The stable fixtures problem with payments

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 :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 (1992,1999,2007) 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the existence of a stable solution and the non-emptiness of the core, which holds in the other models with payments, is no longer valid for our (most general) model.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.

Computing the Nucleolus of Weighted Cooperative Matching Games in Polynomial Time

“…Open Questions Matching Games generalize naturally to b-matching games, where instead the underlying optimization problem is to find an edge subset M with |M ∩δ(v)| ≤ b v for each node v. Biro, Kern, Paulusma, and Wojuteczky [4] showed that the core-separation problem when b v > 2 for some vertex v, is coNP-Hard. Despite this, the complexity of computing the nucleolus of these games is open.…”

“…Kern and Paulusma state the question of computing the nucleolus for general matching games as an important open problem in 2003 [30]. In 2008, Deng and Fang [10] conjectured this problem to be NP-hard, and in 2017 Biró, Kern, Paulusma, and Wojuteczky [4] reaffirmed this problem as an interesting open question. Theorem 1 settles the question, providing a polynomial-time algorithm to compute the nucleolus of a general instance of a weighted cooperative matching game.…”

The Complexity of Matching Games: A Survey

“…Note that 1-assignment games correspond to assignment games. A b-matching game is called a multiple partners matching game in the paper of Biró et al (2018).…”

“…The goal is to decide if (G, b, w) has a (pairwise) stable solution, that is, a solution with no blocking pairs. This problem is called the Stable Fixtures with Payments (SFP) problem (Biró et al, 2018). Restrictions of SFP are called: (Sotomayor, 1992);…”

New and Simple Algorithms for Stable Flow Problems

“…PPAD-hardness [31] is considered a somewhat weaker evidence of intractability than NP-hardness that applies for problems whose decision versions have a 'yes' answer for sure. Note that smf is one of the very few problems in stability [3] where a stable solution exists, but no extension of the Gale-Shapley algorithm is known to solve it -not even a variant with exponential running time.…”