Strongly Stable and Maximum Weakly Stable Noncrossing Matchings

In IWOCA 2019, Ruangwises and Itoh introduced stable noncrossing matchings, where participants of each side are aligned on each of two parallel lines, and no two matching edges are allowed to cross each other. They defined two stability notions, strongly stable noncrossing matching (SSNM) and weakly stable noncrossing matching (WSNM), depending on the strength of blocking pairs. They proved that a WSNM always exists and presented an $$O(n^{2})$$ O ( n 2 ) -time algorithm to find one for an instance with n men and n women. They also posed open questions of the complexities of determining existence of an SSNM and finding a largest WSNM. In this paper, we show that both problems are solvable in polynomial time. Our algorithms are applicable to extensions where preference lists may include ties, except for one case which we show to be NP-complete. This NP-completeness holds even if each person's preference list is of length at most two and ties appear in only men's preference lists. To complement this intractability, we show that the problem is solvable in polynomial time if the length of preference lists of one side is bounded by one (but that of the other side is unbounded).

Pairwise Stable Matching in Large Economies

We formulate a stability notion for two‐sided pairwise matching problems with individually insignificant agents in distributional form. Matchings are formulated as joint distributions over the characteristics of the populations to be matched. Spaces of characteristics can be high‐dimensional and need not be compact. Stable matchings exist with and without transfers, and stable matchings correspond precisely to limits of stable matchings for finite‐agent models. We can embed existing continuum matching models and stability notions with transferable utility as special cases of our model and stability notion. In contrast to finite‐agent matching models, stable matchings exist under a general class of externalities.

Altruism in Coalition Formation Games

Nguyen et al. [2016] introduced altruistic hedonic games in which agents’ utilities depend not only on their own preferences but also on those of their friends in the same coalition. We propose to extend their model to coalition formation games in general, considering also the friends in other coalitions. Comparing the two models, we argue that excluding some friends from the altruistic behavior of an agent is a major disadvantage that comes with the restriction to hedonic games. After introducing our model, we additionally study some common stability notions and provide a computational analysis of the associated verification and existence problems.

Stable and Envy-free Partitions in Hedonic Games

In this paper, we study coalition formation in hedonic games through the fairness criterion of envy-freeness. Since the grand coalition is always envy-free, we focus on the conjunction of envy-freeness with stability notions. We first show that, in symmetric and additively separable hedonic games, an individually stable and justified envy-free partition may not exist and deciding its existence is NP-complete. Then, we prove that the top responsiveness property guarantees the existence of a Pareto optimal, individually stable, and envy-free partition, but it is not sufficient for the conjunction of core stability and envy-freeness. Finally, under bottom responsiveness, we show that deciding the existence of an individually stable and envy-free partition is NP-complete, but a Pareto optimal and justified envy-free partition always exists.

Stability conditions and related filtrations for (G,h)-constellations

Given an infinite reductive algebraic group [Formula: see text], we consider [Formula: see text]-equivariant coherent sheaves with prescribed multiplicities, called [Formula: see text]-constellations, for which two stability notions arise. The first one is analogous to the [Formula: see text]-stability defined for quiver representations by King [Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford Ser.[Formula: see text]2) 45(180) (1994) 515–530] and for [Formula: see text]-constellations by Craw and Ishii [Flops of [Formula: see text]-Hilb and equivalences of derived categories by variation of GIT quotient, Duke Math. J. 124(2) (2004) 259–307], but depending on infinitely many parameters. The second one comes from Geometric Invariant Theory in the construction of a moduli space for [Formula: see text]-constellations, and depends on some finite subset [Formula: see text] of the isomorphy classes of irreducible representations of [Formula: see text]. We show that these two stability notions do not coincide, answering negatively a question raised in [Becker and Terpereau, Moduli spaces of [Formula: see text]-constellations, Transform. Groups 20(2) (2015) 335–366]. Also, we construct Harder–Narasimhan filtrations for [Formula: see text]-constellations with respect to both stability notions (namely, the [Formula: see text]-HN and [Formula: see text]-HN filtrations). Even though these filtrations do not coincide in general, we prove that they are strongly related: the [Formula: see text]-HN filtration is a subfiltration of the [Formula: see text]-HN filtration, and the polygons of the [Formula: see text]-HN filtrations converge to the polygon of the [Formula: see text]-HN filtration when [Formula: see text] grows.