MAJORITY VOTING IN STABLE MARRIAGE PROBL

EM WITH

COUPLES

Using a monotone systems based tournament approach

Tarmo Veskioja, Leo Võhandu

Institute of Informatics, Tallinn Technical University, Raja 15, Tallinn, 12618, Estonia

K

eywords: GDSS, stable matching, voting, tournament, monotone systems, intransitivity

Abstract: Providing centralised matching services can be viewed as a group decision support system (GDSS) for the

participants to reach a stable matching solution. In the original stable marriage problem all the participants

have to rank all members of the opposite party. Two variations for this problem allow for incomplete

preference lists and ties in preferences. If members from one side are allowed to form couples and submit

combined preferences, then the set of stable matchings may be empty (Roth et al., 1990). In that case it is

necessary to use majority voting between matchings in a tournament. We propose a majority voting

tournament method based on monotone systems and a value function for it. The proposed algorithm should

minimize transitivity faults in tournament ranking.

1 INTRODUCTION

Stable marriage problem has attracted a considerable

amount of interest after the problem was first

formulated by Gale and Shapley (Gale et al., 1962).

Many centralized two-sided markets can be

described as variants of the stable marriage problem.

An instance of the original stable marriage

problem (SM) consists of N men and N women,

with each person having a preference list that totally

orders all members of the opposite sex. A man and a

woman form a blocking pair in a matching if both

prefer each other to their current partners. A

matching is stable if it contains no blocking pair. In

every instance of SM there is at least one stable

matching (Gale et al., 1962).

A variant of SM allows for incomplete

preferences. This problem is denoted SMI (Stable

Marriage with Incomplete preferences). The

definition of a blocking pair is extended, so that each

member of the pair prefers the other instead of the

current partner or is currently single and acceptable.

Every instance of SMI has at least one stable

matching, although it may not always be a maximum

cardinality matching. If a player is single in one

stable matching, then that player is single in any

other stable matching.

Another variant of SM allows for ties in the

preferences. This problem is denoted SMT (Stable

Marriage with Ties). In this case the definition of

stability needs to be extended. A man and a woman

form a blocking pair if each strictly prefers the other

to his/her current partner. A matching without such a

blocking pair is called weakly stable. Every instance

of SMT has at least one stable matching.

A variant of SM that allows for both incomplete

preferences and ties in the preferences is denoted

SMTI. In this problem there always exists a weakly

stable matching (Iwama et al., 1999), but the sizes of

stable matchings may vary. Finding maximum

cardinality matching in SMTI is NP-complete

(Iwama et al., 1999) and even the approximation is

APX-hard (Halldórsson et al., 2002).

A hospital-residents assignment (HR),

sometimes also called stable admissions problem or

many-to-one matching, is a variant of SM, where

many residents can be assigned to one hospital and

one resident can fill in only one vacancy. HR

problem can also have relaxations of preferences,

allowing for incomplete preferences and/or ties.

These subproblems are denoted HRI and HRTI

accordingly. Most of the properties of SM, SMI and

SMTI carry over to the HR, HRI and HRTI

problems and algorithms.

Matching is a majority assignment (best-voted

matching) if there is no other matching that is

442

Veskioja T. and Võhandu L. (2004).

MAJORITY VOTING IN STABLE MARRIAGE PROBLEM WITH COUPLES - Using a monotone systems based tournament approach.

In Proceedings of the Sixth International Conference on Enterprise Information Systems, pages 442-447

DOI: 10.5220/0002655904420447

Copyright

c

SciTePress

preferred by a majority of participants to the original

matching. Gärdenfors (Gärdenfors, 1975) observed

that, when the preferences are strict, the set of

majority assignments comprises the set of stable

matchings, thus showing that the notion of majority

assignment is a relaxation of stability (Klijn et al.,

1999). Weakly stable matching is a matching,

possibly having a blocking pair undermining the

stability of a matching, but this blocking pair is not

credible in the sense that one of the partners may

find a more attractive partner with whom he forms

another blocking pair for the original matching

(Klijn et al., 1999). In other words, Klijn and Masso

define an individually rational matching to be

weakly stable if every blocking pair is - in the sense

discussed above - not credible. Clearly, weak

stability is also a relaxation of stability.

