Uniqueness of Equilibrium in Two-sided Matching

This paper analyses a sufficient condition for uniqueness of equilibrium in two-sided matching with non-transferable utility. The condition is easy to interpret, being based on the notion that a person’s characteristics both form the basis of their attraction to the opposite sex, and determine their own sexual preferences.Uniqueness, matching, marriage.

The Eeckhout Condition and the Subgame Perfect Implementation of Stable Matching

We investigate an extensive form sequential matching game of perfect information. We show that the subgame perfect equilibrium of the sequential matching game leads to the unique stable matching when the Eeckhout Condition (2000) for existence of a unique stable matching holds, regardless of the sequence of agents. This result does not extend to preferences that violate the Eeckhout Condition, even if there is a unique stable matching.Matching; unique stable matching; subgame perfect equilibrium

Stable marriages and search frictions

Stable matchings are the primary solution concept for two-sided matching markets with nontransferable utility. We investigate the strategic foundations of stability in a decentralized matching market. Towards this end, we embed the standard marriage markets in a search model with random meetings. We study the limit of steady-state equilibria as exogenous frictions vanish. The main result is that convergence of equilibrium matchings to stable matchings is guaranteed if and only if there is a unique stable matching in the underlying marriage market. Whenever there are multiple stable matchings, sequences of equilibrium matchings converging to unstable, inefficient matchings can be constructed. Thus, vanishing frictions do not guarantee the stability and efficiency of decentralized marriage markets

Defining Equitable Geographic Districts in Road Networks via Stable Matching

We introduce a novel method for defining geographic districts in road networks using stable matching. In this approach, each geographic district is defined in terms of a center, which identifies a location of interest, such as a post office or polling place, and all other network vertices must be labeled with the center to which they are associated. We focus on defining geographic districts that are equitable, in that every district has the same number of vertices and the assignment is stable in terms of geographic distance. That is, there is no unassigned vertex-center pair such that both would prefer each other over their current assignments. We solve this problem using a version of the classic stable matching problem, called symmetric stable matching, in which the preferences of the elements in both sets obey a certain symmetry. In our case, we study a graph-based version of stable matching in which nodes are stably matched to a subset of nodes denoted as centers, prioritized by their shortest-path distances, so that each center is apportioned a certain number of nodes. We show that, for a planar graph or road network with $n$ nodes and $k$ centers, the problem can be solved in $O(n\sqrt{n}\log n)$ time, which improves upon the $O(nk)$ runtime of using the classic Gale-Shapley stable matching algorithm when $k$ is large. Finally, we provide experimental results on road networks for these algorithms and a heuristic algorithm that performs better than the Gale-Shapley algorithm for any range of values of $k$.Comment: 9 pages, 4 figures, to appear in 25th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems (ACM SIGSPATIAL 2017) November 7-10, 2017, Redondo Beach, California, US

Divorce decisions, divorce laws and social norms

This article focuses on the three way relationship between change in divorce law, evolution of divorce rate and evolution of the cultural acceptance of divorce. We consider a heterogeneous population in which individuals differ in terms of the subjective loss they suffer when divorced, this loss being associated with stigmatizing social norms. The proportion of each type of individual evolves endogenously through a cultural transmission process. Divorce law is chosen by majority voting between two alternatives : mutual consent and unilateral divorce. In this framework, evolutions of divorce rate and divorce law may be jointly affected by the cultural dynamics within the society. In particular, we are able to reproduce the fact that divorce rate often raises before a legislation change. Indeed, the shift from consensual to unilateral divorce has an accelerating effect on the increase in divorce rate but is not the driving force behind this evolution.Marriage and divorce, divorce legislation, cultural evolution, social norms.

Percolation in invariant Poisson graphs with i.i.d. degrees

Let each point of a homogeneous Poisson process in R^d independently be equipped with a random number of stubs (half-edges) according to a given probability distribution mu on the positive integers. We consider translation-invariant schemes for perfectly matching the stubs to obtain a simple graph with degree distribution mu. Leaving aside degenerate cases, we prove that for any mu there exist schemes that give only finite components as well as schemes that give infinite components. For a particular matching scheme that is a natural extension of Gale-Shapley stable marriage, we give sufficient conditions on mu for the absence and presence of infinite components

Insertion and deletion tolerance of point processes

We develop a theory of insertion and deletion tolerance for point processes. A process is insertion-tolerant if adding a suitably chosen random point results in a point process that is absolutely continuous in law with respect to the original process. This condition and the related notion of deletion-tolerance are extensions of the so-called finite energy condition for discrete random processes. We prove several equivalent formulations of each condition, including versions involving Palm processes. Certain other seemingly natural variants of the conditions turn out not to be equivalent. We illustrate the concepts in the context of a number of examples, including Gaussian zero processes and randomly perturbed lattices, and we provide applications to continuum percolation and stable matching