A Study of Incentive Compatibility and Stability Issues in Fractional Matchings

01/16/2020 ∙ by Shivika Narang, et al. ∙ 0

Stable matchings have been studied extensively in both economics and computer science literature. However, most of the work considers only integral matchings. The study of stable fractional matchings is fairly recent and moreover, is scarce. This paper reports the first investigation into the important but unexplored topic of incentive compatibility of matching mechanisms to find stable fractional matchings. We focus our attention on matching instances under strict preferences. First, we make the significant observation that there are matching instances for which no mechanism that produces a stable fractional matching is incentive compatible. We then characterize restricted settings of matching instances admitting unique stable fractional matchings. Specifically, we show that there will exist a unique stable fractional matching for a matching instance if and only if the given matching instance satisfies what we call the conditional mutual first preference property (CMFP). For this class of instances, we prove that every mechanism that produces the unique stable fractional matching is (a) incentive compatible and (b) resistant to coalitional manipulations. We provide a polynomial-time algorithm to compute the stable fractional matching as well. The algorithm uses envy-graphs, hitherto unused in the study of stable matchings.

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1 Introduction

Matchings have been studied for several decades now, beginning with Gale and Shapley’s pioneering work [gale1962college]. They introduced the notion of stability and provided algorithms for finding stable matchings. Since then, a considerable amount of work has been carried out on both the theory and applications of stable matchings. Matching mechanisms already in use have also been for their stability and incentive compatibility aspects [Correa2019chile, Baswana2019india, gonczarowski2019matching]. The focus of these studies has often been school choice mechanisms or residency matching mechanisms already in practice [abdulkadirouglu2009strategy, abdulkadirouglu2005boston].

In these familiar settings, nodes are wholly or “integrally” matched. We shall call such matchings as integral matchings. A fractional matchings is essentially a convex combinations of integral matchings. While they do have a lot of practical relevance, fractional matchings are not relevant for settings such as school choice. Consequently, they have only been studied in literature as a means to produce integral matchings. Even papers that explicitly study fractional allocations only use them towards a deeper understanding of integral matchings [roth1993stable, teo1998geometry, sethuraman2006many].

1.1 Fractional Matchings

There are many practical situations where fractional matchings are relevant. Consider for instance, labour markets. Here freelancing experts or professionals can spend different fractions of their time working for various organizations. Labour markets have been mentioned in Caragiannis et al. [caragiannis2019stable], who were the first to study the stability of fractional matchings. Another relevant application would be markets for cloud space where it is not necessary for data to be stored entirely on one server.

Fractional allocations are also relevant for settings where matchings must be done repeatedly. An excellent example comes from the chore division setting. Consider a house shared by multiple people. There are various tasks or chores that must be done around the house. Each person does each task differently and as a consequence generates a different value by performing the task. However each person may not take an equal amount of time or energy when performing a given task. As a result, a person incurs a cost for performing each task. Further, these tasks must be done repeatedly over time and it is perfectly fine for different agents to do a particular task at different times. Consequently, we require a fractional matching that indicates the fractions of times agents perform each task.

Stability is a crucial requirement for fractional matchings as well. For instance, in the chore division setting, it is not desirable to have a division of tasks where there is a person who would incur less cost by always doing another activity and would do it better. Exploring fractional matchings from contexts studied for integral matchings can raise interesting challenges.

Relaxing the integrality constraint in matchings may make the problem harder. For instance, the problem of finding a stable, social welfare maximizing integral matching can be posed as a linear program. However, when we allow for matchings to be fractional, the problem becomes

NP-Hard. This was shown by Caragiannis et al. [caragiannis2019stable]. They also show that by allowing the stable matchings to be fractional, we can make large gains in terms of social welfare. Thus, it is of interest to study fractional matchings and design algorithms to find fractional matchings with desirable properties.

Another crucial requirement for matching mechanisms is incentive compatibility, i.e. the matching mechanism should induce all participating agents to reveal their true preferences. This paper explores the problem of finding incentive compatible mechanisms that produce stable fractional matchings.

1.2 Our Contributions

We focus our study on matching settings where all agents provide their preferences in the form of cardinal valuations. Further, each agent’s valuations are assumed to imply a strict (preference) order of the agents on the other side. We will be defining these terms more formally in Section 3.1. A review of relevant work is presented in Section 2. Section 3.2 provides relevant structural observations on the space of stable fractional matchings. These observations bring out several hurdles to the design of algorithms to find stable fractional matchings which are not integral.

The contribution of this paper is in the important but unexplored topic of incentive compatibility of matching mechanisms to find stable fractional matchings. We focus on matching instances under strict preferences.

