Characterizing convexity of games using marginal vectors

This paper studies the relation between convexity of TU games and marginal vectors. Shapley (1971) andIchiishi (1981) showed that a game is convex if and only if all marginal vectors are core elements. In Rafels,Ybern (1995) it is shown that if all even marginal vectors are core elements, then all odd marginal vectorsare core elements as well, and vice versa. Hence, if all even or all odd marginal vectors are core elements,then the game is convex. In Van Velzen, Hamers and Norde (2002) other sets of marginal vectors areconstructed such that the requirement that these marginal vectors are core elements is a sufficient conditionfor convexity of a game. This construction is based on a neighbour argument, i.e. it is shown that if twospecific neighbours of a marginal vector are core elements, then this marginal vector is a core element aswell. In this way they characterize convexity using a fraction of the total number of marginal vectors.Moreover, they show that this fraction converges to zero.In this paper we use combinatorial arguments to obtain sets of marginal vectors that characterize convexity.We characterize the sets of marginal vectors satisfying this property. Furthermore we present the formula forthe minimum cardinality of sets of marginal vectors that characterize convexity. ReferencesIchiishi T. (1981), Super-Modularity: Applications to Convex Games and the Greedy Algorithm for LP,Journal of Economic Theory 25, 283-286.Rafels C., Ybern N. (1995), Even and Odd Marginal Worth Vectors, Owen's Multilinear Extension andConvex Games, International Journal of Game Theory 24, 113-126.Shapley L. (1971), Cores of Convex Games, International Journal of Game Theory 1, 11-26.Van Velzen B., Hamers H., and Norde H. (2002), Convexity and marginal vectors, CentER, CentERDiscussion Paper 2002-53, Tilburg University, Tilburg, The Netherlands (to appear in: International Journalof Game Theory).