The stable marriage problem (SMP) sm-book is a well-known problem of matching the elements of two sets. Given men and women, where each person expresses a strict ordering over the members of the opposite sex, the problem is to match the men to the women so that there are no two people of opposite sex who would both rather be matched with each other than their current partners. If there are no such people, all the marriages are said to be stable. Gale and Shapley gs proved that it is always possible to solve the SMP and make all marriages stable, and provided a quadratic time algorithm which can be used to find one of two particular but extreme stable marriages, the so-called male optimal or female optimal solution. The Gale-Shapley algorithm has been used in many real-life applications, such as in systems for matching hospitals to resident doctors roth-H and the assignment of primary school students in Singapore to secondary schools revisited. Variants of the stable marriage problem turn up in many domains. For example, the US Navy has a web-based multi-agent system for assigning sailors to ships liebowitz.
One important issue is whether agents have an incentive to tell the truth or can manipulate the result by mis-reporting their preferences. Unfortunately, Roth roth-manip has proved that all stable marriage procedures can be manipulated. He demonstrated a stable marriage problem with 3 men and 3 women which can be manipulated whatever stable marriage procedure we use. This result is in some sense analogous to the classical Gibbard Satterthwaite gibbard; gibbard2 theorem for voting theory, which states that all voting procedures are manipulable under modest assumptions provided we have 3 or more voters. For voting theory, Bartholdi, Tovey and Trick bartholdi proposed that computational complexity might be an escape: whilst manipulation is always possible, there are voting rules where it is NP-hard to find a manipulation.
We might hope that computational complexity might also be a barrier to manipulate stable marriage procedures. Unfortunately, the Gale-Shapley algorithm is computationally easy to manipulate revisited. We identify here stable marriage procedures that are NP-hard to manipulate. This can be considered a first step to understanding if computational complexity might be a barrier to manipulations. Many questions remain to be answered. For example, the preferences met in practice may be highly correlated. Men may have similar preferences for many of the women. Are such profiles computationally difficult to manipulate? As a second example, it has been recently recognised (see, for example, average1; average2) that worst-case results may represent an insufficient barrier against manipulation since they may only apply to problems that are rare. Are there stable marriage procedures which are difficult to manipulate on average?
Another drawback of many stable marriage procedures such as the one proposed by Gale-Shapley is their bias towards one of the two genders. The stable matching returned by the Gale-Shapley algorithm is either male optimal (and the best possible for every man) but female pessimal (that is, the worst possible for every woman), or female optimal but male pessimal. It is often desirable to use stable marriage procedures that are gender neutral masarani. Such procedures return a stable matching that is not affected by swapping the men with the women. The goal of this paper is to study both the complexity of manipulation and gender neutrality in stable marriage procedures, and to design gender neutral procedures that are difficult to manipulate.
It is known that the Gale-Shapley algorithm is computationally easy to manipulate revisited. Our first contribution is to prove that if the male and female preferences have a certain form, it is computationally easy to manipulate any stable marriage procedure. We provide a universal polynomial time manipulation scheme that, under certain conditions on the preferences, guarantees that the manipulator marries his optimal stable partner irrespective of the stable marriage procedure used. On the other hand, our second contribution is to prove that, when the preferences of the men and women are unrestricted, there exist stable marriage procedures which are NP-hard to manipulate.
Our third contribution is to show that any stable marriage procedure can be made gender neutral by means of a simple pre-processing step which may swap the men with the women. This swap can, for instance, be decided by a voting rule. However, this may give a gender neutral stable matching procedure which is easy to manipulate.
Our final contribution is a stable matching procedure which is both gender neutral and NP-hard to manipulate. This procedure uses a voting rule that, considering the male and female preferences, helps to choose between stable matchings. In fact, it picks the stable matching that is most preferred by the most popular men and women. We prove that, if the voting rule used is Single Transferable Vote (STV) handbook-sc, which is NP-hard to manipulate, then the resulting stable matching procedure is both gender neutral and NP-hard to manipulate. We conjecture that other voting rules which are NP-hard to manipulate will give rise to stable matching procedures which are also gender neutral and NP-hard to manipulate. Thus, our approach shows how combining voting rules and stable matching procedures can be beneficial in two ways: by using preferences to discriminate among stable matchings and by providing a possible computational shield against manipulation.