Many markets also require taking into account

some additional constraints - for example in HR a

pair of residents may have formed a couple and

prefer to find a placement at the same hospital, or at

least work in the same city. In this case, the couple

submits rank ordered preferences over acceptable

pairs of hospitals. After acceptable pairs of hospitals

the couple can give rank ordered preferences over

single pairs of hospital – couple member, where one

of the members of the couple is left without a pair.

In this article these mentioned constraints will be

called couple constraints.

Matching markets with couple constraints may

not have any stable matchings (Roth et al., 1990). In

that case it is natural to use majority voting to find

the best matching. We propose a heuristical majority

voting tournament method based on monotone

systems and a value function for it. The proposed

algorithm should minimize transitivity faults in

tournament ranking. Preliminary results (described

in more detail in a paper submitted to a conference

CAISE’04) show that our proposed method

successfully minimizes transitivity faults on all

possible tournament tables of size 5x5. It is feasible

to check the performance of the proposed method on

all tables up to size 7x7 or 8x8, but beyond that it is

only feasible to compare it against other heuristical

methods and on selected tournament tables.

In the next chapter we give the definitions of

domination and the core of a game (Roth et al.,

1990; pages 54-55, 166-167). The definitions are

needed to understand the importance of stability and

the core. Then we use an example of a matching

model with couple constraints from Roth (Roth et

al., 1990) to show that it has intransitivities and

every dominance path of matchings leads to the

cycle of unstable matchings.

In the third chapter we give a simple definition

of a ranking algorithm based on a monotone system,

we describe a specific tournament algorithm and

show that it works on the example.

The fourth chapter is for the conclusions.

2 THE CORE OF A MARRIAGE

GAME

The following are the definitions of domination and

the core of a game (Roth et al., 1990; pages 54-55,

166-167).

Definition 1. For any two feasible game

outcomes x and y, x dominates y if and only if there

exists a coalition of players S such that

(a) every member of the coalition S prefers x to

y; and

(b) the rules of the game give the coalition S the

power to enforce x (over y).

For this reason, if x dominates y, we might

expect that y will not be the outcome of the game.

This leads us to consider the set of undominated

outcomes.

Definition 2. The core of a game is the set of

undominated outcomes.

We can relax the domination conditions of

definition 1, assuming that the coalition can make

side-payments to those players that are indifferent

between outcomes x and y.

Definition 3. For any two feasible game

outcomes x and y, x weakly dominates y if and only

if there exists a coalition of players S such that

(a) every member of the coalition S prefers x at

least as much as y; and

(b) at least one member of the coalition S prefers

x to y; and

(c) the rules of the game give the coalition S the

power to enforce x (over y).

Definition 4. The core of a game defined by

weak domination is the set of weakly undominated

outcomes.

According to the first two definitions the core of

the one-to-one matching market equals the set of

stable matchings (Roth et al., 1990, chapter 3.1,

theorem 3.3). When preferences are strict, the two

cores coincide in the one-to-one matching model,

but not in the many-to-one model. However, when

hospital preferences are responsive (as defined in

Roth et al., 1990, definition 5.2, page 128), and

when preferences over individuals are strict, the set

MAJORITY VOTING IN STABLE MARRIAGE PROBLEM WITH COUPLES

443

of stable matchings coincides with the core defined

by weak domination (Roth et al., 1990; proposition

5.36, page 167). In the many-to-one (or one-to-one)

matching model with couples, the set of stable

matchings and consequently the core may be empty

(Roth et al., 1990, theorem 5.11, page 141). Lets

look at the example that Roth & Sotomayor gave to

illustrate this problem.

