On generalizing shapley's index theory to labelled pseudomanifolds

AbstractFor a bimatrix game one may visualize two bounded polyhedronsX andY, one for each player. OnX × Y one may visualize, as a graphG, the set of almost-complementary points (see text). $$\bar G$$ consists of an even number of nodes, one for each complementary point (one for the origin, others corresponding to extreme points which are equilibrium points); arcs (extreme point paths of almost complementary points); and possibly loops (paths with no equilibrium points).Shapley has shown that one may assign indices (+) and (−) to nodes, and directions called (+) and (−) to arcs or loops in such a way that, leaving a (+) node one moves always in a (+) direction, terminating at a (−) node. Indices and directions for a point are determined knowing only the point.In this paper, these concepts are generalized to labelled pseudomanifolds. An integer labelling of the vertices identifies theG-set of almost-completely labelled simplexes. It is shown that in order for theG-set of any labelling to be directed as above it is necessary and sufficient that the pseudomanifold be orientable.Realized examples for situations of current interest are also developed.