Matching Games: The Least Core and the Nucleolus

Kern, Walter; Paulusma, Daniël

doi:10.1287/moor.28.2.294.14477

Cited by 94 publications

(65 citation statements)

References 18 publications

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“…So far only partial results have been obtained in this direction, namely polynomial-time algorithms for assignment games [43] and for matching games with unit edge weights [27] or, more generally, with edge weights that can be expressed as the sum of positive weights on the incident vertices of the graph G = (N, E) [37], or even more generally, with arbitrary edge weights as long as a certain condition on the least core of a subgraph (with vertex weights) of G is satisfied [18]. Can these results be generalized for b ≤ 2?…”

Section: P1mentioning

confidence: 99%

The Stable Fixtures Problem with Payments

Bíró

Kern

Paulusma

et al. 2016

Graph-Theoretic Concepts in Computer Science

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Abstract. We generalize two well-known game-theoretic models by introducing multiple partners matching games, defined by a graph G = (N, E), with an integer vertex capacity function b and an edge weighting w. The set N consists of a number of players that are to form a set M ⊆ E of 2-player coalitions ij with value w(ij), such that each player i is in at most b(i) coalitions. A payoff vector is a mapping p :A pair of players i, j with ij ∈ E \ M blocks a solution (M, p) if i, j can form, possibly only after withdrawing from one of their existing 2-player coalitions, a new 2-player coalition in which they are mutually better off. A solution is stable if it has no blocking pairs. Our contribution is as follows:-We survey, for the first time, known results on stable solutions in seven basic models and show that our model of multiple partners matching games is the natural model that was missing so far. -We give a polynomial-time algorithm that either finds that a given multiple partners matching game has no stable solution, or obtains a stable solution for it. Previously this result was only known for multiple partners assignment games, which correspond to the case where G is bipartite (Sotomayor, 1992) and for the case where b ≡ 1 (Biró et al., 2012). -We characterize the set of stable solutions of a multiple partners matching game in two different ways and show how this leads to simple proofs for a number of known results of Sotomayor (1992Sotomayor ( ,1999Sotomayor ( ,2007 for multiple partners assignment games and to generalizations of some of these results to multiple partners matching games. -We perform a study on the core of the corresponding cooperative game, where coalitions of any size may be formed. In particular we show that the standard relation between the existence of a stable solution and the non-emptiness of the core, which holds in the other models with payments, is no longer valid for our (most general) model. We also prove that the problem of deciding if an allocation belongs to the core jumps from being polynomial-time solvable for b ≤ 2 to NP-complete for b ≡ 3.

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Section: P1mentioning

confidence: 99%

The Stable Fixtures Problem with Payments

Bíró

Kern

Paulusma

et al. 2016

Graph-Theoretic Concepts in Computer Science

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“…It is known [12] that the nucleolus of a simple matching game can be computed in polynomial time by using the standard procedure of Maschler, Peleg and Shapley [15], after reducing the size of the involved linear programs to be polynomial. This result has been extended to node matching games [20], i.e., matching games defined on a graph G = (N, E) with an edge weighting w that allows a weighting w * : N → R + such that w(uv) = w * (u) + w * (v) for all uv ∈ E; note that every simple matching game is a node matching game by choosing w * ≡ 1 2 .…”

Section: Theorem 1 ([10])mentioning

confidence: 99%

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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“…Recently Greco et al [13] provided a non-trivial upper bound for its complexity. On the other hand there are known polynomial time algorithms for computing the nucleolus of important families of cooperative games, like standard tree [21], assignment [31], matching [16] and bankruptcy games [2]. In addition, Kuipers [18] and Arin and Inarra [1] developed methods to compute the nucleolus for convex games.…”

Section: Introductionmentioning

confidence: 99%

Characterization sets for the nucleolus in balanced games

Solymosi

Sziklai

2016

Operations Research Letters

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We provide a new modus operandi for the computation of the nucleolus in cooperative games with transferable utility. Using the concept of dual game we extend the theory of characterization sets. Dually essential and if the game is monotonic dually saturated coalitions determine both the core and the nucleolus whenever the core is non-empty. We show how these two sets are related to the existing characterization sets. In particular we prove that if the grand coalition is vital then the intersection of essential and dually essential coalitions forms a characterization set itself.

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Matching Games: The Least Core and the Nucleolus

Cited by 94 publications

References 18 publications

The Stable Fixtures Problem with Payments

The Stable Fixtures Problem with Payments

Computing solutions for matching games

Characterization sets for the nucleolus in balanced games

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