2.1 An empty core example of many-

to-one model with couples

The following example is taken from Roth &

Sotomayor (Roth et. al., 1990; theorem 5.11, page

141).

Consider the market with hospitals H = {H

1

, H

2

,

H

3

, H

4

} each of which offers exactly one position

and each of which has strict preferences over

students S = {s

1

, s

2

, s

3

, s

4

} as given in Table 1. The

students consist of two married couples, {s

1

, s

2

} and

{s

3

, s

4

}. Each couple has strict preferences over

ordered pairs of hospitals, as given in Table 1.

Table 1: Preferences of hospitals and couples

Hospitals' rank

orders

Couples' rank

orders

H

1

H

2

H

3

H

4

{s

1

,s

2

} {s

3

,s

4

}

s

4

s

4

s

2

s

2

H

1

H

2

H

4

H

2

s

2

s

3

s

3

s

4

H

4

H

1

H

4

H

3

s

1

s

2

s

1

s

1

H

4

H

3

H

4

H

1

s

3

s

1

s

4

s

3

H

4

H

2

H

3

H

1

H

1

H

4

H

3

H

2

H

1

H

3

H

3

H

4

H

3

H

4

H

2

H

4

H

3

H

1

H

2

H

1

H

3

H

2

H

2

H

3

H

2

H

3

H

1

H

2

H

2

H

4

H

1

H

4

H

2

H

1

H

1

H

3

Thus couple {s

1

, s

2

} has as its first choice that s

1

be matched with H

1

and s

2

with H

2

, and has its last

choice that s

1

be matched with H

2

and s

2

with H

1

.

The 24 individually rational matchings of students to

hospitals are listed in Table 2, along with the reason

that each such matching is unstable.

Table 2: Every matching is unstable

Matching

H

1

H

2

H

3

H

4

Unstable with

respect to

1

s

1

s

2

s

3

s

4

s

4

,H

2

2

s

1

s

2

s

4

s

3

s

4

,H

2

3

s

1

s

3

s

2

s

4

s

2

,H

4

4

s

1

s

3

s

4

s

2

s

4

,H

1

5

s

1

s

4

s

2

s

3

s

2

,H

4

6

s

1

s

4

s

3

s

2

s

4

,H

1

7

s

2

s

1

s

3

s

4

s

4

,H

1

8

s

2

s

1

s

4

s

3

s

4

,H

2

9

s

2

s

3

s

1

s

4

s

2

,H

4

10

s

2

s

3

s

4

s

1

s

4

,H

1

11

s

2

s

4

s

1

s

3

s

2

,H

4

12

s

2

s

4

s

3

s

1

s

4

,H

1

13

s

3

s

1

s

2

s

4

s

4

,H

2

14

s

3

s

1

s

4

s

2

s

2

,H

3

15

s

3

s

2

s

1

s

4

s

2

,H

4

16

s

3

s

2

s

4

s

1

s

2

,H

3

17

s

3

s

4

s

1

s

2

s

1

,H

1

18

s

3

s

4

s

2

s

1

s

2

,H

1

19

s

4

s

1

s

2

s

3

s

4

,H

2

20

s

4

s

1

s

3

s

2

s

2

,H

3

21

s

4

s

2

s

1

s

3

s

2

,H

4

22

s

4

s

2

s

3

s

1

s

2

,H

3

23

s

4

s

3

s

1

s

2

s

3

,H

3

24

s

4

s

3

s

2

s

1

s

4

,H

4

Thus matching 1, which assigns student s

i

to

hospital H

i

, i=1,…,4, is unstable because both

hospital H

2

and couple {s

3

, s

4

} would prefer that

student s

4

be matched with H

2

. (This follows since

H

2

prefers s

4

to s

2

, and {s

3

, s

4

} prefers H

3

H

2

to

H

3

H

4

.)

The domination graph between matchings is

shown on Figure 1.

