A generalization of the Shapley-Ichiishi result and its application to convex games

Field: Mathematics (Physical Sciences)

In this paper we present a generalization of the Shapley Ichiishi result for convex games to the class of all exact games Then we discuss two applications to the class of convex games First we show that it can indeed be used to give an alternative proof of the Shapley Ichiishi result Secondly we use our generalization to derive the minimal de ning system of linear inequalities of the class of convex games Introduction Convex games are interesting from more than one perspective First of all results on convex games can be applied to a wide range of practical game models such as unanimity games bankruptcy games O Neill Aumann and Maschler and sequencing games Curiel et al From a theoretical perspective convex games are interesting because the core of a convex game has a particularly regular form as well as several other pleasant properties For example Shapley showed that the core of a convex game is the unique stable set of that game and that it is the convex hull of the marginal vectors of that game Ichiishi showed that the converse of Department of Economics Uppsala University P O Box Uppsala Sweden Department of Mathematics Maastricht University P O Box MD Maastricht The Netherlands y Department of Quantitative Economics Maastricht University P O Box MD Maastricht The Netherlands z Department of Economics Stockholm School of Economics P O Box Stockholm Sweden the latter statement is also true if the core of a game is equal to the convex hull of its marginal vectors then the game is necessarily convex This two way result of Shapley and Ichiishi a game is convex if and only if its marginal vectors are core elements is very useful because it enables us for example to check fairly easily whether a game is convex In this paper we will present a theorem that can be seen as a generalization of the Shapley Ichiishi result We will rst explain the contents of our theorem and then elaborate on its implications for the class of convex games THE GENERALIZATION First we will prove a generalization of the Shapley Ichiishi result on the class of exact games Within this class we can identify equivalence classes of games whose cores have the same combinatorial structure This combinatorial structure can be expressed in terms of the coalitions that de ne extreme points of the core In its turn this way to describe equivalence classes is a generalization of the way orderings of players de ne marginal vectors the extreme points of the core We show that these equivalence classes one of them is the class of convex games can be given in terms of linear inequalities on the polyhedral class of all exact games The proof of this statement is based on a geometrical interpretation of combinatorial equivalence APPLICATIONS TO CONVEX GAMES In the second part of the paper we will give two applications of this result to the class of convex games The rst application is the derivation of the Shapley Ichiishi result We show that the class of exact games associated via the combinatorial equivalence relation with a strictly exact game is precisely the collection of convex games Since for this speci c strictly convex game it can easily be seen that its core equals the convex hull of all its marginal vectors we get an alternative proof of the Shapley Ichiishi theorem A second application is the derivation of the minimal de ning system of linear inequalities of the class of convex games The set of linear inequalities we get from our generalization when applied to the collection of convex games is a strict subset of the system of linear inequalities that is normally used to de ne convexity We will show that in this speci c setting it is even the minimal system of linear inequalities each inequality in this system is facet de ning So leaving it out would enlarge the set of games satisfying the remaining system of inequalities