Computing solutions for matching games

Abstract

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.

Solutions for the Stable Roommates Problem with Payments

“…This problem is well known to be polynomial-time solvable (cf. [6]); recently, an O(nm+n 2 log n) time algorithm for weighted graphs on n vertices and m edges has been given [3].…”

Analysis of stochastic matching markets

“…This implies that any ergodic set (which is called absorbing set in [14] and [20]) of the corresponding Markov chain must contain some of the above matchings. Constructing this instance in PRISM, we find there are 308 matchings and a single ergodic set which consists of the matchings {M 4 , M 5 , M 6 , M 7 }, where M 7 = {(1, 2), (3,8), (4,6), (5, 7)}. This corresponds to the results presented in [20].…”

“…The h 1 -stable matchings are: (4,5), (6, 7)} while the h 2 -stable matchings are: (4,6), (5, 7)} Theorem 1 states that, starting from any matching, we can always reach one of these matchings by successively satisfying blocking pairs. This implies that any ergodic set (which is called absorbing set in [14] and [20]) of the corresponding Markov chain must contain some of the above matchings.…”

“…Yet, one could investigate the structure of absorbing and ergodic states for special classes of coalition formation games, and analyse particular games in a similar framework as we did here, using PRISM. Furthermore, the same questions can be asked for cooperative games with transferable utilities as well, such as the stable matching problem with payments [6], where the agents who are matched together may share the value of their cooperation between themselves. 12 Finally, more gen-eral network formation games and matching problems could also be analysed with this technique.…”

“…Let M contain every edge that has weight 1 in h, every second edge from each even half-weighted cycle of h (if there were any), and k (disjoint) edges from each odd half-weighted cycle of length 2k + 1 in h. 6 They show that, starting from an arbitrary matching, one can get an h-stable matching by successively satisfying blocking pairs for a given instance of sr. Note that for every solvable instance of sr the set of h-stable matchings is equivalent to the set of stable matchings, thus the above result generalises the theorem of 4 An explanation for the first observation is the result of Blum and Rothblum [7] which demonstrates that, when using the Roth-Vande Vate algorithm, the last agent to arrive always gets their best stable partner (an alternative proof of this result is given by Biró et al [4]).…”