The stable marriage problem (SMP) is the problem of finding a a matching between the elements of two sets. More precisely, given men and women, where each person strictly orders all members of the opposite gender, we wish to marry the men to the women such that there are no two people of opposite sex who would both rather be married to each other than their current partners. If there are no such people, all the marriages are stable.
2.1 The Gale-Shapley algorithm
The Gale-Shapley algorithm gs is a well-known algorithm to solve the SMP problem. It involves a number of rounds where each un-engaged man “proposes" to his most-preferred woman to whom he has not yet proposed. Each woman then considers all her suitors and tells the one she most prefers “maybe" and all the rest of them “No". She is then provisionally “engaged". In each subsequent round, each un-engaged man proposes to one woman to whom he has not yet proposed (the woman may or may not already be engaged), and the women once again reply with one “maybe" and reject the rest. This may mean that already-engaged women can “trade up", and already-engaged men can be “jilted".
This algorithm needs a number of steps that is quadratic in , and it guarantees that:
If the number of men and women coincide, and all participants express a linear order over all the members of the other group, everyone gets married. Once a woman becomes engaged, she is always engaged to someone. So, at the end, there cannot be a man and a woman both un-engaged, as he must have proposed to her at some point (since a man will eventually propose to every woman, if necessary) and, being un-engaged, she would have to have said yes.
The marriages are stable. Let Alice be a woman and Bob be a man. Suppose they are each married, but not to each other. Upon completion of the algorithm, it is not possible for both Alice and Bob to prefer each other over their current partners. If Bob prefers Alice to his current partner, he must have proposed to Alice before he proposed to his current partner. If Alice accepted his proposal, yet is not married to him at the end, she must have dumped him for someone she likes more, and therefore doesn’t like Bob more than her current partner. If Alice rejected his proposal, she was already with someone she liked more than Bob.
Note that the pairing generated by the Gale-Shapley algorithm is male optimal, i.e., every man is paired with his highest ranked feasible partner, and female-pessimal, i.e., each female is paired with her lowest ranked feasible partner. It would be the reverse, of course, if the roles of male and female participants in the algorithm were interchanged.
Given men and women, a profile is a sequence of strict total orders, over the men and over the women. In a profile, every woman ranks all the men, and every man ranks all the women.
Assume . Let and be respectively the set of women and men. The following sequence of strict total orders defines a profile:
(i.e., the man prefers the woman to to ),
For this profile, the Gale-Shapley algorithm returns the male optimal solution . On the other hand, the female optimal solution is .
2.2 Gender neutrality and non-manipulability
A desirable property of a stable marriage procedure is gender neutrality. A stable marriage procedure is gender neutral masarani if and only if when we swap the men with the women, we get the same result. A related property, called peer indifference masarani, holds if the result is not affected by the order in which the members of the same sex are considered. The Gale-Shapley procedure is peer indifferent but it is not gender neutral. In fact, if we swap men and women in Example 1, we obtain the female optimal solution rather than the male optimal one.
Another useful property of a stable marriage procedure is its resistance to manipulation. In fact, it would be desirable that lying would not lead to better results for the lier. A stable marriage procedure is manipulable if there is a way for one person to mis-report their preferences and obtain a result which is better than the one they would have obtained with their true preferences.
Roth roth-manip has proven that stable marriage procedures can always be manipulated, i.e, that no stable marriage procedures exist which always yields a stable outcome and give agents the incentive to reveal their true preferences. He demonstrated a 3 men, 3 women profile which can be manipulated whatever stable marriage procedure we use. A similar result in a different context is the one by Gibbard and Satterthwaite gibbard; gibbard2, that proves that all voting procedures handbook-sc are manipulable under some modest assumptions. In this context, Bartholdi, Tovey and Trick bartholdi proposed that computational complexity might be an escape: whilst manipulation is always possible, there are rules like Single Transferable Vote (STV) where it is NP-hard to find a manipulation stvhard. This resistance to manipulation arises from the difficulty of inverting the voting rule and does not depend on other assumptions like the difficulty of discovering the preferences of the other voters. In this paper, we study whether computational complexity may also be an escape from the manipulability of stable marriage procedures. Our results are only initial steps to a more complete understanding of the computational complexity of manipulating stable matching procedures. As mentioned before, NP-hardness results only address the worst case and may not apply to preferences met in practice.
