The nucleon of cooperative games and an algorithm for matching games

Faigle, Ulrich; Kern, Walter; Fekete, Sándor P.; Hochstättler, Winfried

doi:10.1007/bf02680558

Cited by 37 publications

(40 citation statements)

References 12 publications

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“…This condition is sufficient to compute the nucleolus of a cooperative game in polynomial time if the core is nonempty [9]. It also stems from the result that an imputation in the nucleon can be computed in polynomial time for matching games, as shown by Faigle et al [8]. The nucleon is a solution concept similar to the nucleolus.…”

Section: Theorem 1 ([10])mentioning

confidence: 97%

Computing solutions for matching games

2011

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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.

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Section: Theorem 1 ([10])mentioning

confidence: 97%

Computing solutions for matching games

2011

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“…A multiplicative variant of the nucleolus, the so-called nucleon, has been introduced in Faigle et al [8]. Assuming v ≥ 0 (and v-worths representing profits) this paper proposes to solve…”

Section: The Nucleon Of Simple Flow Gamesmentioning

confidence: 99%

“…See Owen [23] for a survey. Multiplicative variants of the nucleolus are the nucleon (Faigle et al [8]) (also called the proportional nucleolus (Young et al [34])), and the per-capita nucleolus (Grotte [17], Young [33]) (also called the weak nucleolus (Shapley [27])). Both variants can be more natural to model situations in which taxation is imposed proportionally to the worth (e.g., interest or sales tax).…”

Section: Introduction a Cooperative Game Is Given By A Set E Of Playmentioning

confidence: 99%

“…As an illustration, consider the earlier mentioned matching games and min-cost spanning tree games. In Faigle et al [7] it has been shown that testing membership in the core of min-cost spanning tree games is NP-hard, while this can be done in polynomial time for matching games (Faigle et al [8]). For assignment games (Solymosi and Raghavan [29]) and simple matching games (i.e., with unit edge weights) (Kern and Paulusma [21]), the nucleolus can be computed in polynomial time.…”

Section: Introduction a Cooperative Game Is Given By A Set E Of Playmentioning

confidence: 99%

“…For assignment games (Solymosi and Raghavan [29]) and simple matching games (i.e., with unit edge weights) (Kern and Paulusma [21]), the nucleolus can be computed in polynomial time. For (general) matching games, an allocation in the nucleon (Faigle et al [8]) can be computed in polynomial time. For min-cost spanning tree games, computing the nucleolus (Faigle et al [9]) and computing an allocation in the nucleon (Faigle et al [6]) are NP-hard problems.…”

Section: Introduction a Cooperative Game Is Given By A Set E Of Playmentioning

confidence: 99%

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On the Core and f-Nucleolus of Flow Games

Kern

Paulusma

2009

Mathematics of OR

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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. INFORMS holds copyright to this article and distributed this copy as a courtesy to the author(s). Additional information, including rights and permission policies, is available at https://www.informs.org/. show that the nucleolus of simple flow games (where all edge capacities are equal to one) can be computed in polynomial time. Our main result is a combinatorial method based on eliminating redundant s-t path constraints such that only a polynomial number of constraints remains. This leads to efficient algorithms for computing the core, nucleolus, and nucleon of simple flow games. Deng et al. also prove that computing the nucleolus for (general) flow games is NP-hard. We generalize this by proving that computing the f -nucleolus of flow games is NP-hard for all priority functions f that satisfy f A > 0 for all coalitions A with worth v A > 0 (so, including the priority functions corresponding to the nucleolus, nucleon, and per-capita nucleolus). MATHEMATICS OF OPERATIONS RESEARCHVol. 34, No. 4, November 2009, pp. 981-991 issn 0364-765X eissn 1526-5471 09 3404 0981 informs ® doi 10.1287

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Computing the Nucleolus of Weighted Cooperative Matching Games in Polynomial Time

Könemann

Pashkovich

Toth

2019

Integer Programming and Combinatorial Optimization

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We provide an efficient algorithm for computing the nucleolus for an instance of a weighted cooperative matching game. This resolves a long-standing open question posed in [Faigle, Kern, Fekete, Hochstättler, Mathematical Programming, 1998].Let ε * be the optimum value of (P ), and define P (ε * ) to be the set of allocations x such that (x, ε * ) is feasible for (P ). The set P (ε * ) is known as the leastcore [39] of the given cooperative game, and the special case when ε * = 0, P (0) is the well-known core [25] of (V, ν). Intuitively, allocations in the core describe payoffs in which no coalition of players could profitably deviate from the grand coalition V . Why stop at maximizing the bottleneck excess? Consider an allocation which, subject to maximizing the smallest excess, maximizes the second smallest excess, and subject to that maximizes the third smallest excess, and so on. This process of successively optimizing the excess of the worst-off coalitions yields our primary object of interest, the nucleolus. For an allocation x ∈ R V , let θ(x) ∈ R 2 V −2 be the vector obtained by sorting the list of excess values x(S) − ν(S) for any ∅ = S ⊂ V in non-decreasing order 3 . The nucleolus, denoted η(V, ν) and defined by Schmeidler [45], is the unique allocation that lexicographically maximizes θ(x): η(V, ν) := arg lex max{θ(x) : x ∈ P (ε * )}.We refer the reader to Appendix B for an example instance of the weighted matching game with its nucleolus. We now have sufficient terminology to state our main result:Theorem 1. Given a graph G = (V, E) and weights w : E → R, the nucleolus η(V, ν) of the corresponding weighted matching game can be computed in polynomial time.

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The nucleon of cooperative games and an algorithm for matching games

Cited by 37 publications

References 12 publications

Computing solutions for matching games

Computing solutions for matching games

On the Core and f-Nucleolus of Flow Games

Computing the Nucleolus of Weighted Cooperative Matching Games in Polynomial Time

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