set of four-person balanced games

It is well known that in three-person transferable-utility cooperative games the bargaining set ℳⁱ ₁ and the core coincide for any coalition structure, provided the latter solution is not empty. In contrast, five-person totally-balanced games are discussed in the literature in which the bargaining set ℳⁱ ₁ (for the grand coalition) is larger then the core. This paper answers the equivalence question in the remaining four-person case. We prove that in any four-person game and for arbitrary coalition structure, whenever the core is not empty, it coincides with the bargaining set ℳⁱ ₁. Our discussion employs a generalization of balancedness to games with coalition structures. Received: August 2001/Revised version: April 2002     

The core of a cooperative game without side payments

For cooperative games without side payments, there are several types of conditions which guarantee nonemptiness of the core, for example balancedness and convexity. In the present paper, a general condition for nonempty core is introduced which includes the known ones as special cases. Moreover, it is shown that every game with nonempty core satisfies this condition.