Analysis of stochastic matching markets

Suppose that the agents of a matching market contact each other randomly and form new pairs if is in their interest. Does such a process always converge to a stable matching if one exists? If so, how quickly? Are some stable matchings more likely to be obtained by this process than others? In this paper we are going to provide answers to these and similar questions, posed by economists and computer scientists. In the first part of the paper we give an alternative proof for the theorems by Diamantoudi et al. and Inarra et al., which imply that the corresponding stochastic processes are absorbing Markov chains. The second part of the paper proposes new techniques to analyse the behaviour of matching markets. We introduce the Stable Marriage and Stable Roommates Automaton and show how the probabilistic model checking tool PRISM may be used to predict the outcomes of stochastic interactions between myopic agents. In particular, we demonstrate how one can calculate the probabilities of reaching different matchings in a decentralised market and determine the expected convergence time of the stochastic process concerned. We illustrate the usage of this technique by studying some well-known marriage and roommates instances and randomly generated instances.     

The stable fixtures problem—A many-to-many extension of stable roommates

We study a many-to-many generalisation of the well-known stable roommates problem in which each participant seeks to be matched with a number of others. We present a linear-time algorithm that determines whether a stable matching exists, and if so, returns one such matching.     

The stable crews problem

In this paper the classical stable roommates problem is generalized to situations when the two partners in a pair perform different roles. We propose an efficient algorithm to decide the existence of a stable matching in this problem.     

Individual and collective opportunities

We consider the bargaining problem in the context of a variable number of agents. When new agents enter the scene but the opportunities open to the enlarged group do not expand, some solutions paradoxically may recommend that some of the agents originally present gain. We propose a quantitative measure of the extent to which a solution allows this phenomenon to occur and we rank the major solutions on that basis. The Kalai-Smorodinsky solution performs better than all weakly pareto-optimal and anonymous solution, and in particular strictly better than the Nash solution. However, the two solutions are equivalent when it is the opportunities for gains offered to initial groups, instead of individuals, that are being compared.     

Single-payoff farsighted stable sets in strategic games with dominant punishment strategies

We investigate farsighted stable sets in a class of strategic games with dominant punishment strategies. In this class of games, each player has a strategy that uniformly minimizes the other players’ payoffs for any given strategies chosen by these other players. We particularly investigate a special class of farsighted stable sets, each of which consists of strategy profiles yielding a single payoff vector. We call such a farsighted stable set as a single-payoff farsighted stable set. We propose a concept called an inclusive set that completely characterizes single-payoff farsighted stable sets in strategic games with dominant punishment strategies. We also show that the set of payoff vectors yielded by single-payoff farsighted stable sets is closely related to the strict \(\alpha \)-core in a strategic game. Furthermore, we apply the results to strategic games where each player has two strategies and strategic games associated with some market models.     

An upper bound for the solvability probability of a random stable roommates instance

It is well-known that not all instances of the stable roommates problem admit a stable matching. Here we establish the first nontrivial upper bound on the limiting behavior of P_n, the probability that a random roommates instance of size n has a stable matching, namely, lim _n→∞ Pn? e^1/2/2 (=0.8244…). © 1994 John Wiley & Sons, Inc.     

Monotonicity of stable solutions in shadow systems

A shadow system appears as a limit of a reaction-diffusion system in which some components have infinite diffusivity. We investigate the spatial structure of its stable solutions. It is known that, unlike scalar reaction-diffusion equations, some shadow systems may have stable nonconstant (monotone) solutions. On the other hand, it is also known that in autonomous shadow systems any nonconstant non-monotone stationary solution is necessarily unstable. In this paper, it is shown in a general setting that any stable bounded (not necessarily stationary) solution is asymptotically homogeneous or eventually monotone in .

     

Persistency of linear programming relaxations for the stable set problem

The Nemhauser–Trotter theorem states that the standard linear programming (LP) formulation for the stable set problem has a remarkable property, also known as (weak) persistency: for every optimal LP solution that assigns integer values to some variables, there exists an optimal integer solution in which these variables retain the same values. While the standard LP is defined by only non-negativity and edge constraints, a variety of other LP formulations have been studied and one may wonder whether any of them has this property as well. We show that any other formulation that satisfies mild conditions cannot have the persistency property on all graphs, unless it is always equal to the stable set polytope.