If we study the dominance between these

matchings, it becomes clear that every dominance

path leads to the following cycle of unstable

Figure 1: Domination graph

ICEIS 2004 - ARTIFICIAL INTELLIGENCE AND DECISION SUPPORT SYSTEMS

444

matchings {6, 20, 19, 5} (in the order of

dominance), entering the cycle from one of the

matchings 6, 20 or 5.

Roth & Sotomayor (Roth et al., 1990, page 142)

formulated an open question whether there exist

plausible restrictions on the preferences of the

couples that would insure that stable matchings

always exist. We suggest that studying these

restrictions in the context of minimizing transitivity

faults is a more fruitful approach. If there are few

enough transitivity faults left in the tournament

ranking, then at some point stable matchings should

emerge.

As soon as the players of the marriage game

realize that there is no stable outcome, they start

looking for a way out of this vicious cycle, at least in

a cooperative game. In doing that the players will

start seeking coalitions to reach an outcome through

majority voting. The existence of cyclic domination

also means the existence of intransitivity. So to

reach an outcome, the players have to vote between

pairwise matchings as in a tournament.

Note that the only way to avoid a tournament is

to not let the existence of the cycle to become

common knowledge. The blocking pair (or one of

them) to the last matching in the cycle can choose to

not form a pair by themselves, but to seek coalition

partners to seek out the best matching in the cycle

for the coalition and to dominate over other

matchings.

Lets look at the tournament between the

matchings. An algorithm based on monotone

systems is used for the tournament.

3 FINDING BEST SOLUTIONS

WITH A TOURNAMENT

The aim of the tournament method is to minimize

transitivity inconsistencies in a ranking. Transitivity

requires that if solution a is better than solution b

and solution b is better than solution c, then solution

a must be better than solution c. With special

constraints (for example permitting couples to

submit combined preferences) this transitivity may

not always hold in a marriage model.

The first criterion for ranking is to minimize the

number of transitivity faults. If there are several

different rankings with the same minimal number of

transitivity faults, then a second-order criterion must

be used. The minimal transitivity inconsistency

ranking in a tournament problem is NP-hard, that is

why an efficient heuristic tournament method is

needed. One class of good heuristic methods for this

problem is based on monotone systems (Mullat,

1976; Võhandu, 1989, 1990).

3.1 An algorithm based on a

monotone system

Definition 5. (A weakly) monotone system is a

system built on a set of objects, such that

(a) objects are weighted by a value function

(b) after removal of one object from the set all

the weights of other objects still in the set

change monotonically in one direction

(increase or decrease) or stay on the same

level.

Algorithms based on such a simple monotone

system work as follows:

Step 1. Evaluate all objects in the set.

Step 2. Find the weakest object (with the

smallest (largest) weight), and remove it from the

set. If there are several weakest objects, then

recursively apply the tournament algorithm to the set

of weakest objects. If at any stage of the recursion

any object was removed from the set of weakest

objects, then backtrack. If the set of weakest objects

still contains more than one object, then compare the

weights from the previous iteration and choose an

object that is more similar to the previously removed

object. If the weights in all the previous iterations

are the same, then according to the value function

these objects are equivalent and we can remove any

one of those (usually the first object will be

removed).

Step 3. If there are still objects in the set, then

continue from Step 1.

Any given algorithm always removes the object

with the smallest weight, or the largest weight.

Algorithm cannot change the choice function (min,

max) during the course of action. Value function can

be chosen relatively freely, as long as it satisfies

monotonicity condition. The sequence of removal of

objects constitutes object ranking.

3.2 Tournament method based on a

monotone system

To construct a tournament method for the stable

marriage problem we need to define a value function

and an ordered set of object removal criteria.

In a majority voting, all the players have to vote

(pairwise) between the matchings. Voting results

MAJORITY VOTING IN STABLE MARRIAGE PROBLEM WITH COUPLES

445

constitute the voting table v. Voting table for the

cycle of unstable matchings {6, 20, 19, 5} is given

in Table 3.

