ON THE INVARIANCE OF MALE OPTIMAL STABLE MATCHING

O N T H E I N V A R I A N C E O F M A L E O P T I M A L S T A B L E M A T C H I N G

R. T. KUO and S. S. TSENG

Institute of Computer Science and Institute of Computer and

Information Engineering Information Science

National Chiao Tung University National Chiao Tung University

Hsinchu, Taiwan 30050, Republic of China. Hsinchu, Taiwan 30050, Republic of China.

Abstract.

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.

CR Categories: F.2.0, G.2.1.

Keywords: Stable matching problem, assignment, preference, invariance.

1. Introduction.

Gale and Shapley [1] introduced and solved the stable matching problem which is a kind of assignment game problem. That problem involves two disjoint sets of equal cardinality n, the men and the women. Each person ranks all members of the opposite sex in order of preference. A stable matching is defined as a complete matching between men and women with the property that no man and woman who are not partners both prefer each other to their actual partners under the matching.

Several stable matching algorithms [1, 5, 6, 7] were proposed to solve the problem by returning the male (or female) optimal stable matching as its answer. Any sequential one of those algorithms will be called a Stable-Matching Algorithm in this paper.

The original stable matching problem has a restriction on the static requirements of the preference lists. The purpose of this paper is to relax this requirement by allowing the dynamic adjustment of the stable matching problem instance after the male optimal stable matching has been found. This would lead to the dynamic stable matching problem. Note that the male optimal stable matching for the adjusted stable matching instance is the solution of the problem. The purpose of this paper is

ON THE INVARIANCE OF MALE OPTIMAL STABLE MATCHING 593 not to find a solution for this problem. O u r main concern is: What kind of alterations can be made to the preferences so that the male optimal stable matching for the resulting instance is identical to the male optimal stable matching for the original? That is, what conditions will lead to the invariance of the male optimal stable matching?

In Section 2 we recall briefly the definitions and some background results of the stable matching problem; in Section 3 we show a set of sufficient conditions on the preference lists, under which the male optimal stable matching is preserved. In Section 4 concluding remarks will be given.

2. Definitions and background results.

An instance of the stable matching problem consists of a set M of n men and a set W of n women. Each member in these two sets has a total rank-ordered preference list of the n people of the opposite sex. F o r convenience, let P(m~) and

P(wi) denote

the preference lists for any man m~ and any w o m a n w j, respectively. When no confusion will occur, Pi will sometimes be used to represent P(mi) or P(w~). Let P(M) and

P(W)

denote the sets of preference lists for all members in M and in W,, respectively. Let the ranking function r(m~, ws) denote the position of w o m a n w s in

P(m~) and

r(wj, mi) denote the position of man m~ in

P(ws). F o r each man mi, let

w s ~ w~ denote that m~ prefers woman wj to woman

Wk. Similarly, for each woman

wi, let m s ~ mk denote that w~ prefers man m s to man m k. Note that the statement "m~ prefers w s to Wk to W[' can be written as w s ~ Wk ~ W~ or m~: wj ~ Wk ~" WZ or more simply m~: W sWk Wz. Let the preference

pattern P = ( P( M), P( W)) = ( P(m O ... P(mn),

P(wl) ... P(w,)) denote the vector of preference lists of all members in M u W, and

let (M, W; P) denote a stable matching problem instance.

A matching # is a one-to-one mapping of the men and the women, i.e., an invertible

function p: M ~ Wsuch that #(mi) is the woman matched with man mi and/~- l(w~) is the man matched with woman wj. The pair

(m, w) blocks

the matching p if m and w are not matched by # but prefer each other to their respective partners given in #. A matching is stable if it is not blocked by any pair. A matching that is not stable is called unstable.

The algorithm proposed by Gale and Shapley [1] finds the Male Optimal Stable

Matching (MOSM) #m, although it can, by relabeling, also find the female optimal

stable matching.

Note that the sequential Stable-Matching algorithms for a solution to a stable matching instance [1, 5, 6] are based on a sequence of proposals from the men to the women. It is shown [5] that the sequence of proposals ends with every w o m a n holding a unique proposal, and that the proposals held constitute a stable matching which is the

MOSM.

A similar outcome results if the roles of males and females are reversed, in which case the resulting stable matching that is the female optimal stable matching may or may not be the same as that obtained from the male proposal sequence. It is pointed out in [5] that the M O S M is female pessimistic, i.e.,

for every woman w, there is no stable matching in which w is matched to a man she prefers less than her partner in the M O S M . Similarly, the female optimal stable matching is male pessimistic. In other words, the achievement of best potential partners by the members of one sex results in the members of the opposite sex having their worst potential partners.

