Stable outcomes of the roommate game with transferable utility

Eriksson, Kimmo; Karlander, Johan

doi:10.1007/s001820000058

Cited by 39 publications

(37 citation statements)

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“…[5,7,20]. Here, a cover of a graph G = (N, E) with edge weighting w is a vertex mapping c : N → R + such that c(u) + c(v) ≥ w(uv) for each edge uv ∈ E. The weight of c is defined as c(N ) = u∈N c(u).…”

Section: Theorem 1 ([10])mentioning

confidence: 99%

“…This problem is polynomially solvable by the following observation which is well known (cf. Eriksson and Karlander [7]) and easy to verify.…”

Section: Theorem 1 ([10])mentioning

confidence: 99%

“…Eriksson and Karlander [7] characterize stable solutions for instances of the problem Stable Roommates with Payments in terms of forbidden minors. By Observation 2 we can apply Proposition 1 to find an alternative characterization, namely that a weighted graph G has a stable solution if and only if the maximum weight of a matching in G is equal to the maximum weight of a half-matching in G. …”

Section: Theorem 3 ([2]mentioning

confidence: 99%

“…This is the example Eriksson and Karlander [7] used to illustrate the stable roommates problem with payments.…”

Section: Introductionmentioning

confidence: 99%

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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: 99%

“…This problem is polynomially solvable by the following observation which is well known (cf. Eriksson and Karlander [7]) and easy to verify.…”

Section: Theorem 1 ([10])mentioning

confidence: 99%

Section: Theorem 3 ([2]mentioning

confidence: 99%

“…This is the example Eriksson and Karlander [7] used to illustrate the stable roommates problem with payments.…”

Section: Introductionmentioning

confidence: 99%

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Computing solutions for matching games

2011

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“…The general (nonbipartite) case is studied in Deng et al (1999), characterizing when the core is nonempty. The extreme points of the core (in case this is nonempty) are characterized in Eriksson and Karlander (2001). (This paper also touches a "nontransferable utility" version of the roommate problem and provides several references.)…”

mentioning

confidence: 99%

Matching Games: The Least Core and the Nucleolus

Kern

Paulusma

2003

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. 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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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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Stable outcomes of the roommate game with transferable utility

Cited by 39 publications

References 0 publications

Computing solutions for matching games

Computing solutions for matching games

Matching Games: The Least Core and the Nucleolus

Computing the Nucleolus of Weighted Cooperative Matching Games in Polynomial Time

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