![](bg6.png)
on the subset of individually rational matching,
which always includes the cycle of unstable
matchings. If we look at the following dominance
path {18; 12; 22; 24; 3; 4; 23; 20; 19; 5; 6}, then the
minimum number of transitivity faults is 2 and there
are several rankings with that number of faults. Our
tournament method gives a following ranking (12;
24; 22; 6; 5; 3; 18; 23; 19; 20; 4). There are only two
transitivity faults. As we can see, if the matching 12
is included (and all subsequent matchings along the
dominance path to the cycle) in the tournament, it
always wins.
In the complete information game the matchings
need not even be restricted to the cycle and the path
leading to the cycle, but all the matchings in the
majority voting are “fair game”. If we were to
include all matchings in the tournament, then the
ranking order in our example would be (12; 24; 5; 2;
18; 22; 6; 11; 20; 1; 3; 23; 10; 17; 9; 21; 19; 7; 4; 15;
13; 16; 14; 8). The number of transitivity faults is
13.
If the stable marriage model includes couples,
then the complexity of finding if there exists a stable
matching is NP-complete and “logspace P-hard”
(Ronn 1986, 1987). So for large markets with
couples it may not always be practical to find a
stable matching even when one exists. In this case a
probabilistic matching algorithm can be used to find
a stable matching or a cycle of unstable matchings.
One promising approach would also be using a
genetic algorithm together with majority voting
tournaments to search for the best matching.
4 CONCLUSION
We have described a matching model, where
intransitivities may arise and for this situation we
have proposed using majority voting in a
tournament.
We have also proposed a tournament method
based on monotone systems and a value function for
it. The proposed algorithm should minimize
transitivity faults in tournament ranking and
experimental results show that it does that on tables
up to size 5x5. The proposed method has a
maximum time complexity of O(N
3
) and average
time complexity between O(N
2
) and O(N
3
), thus
enabling to use it on tournament tables of up to (tens
of) thousands of objects.
One open question regarding our proposed
solution is how to select matchings for the
tournament. We have formulated several alternative
answers for that question.
ACKNOWLEDGEMENT
This work was partially supported by ESF Grant
4844.
REFERENCES
Gale, D., Shapley, L., 1962. College Admissions and the
Stability of Marriage. In American Mathematical
Monthly, 69(1), pp.9-15.
Gusfield, D., Irwing, R. W., 1989. Stable Marriage
Problem: Structure and Algorithms, MIT Press Series
in the Foundations of Computing, MIT Press,
Cambridge, Massachusetts.
Gärdenfors, P., 1975. Match Making: Assignments Based
on Bilateral Preferences, In Behavioral Science, 20,
pp.166-173.
Halldórsson, M., Iwama, K., Miyazaki, S., Morita, Y.,
2002. Inapproximability results on stable marriage
problems, In Proceedings of LATIN 2002: the Latin-
American Theoretical INformatics symposium, volume
2286 of Lecture Notes in Computer Science, pp. 554-
568. Springer-Verlag.
Iwama, K., Manlove, D., Miyazaki, S., Morita, Y., 1999.
Stable marriage with incomplete lists and ties, In Proc.
ICALP'99, pp.443-452.
Klijn,F., Masso,J., 2003. Weak stability and a bargaining
set for the marriage model, in Games and Economic
Behavior, 42, pp. 91-100.
Mullat, I., 1976. Extremal Subsystem of Monotone
Systems. In Automation and Remote Control, 5, pp.
130-139; 8, pp. 169-178 (in Russian)
Ronn, E., 1986. On the complexity of stable matchings
with and without ties, Ph.D. diss., Yale University.
Ronn, E., 1987. NP-complete stable matching problems,
Computer Science Department, Technion – Israel
Institute of Technology. Mimeo
Roth, A., Sotomayor, M., 1990. Two-sided matching : a
study in game-theoretic modeling and analysis,
Cambridge University Press, Econometric Society
Monographs, no.18, Cambridge, Massachusetts, USA.
Võhandu, L., 1989. Fast Methods in Exploratory Data
Analysis, In Proceedings of Tallinn Technical
University, Tallinn, No. 705, pp. 3-13.
Võhandu, L., 1990. Best Orderings in Tournaments, In
Proceedings of Tallinn Technical University, Tallinn,
No. 723, pp. 8-14.
MAJORITY VOTING IN STABLE MARRIAGE PROBLEM WITH COUPLES
447