3 Manipulating stable marriage procedures
A manipulation attempt by a participant is the mis-reporting of ’s preferences. A manipulation attempt is unsuccessful if the resulting marriage for is strictly worse than the marriage obtained telling the truth. Otherwise, it is said to be successful. A stable marriage procedure is manipulable if there is a profile with a successful manipulation attempt from a participant.
The Gale-Shapley procedure, which depending on how it is defined returns either the male optimal or the female optimal solutions, is computationally easy to manipulate revisited. However, besides these two extreme solutions, there may be many other stable matchings. Several procedures have been defined to return some of these other stable matchings gusfield. Our first contribution is to show that, under certain conditions on the shape of the male and female preferences, any stable marriage procedure is computationally easy to manipulate.
Consider a profile and a woman in such a profile. Let be the male optimal partner for in , and be the female optimal partner for in . Profile is said to be universally manipulable by if the following conditions hold:
in the men-proposing Gale-Shapley algorithm, receives more than one proposal;
there exists a woman such that is the male optimal partner for in ;
prefers to ;
’s preferences are ;
’s preferences .
Consider any stable marriage procedure and any woman . There is a polynomial manipulation scheme that, for any profile which is universally manipulable by , produces the female optimal partner for . Otherwise, it produces the same partner.
Consider the manipulation attempt that moves the male optimal partner of to the lower end of ’s preference ordering, obtaining the new profile . Consider now the behaviour of the men-proposing Gale-Shapley algorithm on and . Two cases are possible for : is proposed to only by man , or it is proposed to also by some other man . In this second case, it must be prefers to since is the male optimal partner for .
If is proposed to by and also by some , then, when compares the two proposals, in she will decide for , while in she will decide for . At this point, in , will have to propose to the next best woman for him, that is, , and she will accept because of the assumptions on her preference ordering. This means that (who was married to in ) now in has to propose to his next best choice, that is, , who will accept, since prefers to . So, in , the male optimal partner for , as well as her female optimal partner, is . This means that there is only one stable partner for in . Therefore, any stable marriage procedure must return as the partner for .
Thus, if woman wants to manipulate a stable marriage procedure, she can check if the profile is universally manipulable by her. This involves simulating the Gale-Shapley algorithm to see whether she is proposed by only or also by some other man. In the former case, she will not do the manipulation. Otherwise, she will move to the far right it and she will get her female optimal partner, whatever stable marriage procedure is used. This procedure is polynomial since the Gale-Shapley algorithm takes quadratic time to run.
In a setting with 3 men and 3 women, consider the profile In this profile, the male optimal solution is This profile is universally manipulable by . In fact, woman can successfully manipulate by moving after , and obtaining the marriage , thus getting her female optimal partner. Notice that this holds no matter what stable marriage procedure is used. This same profile is not universally manipulable by or , since they receive just one proposal in the men-proposing Gale-Shapley algorithm. In fact, woman cannot manipulate: trying to move after gets a worse result. Also, woman cannot manipulate since her male optimal partner is her least preferred man.
Restricting to universally manipulable profiles makes manipulation computationally easy. On the other hand, if we allow all possible profiles, there are stable marriage procedures that are NP-hard to manipulate. The intuition is simple. We construct a stable marriage procedure that is computationally easy to compute but NP-hard to invert. To manipulate, a man or a woman will essentially need to be able to invert the procedure to choose between the exponential number of possible preference orderings. Hence, the constructed stable marriage procedure will be NP-hard to manipulate. The stable marriage procedure used in this proof is somewhat “artificial”. However, we will later propose a stable marriage procedure which is more natural while remaining NP-hard to manipulate. This procedure selects the stable matching that is most preferred by the most popular men and women. It is an interesting open question to devise other stable marriage procedures which are “natural” and computationally difficult to manipulate.
There exist stable marriage procedures for which deciding the existence of a successful manipulation is NP-complete.