Abstract
For 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.
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This research was supported by the National Science Foundation Grant NSF GP-32844X.
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Lemke, C.E., Grotzinger, S.J. On generalizing shapley's index theory to labelled pseudomanifolds. Mathematical Programming 10, 245–262 (1976). https://doi.org/10.1007/BF01580670
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DOI: https://doi.org/10.1007/BF01580670