# Digraph competitions and cooperative games

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~ ~r~ Discussion for a er8414 ~~c Research1994 NR.24 N lln III N IIII I I Ih M II I IIII I I IIII IIIII I II i , Center for Economic Research No. 9424 DIGRáPH COMPETITIONS AND COOPERATIVE GAMES by René van den Brink and Peter Borm February 1994 ISSN 0924-7815 D K.U.B. - BIBLIOTHEEK TILBURG Digraph Competitions and Cooperative Games~` René van den Brinkt Peter Borm Depa.rtment of Econometrics Tilburg University P.O. Box 90153 5000 LE Tilburg The Netherlands February 1994 'The authors would like to thank Vincent Feltkamp, Rob Gilles and Stef Tqa for their uaefulremarks. 1This author is financially supported by the Netherlande Organization for Scientific Reecarch (NWO), grant 950-228-022 I Abstract A dominance structure or competition between a set of players can be mod- clled by means of a directed graph (N, D). Here, the set of nodes N represents a set of players, and (i, j) E D means that player i dominates or defeats player j. We introduce the conservative score mapping which assigna to every digraph a cooperative game urith transJeraóle utilitiea. Solution concepts for these scote games can be interpreted as a way to evaluate the underlying dominance struc- ture or competition, i.e., as a way to rank the players. We provide characterizations of the score mapping and of the aubclass of score games. It is shown that a score game is convex, and its core is equal to the convex hull oC the set of score vectors corresponding to those subgraphs in which each player that is defeated in the original digraph now is defeated exactly once. (;onsequently, all marqina! vectors aze such score vectors. However, in general the converse need not hold, and the class of digraphs for which it dces is characterized. Flrrthermore, the Shapley vnlue of a score game, which by definition is the mean of all marginal vectors, turns out to be also the mean of all score vectors. 1 Introduction A situation in which a set of players can obtain certain payoffs by cooperati

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