Two fundamental implications of the male proposal sequence, implicit in [5-1, are (i) if m proposes to w then there is no stable matching in which m has a better

partner than w, and

(ii) if w receives a proposal from m then there is no stable matching in which w has a worse partner than m.

F r o m these observations, it is shown [3] that we should explicitly remove m from w's list, and w from m's, if w receives a proposal from someone she likes better than m. Note that when man m is temporarily accepted by woman w, all the men in woman w's preference list who are lower than man m can be eliminated. The resulting lists are referred to as the male-oriented shortlists, for the given problem instance.

In the context of the male-oriented shortlists, a male-oriented rotation [3, 4] exposed in/~ is a sequence

r = (m0, w0) . . . (rap_ 1, wp_ 1)

of pairs from/~ such that, for each i (0 < i < p - 1) (i) w~ is first in mi's shortlist, and (ii) wi + x is second in m~'s shortlist (i + 1 taken modulo p).

Similarly, we can, by relabeling, also define the female-oriented rotation exposed in/~-x. Note that for a given matching there may be many or there may be no exposed rotations.

It is pointed out in [3] that if the first entries in the male-oriented shortlists do not specify the Pw, then at least one rotation must be exposed. The main significance of such a rotation lies in the fact that if, in p,,, each m~ exchanges his partner wi for

w~i+l)modp then the resulting matching is also stable. If, given the rotation r = (too, wo) . . . . , (rap_ 1, wp_ 1), each successor y of mi- 1 in w's shortlists is removed,

together with the corresponding appearance of w~ in y's list, for each i (0 < i < p - 1, i - 1 taken modulo p), then the rotation r is said to have been eliminated. This process is referred to as eliminating a rotation. It is proved in [3] that every stable matching for a given stable matching instance may be obtained from the shortlists by eliminating some sequence of zero or more exposed rotations.

Let us briefly review the Gale-Shapley algorithm [1, 2] here. A L G O R I T H M Gale-Shapley.

Gale-Shapley (M, IV,, P: #,,)

Input: M, W,, and P

Output:

the M O S M #m

At the start of the algorithm, each person is free and becomes engaged during the execution of the algorithm. Once a woman is engaged she never becomes free again,

ON THE INVARIANCE OF MALE OPTIMAL STABLE MATCHING 595 but men can alternate between being free and being engaged. Repeating the follow- ing until all men are engaged:

Choose a free man m, and have m propose to his most preferred woman w on his preferences list, such that w has not already rejected m. If w is free, then w and m become engaged. If w is engaged to man m' (a conflict occurs), then she rejects the

man (m or m') that she least prefers, and becomes, or remains, engaged to the other man. The rejected man becomes, or remains, free.

3. The invariance of the male optimal stable matching.

In view of the dynamization on a given preference pattern, we may ask whether the corresponding male optimal stable matching is still preserved for the new preference pattern. This gives the motivation for finding the conditions on the preference lists which will lead to the invariance of the male optimal stable matching. In the following we will answer the question: Is the MOSM invariant when the

male and female preference lists are subject to r a n d o m permutations without crossover? In other words, can we permute the sublists on either side of the MOSM

on the male or female preference lists?

It is clear that the sublists to the right of the MOSM in the male preference lists

and to the left of the MOSM in the female preference lists cannot effect the MOSM

since they play no part in the Gale-Shapley algorithm. They therefore can be

permuted at will.

As will be shown in the following example, the alterations to the right of the

MOSM in the female preference lists can cause the MOSM to change.

EXAMPLE. Given ({ 1 .. . . . 8}, { 1 .. . . . 8}; P) with the preference pattern P defined by M: Male's preference lists W: Female's preference lists

1: 5 8 2 3 1 4 7 6 1: 6 1 8 7 4 3 2 5 2: 2 1 3 8 7 5 4 6 2: 3 7 4 5 8 2 6 1 3: 7 3 1 4 8 6 2 5 3: 8 6 1 2 7 4 3 5 4: 4 7 1 3 6 2 5 8 4: 2 1 6 7 8 3 5 4 5: 1 3 8 2 6 4 5 7 5: 4 3 8 5 1 2 7 6 6: 2 8 6 7 3 1 4 5 6: 7 4 5 3 6 1 2 8 7: 5 3 4 7 1 6 8 2 7: 1 7 6 8 2 4 5 3 8: 4 8 2 5 7 6 3 1 8: 6 5 1 4 2 3 7 8.