     

Computational complexity of stable partitions with B-preferences

Consider a special stable partition problem in which the player's preferences over sets to which she could belong are identical with her preferences over the most attractive member of a set and in case of indifference the set of smaller cardinality is preferred. If the preferences of all players over other (individual) players are strict, a strongly stable and a stable partition always exists. However, if ties are present, we show that both the existence problems are NP-complete. These results are very similar to what is known for the stable roommates problem. Received: July 2000/Revised: October 2002 RID="*" ID="*"  This work was supported by the Slovak Agency for Science, contract #1/7465/20 “Combinatorial Structures and Complexity of Algorithms”.     

Scatter search for trainees to software project requirements stable allocation

In this paper, we study a centralized, stable matching scheme, which allocates trainees to software project requirements, to minimize retraining and relocation costs when the preference lists of the project requirements may contain ties of arbitrary lengths. This particular trainees’ assignment problem is important because the allocation decisions not only influence the costs but also impact software project deliverables and intern attrition rates. It is also an NP-hard problem because of the inclusion of the ties, and the costs in the stable allocation model. We, therefore, have designed a GRASP-based scatter search method, to solve the large size instances of our assignment problem efficiently. The GRASP method uses randomized algorithms to generate initial trial solutions. A repair heuristic based on regret minimization idea is designed to convert an unstable solution to a stable solution during an improvement phase. Computational experiments suggest that the proposed algorithm significantly reduces run time compared to the CPLEX, and produces solutions that are at an average 4.5% away from the best CPLEX solutions for the large size problem instances. Moreover, our scatter search method consistently provides better quality solutions than the two state of the art methods from the prior literature.     

Contractually stable networks

We consider situations where players are part of a network and belong to coalitions in a given coalition structure. We propose the concept of contractual stability to predict the networks that are going to emerge at equilibrium when the consent of coalition partners is needed for adding or deleting links. Two different decision rules for consent are analyzed: simple majority and unanimity. We characterize the coalition structures that make the strongly efficient network contractually stable under the unanimity decision rule and the coalition structures that sustain some critical network as contractually stable under the simple majority decision rule and under any decision rule requiring the consent of any proportion of coalition partners. Requiring the consent of coalition members may help to reconcile stability and efficiency in some classical models of network formation.     

A maximum stable matching for the roommates problem

The stable roommates problem is that of matchingn people inton/2 disjoint pairs so that no two persons, who are not paired together, both prefer each other to their respective mates under the matching. Such a matching is called a complete stable matching. It is known that a complete stable matching may not exist. Irving proposed anO(n ²) algorithm that would find one complete stable matching if there is one, or would report that none exists. Since there may not exist any complete stable matching, it is natural to consider the problem of finding a maximum stable matching, i.e., a maximum number of disjoint pairs of persons such that these pairs are stable among themselves. In this paper, we present anO(n ²) algorithm, which is a modified version of Irving's algorithm, that finds a maximum stable matching.This research was supported by National Science Council of Republic of China under grant NSC 79-0408-E009-04.     

Globally stable vaccine-induced eradication of horizontally and vertically transmitted infectious diseases with periodic contact rates and disease-dependent demographic factors in the population

Within the framework of SEIR-like epidemic models, we studied the conditions for the stable eradication of some families of vertically and horizontally transmitted infectious diseases in the case of periodically varying contact rate. By means of Floquet’s theory, we found a condition for the eradication solution to be locally asymptotically stable. We then demonstrated that the same condition guarantees also that this vaccine-induced disease-free solution is globally asymptotically stable. A model with interacting populations is also considered. In the final part of this work, we extended the model by taking into account the variation of population size, the impact of disease-related deaths and reduction of fertility.     

