Dynamic Matching Markets and Voting Paths

We consider a matching market, in which the aim is to maintain a popular matching between a set of applicants and a set of posts, where each applicant has a preference list that ranks some subset of acceptable posts. The setting is dynamic: applicants and posts can enter and leave the market, and applicants can also change their preferences arbitrarily. After any change, the current matching may no longer be popular, in which case, we are required to update it. However, our model demands that we can switch from one matching to another only if there is consensus among the applicants to agree to the switch. Hence, we need to update via a voting path, which is a sequence of matchings, each more popular than its predecessor, that ends in a popular matching. In this paper, we show that, as long as some popular matching exists, there is a 2-step voting path from any given matching to some popular matching. Furthermore, given any popular matching, we show how to find a shortest-length such voting path in linear time.

[1]  Robert W. Irving.  An Efficient Algorithm for the "Stable Roommates" Problem , 1985, J. Algorithms.

[2]  Andrew Postlewaite,et al.  Weak Versus Strong Domination in a Market with Indivisible Goods , 1977 .

[3]  Kurt Mehlhorn,et al.  Rank-maximal matchings , 2004, TALG.

[4]  Julián Mestre.  Weighted Popular Matchings , 2006, ICALP.

[5]  Mohammad Mahdian,et al.  Random popular matchings , 2006, EC '06.

[6]  A. Roth,et al.  Random paths to stability in two-sided matching , 1990 .

[7]  Kurt Mehlhorn,et al.  Pareto Optimality in House Allocation Problems , 2005, ISAAC.

[8]  Martin Dyer,et al.  The Stable Marriage Problem: Structure and Algorithms , 1991 .

[9]  Atila Abdulkadiroglu,et al.  RANDOM SERIAL DICTATORSHIP AND THE CORE FROM RANDOM ENDOWMENTS IN HOUSE ALLOCATION PROBLEMS , 1998 .

[10]  Gerhard J. Woeginger,et al.  The complexity of economic equilibria for house allocation markets , 2003, Inf. Process. Lett..

[11]  Akihisa Tamura,et al.  Transformation from Arbitrary Matchings to Stable Matchings , 1993, J. Comb. Theory, Ser. A.

[12]  Eiichi Miyagawa,et al.  Random paths to stability in the roommate problem , 2004, Games Econ. Behav..

[13]  Lin Zhou.  On a conjecture by gale about one-sided matching problems , 1990 .

[14]  Robert W. Irving,et al.  The Stable marriage problem - structure and algorithms , 1989, Foundations of computing series.

[15]  Yufei Yuan,et al.  Residence exchange wanted: A stable residence exchange problem , 1996 .

[16]  W. Gasarch,et al.  Stable Marriage and its Relation to Other Combinatorial Problems : An Introduction to Algorithm Analysis , 2002 .

[17]  H. Landau.  On dominance relations and the structure of animal societies: III The condition for a score structure , 1953 .

[18]  R. Zeckhauser,et al.  The Efficient Allocation of Individuals to Positions , 1979, Journal of Political Economy.

[19]  P. Gärdenfors.  Match making: Assignments based on bilateral preferences , 1975 .