With the given instance, the MOSM #,. is

Suppose that the male preference pattern is left alone and the female preference pattern is changed to:

1: 6 1 8 7 4 2* 3* 5 2: 3 7 4 5 8 6* 2* 1 3: 8 6 1 2 7 4 5* 3* 4: 2 1 6 7 8 3 5 4 5: 4 3 8 5 7* 1" 2* 6 6: 7 4 5 3 6 1 2 8 7: 1 7 6 8 2 4 5 3 8: 6 5 1 4 2 3 7 8.

Note that only those figures marked with *'s are changed from the original. The

M O S M for the resulting instance becomes {(1, 8), (2, 1), (3, 6), (4, 7), (5, 3), (6, 2), (7, 5),

(8,4)}.

All that remains is to show that the permutations to the left of the M O S M in the male preference lists leave the M O S M invariant.

THEOREM. Given any arbitrary instance (M, W; P) and its M O S M It,n, the inter-

change operation applied on a sublist to the left of the M O S M in the male preference lists cannot effect the MOSM.

PROOF. We first show that such interchange leave the M O S M as a stable match- ing. This is straightforward because any pair which caused instability in the new instance would have caused instability in the original M O S M and this is a contradic- tion because the original M O S M is stable.

Next, we claim that there is no female-oriented rotation exposed in the original

M O S M in the female preference pattern of new instance. Suppose, to the contrary,

that there exists such a rotation r. By the definition of the new instance, the rotation r must also be a female-oriented rotation exposed in M O S M in the female preference pattern for (M, W; P). This implies that the better potential partners of the male could be achieved by the elimination of r from the female-oriented shortlists corresponding to the M O S M in the female preference pattern for the (M, W; P), and contradicts that #m is the M O S M for (M, W; P). Thus there does not exist any female-oriented rotation exposed in the original M O S M for the female preference pattern of the new instance. Combining this with our claim that the original M O S M is stable for the new instance, we conclude that the original M O S M is also the

M O S M for the new instance.

The proof is now complete. •

COROLLARY. Given any arbitrary instance (M, W;P) and its M O S M Itm, any

permutation of a sublist to the left of the M O S M in the male preference lists cannot effect the MOSM.

ON THE INVARIANCE OF MALE OPTIMAL STABLE MATCHING 597 PROOF. It is well known that a permutation can be made up of a sequence of appropriate interchanging operations. By repeatedly applying the theorem above we can easily show that the permutation of a sublisst to the left of the M O S M in the male preference lists leave the M O S M invariant. Thus, the corollary follows. •

Further, by repeated application of the above corollary, it is easy to show that the permutations to the left of the M O S M in the male preference lists leave the M O S M invariant.

4. Concluding remarks.

In this paper, we have presented 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. The following is an outline of our results. (1) The preference (2) The preference (3) The preference (4) The permutations of the lists cannot effect the permutations of the lists cannot effect the permutations of the

sublists to the left of the M O S M in the male

M O S M .

sublists to the right of the M O S M in the male

M O S M .

sublists to the left of the M O S M in the female lists cannot effect the M O S M .

permutations of the sublists to the right of the M O S M in the female preference lists can cause the M O S M to change.

Note that the optimality-preserving conditions given in this paper are useful in the study of the stable matching problem with dynamic preference orderings.

In closing this remark, let us pose two interesting problems. The first one is an extension of the results given in this paper, i.e. the consideration of the effects of crossover on the invariance of the M O S M . The second one is stated as follows: Given any stable matching instance I = (M, W; P) and its M O S M #,,, does there exist any coalition C of members from M such that

[r(m, #re(m)) -

r(m,

p~(m))] > O,

m ~ C

r(m, I~m(m)) >- r(m, #'(m)),

V

m ~ C, and

r(m,#m(m)) = r(m,#'~(m)), V m ~ M - C

under the conditions that some members i n C give false preferences, where #" is the

M O S M for the instance corresponding to the falsified preference pattern? How identify one, if it exists? Note that the results given in this paper have shown that there is no effect on the M O S M if those members in C give their falsified preferences satisfying the optimality-preserving conditions.

Acknowledgement.

The authors are grateful for the valuable and constructive comments made by the anonymous referees.

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