Sufficient conditions for unique stable sets in three agent pillage games

Pillage games (Jordan, 2006a) have two features that make them richer than cooperative games in either characteristic or partition function form: they allow power externalities between coalitions; they allow resources to contribute to coalitions’ power as well as to their utility. Extending von Neumann and Morgenstern’s analysis of three agent games in characteristic function form to anonymous pillage games, we characterise the core for any number of agents; for three agents, all anonymous pillage games with an empty core represent the same dominance relation. When a stable set exists, and the game also satisfies a continuity and a responsiveness axiom, it is unique and contains no more than 15 elements, a tight bound. By contrast, stable sets in three agent games in characteristic or partition function form may not be unique, and may contain continua. Finally, we provide an algorithm for computing the stable set, and can easily decide non-existence. Thus, in addition to offering attractive modelling possibilities, pillage games seem well behaved and analytically tractable, overcoming a difficulty that has long impeded use of cooperative game theory’s flexibility.     

Asymmetric equilibria in dynamic two-sided matching markets with independent preferences

A fundamental fact in two-sided matching is that if a market allows several stable outcomes, then one is optimal for all men in the sense that no man would prefer another stable outcome. We study a related phenomenon of asymmetric equilibria in a dynamic market where agents enter and search for a mate for at most n rounds before exiting again. Assuming independent preferences, we find that this game has multiple equilibria, some of which are highly asymmetric between sexes. We also investigate how the set of equilibria depends on a sex difference in the outside option of not being mated at all.     

On the invariance of male optimal stable matching

The stable matching problem is that of matching two sets of agents in such a manner that no two unmatched agents prefer each other to their actual partners under the matching. In this paper, we present a set of sufficient conditions on the preference lists of any given stable matching instance, under which the optimality of the original male optimal stable matching is still preserved.     

Heuristics for Large Constrained Vehicle Routing Problems

This paper presents a heuristic for solving very large routing problems (thousands of customers and hundreds of vehicles) with side constraints such as time windows. When applied to traditional benchmarks (Solomon's), we obtain high quality results with short resolution time (a few seconds). We also introduce a LDS (Limited Discrepancy Search) variation that produces state-of-the-art results. The heart of this heuristic is a combination of a look-ahead insertion algorithm, an incremental local optimization scheme and a constraint solver for constrained traveling salesman problems. The incrementality means that instead of visiting some large neighborhood after an initial solution has been found, a limited number of moves is examined, after each insertion, on the partial solution. This incremental version is not only faster, it also yields better results than using local optimization once a full solution has been built. We also show how additional constraints can be used in order to guide the insertion process. Because of its use of separate CP (Constraint Programming) modules, this method is flexible and may be used to solve large dispatching problems that include many additional constraints such as setup times (asymmetrical distance) or skill matching.     

Farsighted stable sets in Hotelling’s location games

We apply the farsighted stable set to two versions of Hotelling’s location games: one with a linear market and another with a circular market. It is shown that there always exists a farsighted stable set in both games, which consists of location profiles that yield equal payoff to all players. This stable set contains location profiles that reflect minimum differentiation as well as those profiles that reflect local monopoly. These results are in contrast to those obtained in the literature that use some variant of Nash equilibrium. While this stable set is unique when the number of players is two, uniqueness no longer holds for both models when the number of players is at least three.     

Trading water along a river

A set of agents is located along a river. Each agent consumes certain amount of water he receives from his part of the river basin and may sell certain amount to his downstream agent if it is mutually beneficial. Water trading is restricted to two neighboring agents and an agent can only pass water to his downstream agent. We ask if this restricted trade to neighboring agents can implement an efficient allocation of water. We show that the efficient allocation of water can be achieved through the process of downstream bilateral trading. Specifically, we show that this one way “downstream” trading process implements the unique efficient allocation as well as a welfare distribution. We also show that the welfare distribution is in the core of the associated game of the problem. Moreover, we show that the coalition of agents upstream any agent obtains more welfare with the bilateral trading than with the downstream incremental distribution proposed by Ambec and Sprumont (2002) and less than with the upstream incremental distribution proposed by [Ambec and Ehlers, 2008a] and [Ambec and Ehlers, 2008b].