Cooperative games are considered where only those coalitions of players are feasible that respect a given precedence structure on the set of players. Strengthening the classical symmetry axiom, we obtain three axioms that give rise to a unique Shapley value in this model. The Shapley value is seen to reflect the expected marginal contribution of a player to a feasible random coalition, which allows us to evaluate the Shapley value nondeterministically. We show that every exact algorithm for the Shapley value requires an exponential number of operations already in the classical case and that even restriction to simple games is # P-hard in general. Furthermore, we outline how the multi-choice cooperative games of Hsiao and Raghavan can be treated in our context, which leads to a Shapley value that does not depend on pre-assigned weights. Finally, the relationship between the Shapley value and the permission value of Gilles, Owen and van den Brink is ,discussed. Both refer to formally similar models of cooperative games but reflect complementary interpretations of the precedence constraints and thus give rise to fundamentally different solution concepts.
AbsztraktA párosításjáték egy kooperatív játék (N,v), amely egy G=(N,E) gráffal és w: E R + élsúlyokkal definiálható. N jelöli a játékosok halmazát és egy S N koalíció értéke megegyezik az S csúcshalmaz által feszített részgráfban található maximális párosítás súlyával. A cikkben először adunk egy O(nm+n 2 log n) idejű algoritmust, amely egy adott párosításjátékra eldönti, hogy üres-e a magja, és ha nem üres, akkor talál egy magbeli elosztást (ahol n a gráf csúcsainak számát, m pedig az élek számát jelöli). Ez algoritmusjavítást jelent a korábbi ellipszoid módszeren alapuló megoldáshoz képest, amelyet a szobatárs probléma kifizetéses változatára adtak. Ezután azt is megmutatjuk, hogy nem üres maggal rendelkező játékok esetén a játék nukleolusza O(n 4 ) futásidőben kiszámítható.Ez az eredmény Solymosi és Raghavan hozzárendelési játékokra adott eredményét általánosítja. Végül azt látjuk be, hogy NP-nehéz feladat egy olyan elosztást találni, amelyre a blokkoló párok száma minimális, még akkor is, ha az élek egységnyi súlyúak. Viszont azt is megmutatjuk, hogy minimális blokkoló értékű elosztást polinomiális időben lehet találni.Kulcsszavak: párosítás játék; nukleolusz; kooperatív játékelmélet.JEL kódok: C61, C63, C71, C78Computing solutions for matching games Abstract. A matching game is a cooperative game (N, v) defined on a graph G = (N, E) with an edge weighting w : E → R+. The player set is N and the value of a coalition S ⊆ N is defined as the maximum weight of a matching in the subgraph induced by S. First we present an O(nm + n 2 log n) algorithm that tests if the core of a matching game defined on a weighted graph with n vertices and m edges is nonempty and that computes a core member if the core is nonempty. This algorithm improves previous work based on the ellipsoid method and can also be used to compute stable solutions for instances of the stable roommates problem with payments. Second we show that the nucleolus of an n-player matching game with a nonempty core can be computed in O(n 4 ) time. This generalizes the corresponding result of Solymosi and Raghavan for assignment games. Third we prove that is NP-hard to determine an imputation with minimum number of blocking pairs, even for matching games with unit edge weights, whereas the problem of determining an imputation with minimum total blocking value is shown to be polynomial-time solvable for general matching games.
Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. MATCHING GAMES: THE LEAST CORE AND THE NUCLEOLUS WALTER KERN and DANIËL PAULUSMAA matching game is a cooperative game defined by a graph G = N E . The player set is N and the value of a coalition S ⊆ N is defined as the size of a maximum matching in the subgraph induced by S. We show that the nucleolus of such games can be computed efficiently. The result is based on an alternative characterization of the least core, which may be of independent interest. The general case of weighted matching games remains unsolved.
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