Random paths to stability in the roommate problem (original) (raw)
Related papers
Random paths to P-stability in the roommate problem
International Journal of Game Theory, 2008
For solvable roommate problems with strict preferences Diamantoudi et al. (Games Econ Behav 48: 18-28, 2004) show that for any unstable matching, there exists a finite sequence of successive myopic blocking pairs leading to a stable matching. In this paper, we define P-stable matchings associated with stable partitions and, by using a proposal-rejection procedure, generalize the previous result for the entire class of roommate problems.
“Almost Stable” Matchings in the Roommates Problem
Lecture Notes in Computer Science, 2006
An instance of the classical Stable Roommates problem (sr) need not admit a stable matching. This motivates the problem of finding a matching that is "as stable as possible", i.e. admits the fewest number of blocking pairs. In this paper we prove that, given an sr instance with n agents, in which all preference lists are complete, the problem of finding a matching with the fewest number of blocking pairs is NP-hard and not approximable within n 1 2 −ε , for any ε > 0, unless P=NP. If the preference lists contain ties, we improve this result to n 1−ε . Also, we show that, given an integer K and an sr instance I in which all preference lists are complete, the problem of deciding whether I admits a matching with exactly K blocking pairs is NP-complete. By contrast, if K is constant, we give a polynomial-time algorithm that finds a matching with at most (or exactly) K blocking pairs, or reports that no such matching exists. Finally, we give upper and lower bounds for the minimum number of blocking pairs over all matchings in terms of some properties of a stable partition, given an sr instance I.
The dynamics of stable matchings and half-matchings for the stable marriage and roommates problems
International Journal of Game Theory, 2008
We study the dynamics of stable marriage and stable roommates markets. Our main tool is the incremental algorithm of Roth and Vande Vate and its generalization by Tan and Hsueh. Beyond proposing alternative proofs for known results, we also generalize some of them to the nonbipartite case. In particular, we show that the lastcomer gets his best stable partner in both incremental algorithms. Consequently, we confirm that it is better to arrive later than earlier to a stable roommates market. We also prove that when the equilibrium is restored after the arrival of a new agent, some agents will be better off under any stable solution for the new market than at any stable solution for the original market. We also propose a procedure to find these agents.
The stability of the roommate problem revisited
2008
The lack of stability in some matching problems suggests that alternative solution concepts to the core might be applied to find predictable matchings. We propose the absorbing sets as a solution for the class of roommate problems with strict preferences. This solution, which always exists, either gives the matchings in the core or predicts some other matchings when the core is empty. Furthermore, it satisfies an interesting property of outer stability. We also characterize the absorbing sets, determine their number and, in case of multiplicity, we find that they all share a similar structure.
Research Papers in Economics, 2016
We approach the roommate problem by focusing on well-behaved matchings, which are those individually rational matchings whose blocking pairs, if any, are formed with unmatched agents. We show that the set of stable matchings is non-empty if and only if no well-behaved and unstable matching is Pareto optimal among all well-behaved matchings. The economic intuition underlying this condition is that blocking can be done so that the transactions at any well-behaved and unstable matching need not be undone as agents reach the the set of stable matchings. We also give a sufficient condition on the preferences of the agents for the non-emptiness of the set of stable matchings. New properties of economic interest are proved
The "Stable Roommates" Problem with Random Preferences
The Annals of Probability, 1993
The general stable roommates problem with n agents has time and space complexity O(n 2). Random instances can be solved faster and with less memory, however. We introduce an algorithm that has average time and space complexity O(n 3 2) for random instances. We use this algorithm to simulate large instances of the stable roommates problem and to measure the probabilty pn that a random instance of size n admits a stable matching. Our data supports the conjecture that pn = Θ(n −1/4).
Stable Matchings for the Room-mates Problem
SSRN Electronic Journal, 2000
We show that, given two matchings for a room-mates problem of which say the second is stable, and given a non-empty subset of agents S if (a) no agent in S prefers the first matching to the second, and (b) no agent in S and his room-mate in S under the second matching prefer each other to their respective room-mates in the first matching, then no room-mate of an agent in S prefers the second matching to the first. This result is a strengthening of a result originally due to Knuth (1976). In a paper by Sasaki and Toda (1992) it is shown that if a marriage problem has more than one stable matchings, then given any one stable matching, it is possible to add agents and thereby obtain exactly one stable matching, whose restriction over the original set of agents, coincides with the given stable matching. We are able to extend this result here to the domain of room-mates problems. We also extend a result due to Roth and Sotomayor (1990) originally established for two-sided matching problems in the following manner: If in a room-mates problem, the number of agents increases, then given any stable matching for the old problem and any stable matching for the new one, there is at least one agent who is acceptable to this new agent who prefers the new matching to the old one and his room-mate under the new matching prefers the old matching to the new one. Sasaki and Toda (1992) shows that the solution correspondence which selects the set of all stable matchings, satisfies Pareto Optimality, Anonymity, Consistency and Converse Consistency on the domain of marriage problems. We show here that if a solution correspondence satisfying Consistency and Converse Consistency agrees with the solution correspondence comprising stable matchings for all room-mates problems involving four or fewer agents, then it must agree with the solution correspondence comprising stable matchings for all room-mates problems.
Stable Roommates Problem with Random Preferences
2014
The stable roommates problem with n agents has worst case complexity O(n^2) in time and space. Random instances can be solved faster and with less memory, however. We introduce an algorithm that has average time and space complexity O(n^3/2) for random instances. We use this algorithm to simulate large instances of the stable roommates problem and to measure the probabilty p_n that a random instance of size n admits a stable matching. Our data supports the conjecture that p_n = Θ(n^-1/4).
Journal of Statistical Mechanics: Theory and Experiment, 2005
The stable matching problem is a prototype model in economics and social sciences where agents act selfishly to optimize their own satisfaction, subject to mutually conflicting constraints. A stable matching is a pairing of adjacent vertices in a graph such that no unpaired vertices prefer each other to their partners under the matching. The problem of finding stable matchings is known as the stable marriage problem (on bipartite graphs) or as the stable roommates problem (on the complete graph). It is well known that not all instances on non-bipartite graphs admit a stable matching. Here we present numerical results for the probability that a graph with n vertices and random preference relations admits a stable matching. In particular we find that this probability decays algebraically on graphs with connectivity Θ(n) and exponentially on regular grids. On finite connectivity Erdös-Rényi graphs the probability converges to a value larger than zero. On the basis of the numerical results and some heuristic reasoning we formulate five conjectures on the asymptotic properties of random stable matchings.
The Stable Roommates Problem with Globally-Ranked Pairs
Lecture Notes in Computer Science, 2007
We introduce a restriction of the stable roommates problem in which roommate pairs are ranked globally. In contrast to the unrestricted problem, weakly stable matchings are guaranteed to exist, and additionally, can be found in polynomial time. However, it is still the case that strongly stable matchings may not exist, and so we consider the complexity of finding weakly stable matchings with various desirable properties. In particular, we present a polynomial-time algorithm to find a rankmaximal (weakly stable) matching. This is the first generalization of the algorithm due to Irving et al.