Table 3: Voting table

v 5 6 19

20

5 3 5 6

6 3 4 3

19

1 4 5

20

2 3 1

The tournament table t is computed based on the

voting table v. We compare votes of all pairs of

matchings v

rc

and v

cr

and make the following

transformations:

If v

rc

< v

cr

then t

rc

=0, t

cr

=1;

If v

rc

> v

cr

then t

rc

=1, t

cr

=0;

If v

rc

= v

cr

then t

rc

=0, t

cr

=0.

The tournament table for the cycle of unstable

matchings {6, 20, 19, 5} is given in Table 4.

Our proposed method makes use of both the

number of wins (rowsums) and losses (column

sums). The method iteratively finds the weakest

object, removes it from the tournament table and

adds it to the ranking. If we were to remove one

matching, then the winning points (and also the

losing points) of all the other matchings will

decrease or stay on the same level, so the system is a

(weakly) monotone one.

The process of finding the weakest object to

remove is also iterative. In one iteration the number

of wins and losses in the remaining subset of the

weakest objects are calculated at first. The weakest

objects are selected by the minimum number of wins

and then by the maximum number of losses. This

iterative minimax selection is used until either only

one weakest object remains or the last minimax

selection was not able to reduce the number of

weakest objects. In the latter case the last remaining

weakest object in the original ranking is removed.

The last remaining object in the tournament table is

the winner.

Table 4: Tournament table

t 5 6 19

20 Iter1 Iter2 Iter3 Iter4

5 0 1 1 2 1 0 0

6 0 0 0 0 0 0

19 0 0 1 1 0

20 0 0 0 0

Wins

Iter1

0 0 1 2

Iter2

0 0 1

Iter3

0 0

Iter4

0 Losses

In the first iteration matchings 6 and 20 have no

wins, but the number of losses are 0 and 2

accordingly. Matching 20 is removed first based on

the number of losses. Values from column 20 are

subtracted from the winning points (row sums) of

remaining matchings. Values from row 20 are

subtracted from the losses (column sums) of

remaining matchings.

In the second iteration matchings 6 and 19 have

no wins. Based on the number of losses (0 and 1)

matching 19 will be removed. Wins and losses of the

remaining matchings are recalculated.

In the third iteration matchings 5 and 6 have no

wins. Voting between them gave a draw. Both have

no losses, since voting between them gave a draw.

One way to differentiate between the two matchings

is to look at the wins (and then losses) before the

first iterations. Matching 5 had one win in the

previous iteration, so matching 6 has to be removed

first and matching 5 will be removed last.

The obtained tournament ranking is (5, 6, 19, 20)

and matching 5 is the best matching. Note that

simple majority voting does not always produce

transitivity faults in the cycle of unstable matchings,

since even if one matching is dominated by the other

in the sense of stability, the voting between the two

matchings may still be a draw. One can, however,

define a rule that if voting between two matchings

gives a draw then the second criterion to decide the

better one is the domination. Clearly, such a rule

introduces intransitivities inside the cycle of

unstable matchings.

The proposed method has been tested to give a

ranking with minimum number of transitivity faults

on all tournament tables (including ties) up to size

5x5 (results described in more detail in a paper

submitted to a conference CAISE’04). 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.

3.3 How to select matchings for the

tournament

When using majority voting in a full tournament one

has to have a relatively small set of matchings (up to

thousands or tens of thousands). Since the number of

individually rational matchings is combinatorial, the

selection of matchings for majority voting

tournament becomes critical.

One solution is to hold a tournament between the

set of matchings in the cycle of unstable matchings.

A stable matching searching algorithm can be used

to find the cycle.

It would be interesting to know whether the

outcome of the majority voting tournament depends

ICEIS 2004 - ARTIFICIAL INTELLIGENCE AND DECISION SUPPORT SYSTEMS

446

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.

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