  • First, in Section 4, we make the significant observation that there are matching instances for which no mechanism that produces a stable fractional matching is incentive compatible.

  • We then characterize, in Section 4.2, restricted settings of matching instances that admit unique stable fractional matchings. Specifically, we show that there will exist a unique stable fractional matching if and only if the given matching instance satisfies what we call the conditioned mutual first preference property (CMFP).

  • When the input is restricted to the above class of instances, every mechanism that produces the unique stable fractional matching will be incentive compatible. Furthermore, the output of this mechanism is resistant to coalitional manipulations for the above setting.

  • We next provide, in Section 4.3, the first algorithm to compute stable fractional matchings which are not integral. Our polynomial-time algorithm makes intelligent use of envy-graphs, hitherto unused in the stable matchings literature.

2 Relevant Work

Work in the study of matchings began with Gale and Shapley’s seminal work [gale1962college]. They considered a setting with men and women where each woman had preference list over the set of all men and vice versa. They defined the notion of stability and showed that in the stable marriage problem, a stable matching always exists. Our work also considers the same setting, with the restriction that the preferences are available in cardinal form. Gale and Shapley’s original paper initiated decades of work into the rich field of matchings, studying various aspects of stable matchings in particular. Roth established that no stable (integral) matching mechanism is incentive compatible for all agents [roth1982economics]. He showed however that the Gale Shapley algorithm is optimal and incentive compatible for the proposing side. Kojima and Pathak [kojima2009incentives] studied incentive compatibility of the college admissions problem. They gave regularity conditions under which the fraction of the population with an incentive to lie goes to 0.

A part of the matchings literature is devoted to the manipulation of the mechanism given by Gale and Shapley. Teo et al. [teo2001gale] gave a polynomial-time algorithm to compute the optimal manipulation of each woman. Here the underlying mechanism implements the Gale Shapley algorithm with men proposing. Vaish and Garg [vaish2017manipulating] show that the matching returned by the Gale Shapley algorithm on the optimal manipulation is also stable under the true preferences. They further show that for each woman there is an optimal inconspicuous manipulation111An inconspicuous manipulation is one where the position of exactly one agent is changed in the preference list.. Further, the lattice of stable matchings under this manipulation is contained in the lattice of stable matchings on the true preferences. Shen et al. [shen2018coalition] give a polynomial-time algorithm for each coalition to find their optimal manipulation. They further show that any such manipulation can be made inconspicuous.

Gale and Shapley’s work while having profound consequences gave no intuition into the structure of the space of stable matchings. As a result, much of the initial work on matchings tried to explore this. Work by Roth et al. [roth1993stable] and Teo and Sethuraman [teo1998geometry] used linear programming formulations to capture and analyze the stable marriage problem. With linear programming formulations, fractional matchings also become an important part of the analysis. Both establish the integrality of the polytope. The fractional allocations studied in both papers are said to be stable if the closest integral matching is stable. In general, in the majority of literature a fractional matching is said to be stable if the matchings in its support are stable. Most matching algorithms which find fractional matchings use them to round to an integral matching.

An important setback in the initial analysis of fractional matchings was that the preferences available are ordinal and not cardinal222That is, agents’ preferences were given in the form of preference lists and not numeric values.. Consequently, there was no non-trivial way to define the stability of a fractional matching. Note that, a fractional matching is a convex combination of integral matchings. Caragiannis et al. [caragiannis2019stable] overcome this by considering a stable matching setting where cardinal valuations are available. Thus, it is now possible to analyse the social welfare. In many instances fractional matchings are able achieve much higher social welfare as compared to integral matchings. Consequently, they aim to find a stable fractional matching which maximizes social welfare. They call this problem as .

The authors give a series of structural inferences about the space of stable fractional matchings, some algorithmic results and finally concluding by showing hardness of approximation for . An important observation is that the space of stable fractional matchings is not convex. The authors show a simple example where the convex combinations of two stable integral matchings is not stable333This further shows that fractional allocations in the stable matchings polytope studied in [roth1993stable]and [teo1998geometry] need not in fact be stable themselves. In the followup work in [sethuraman2006many] the authors call the fractional allocations fractional stable matchings indicating that they are not discussing the stability of these fractional matchings but are only interested in their being the convex combination of stable matchings.. It is relevant to note that Caragiannis et al., in fact, do not give an algorithm to find a stable fractional matching which is not integral, whenever one exists, irrespective of the social welfare.

Our work addresses this notable gap. We design a polynomial-time algorithm to find a stable fractional matching, which is not integral, whenever one exists, under strict preference orders. This condition, while being realistic and only mildly restrictive for practical purposes, is critical to our construction. This in turn also helps us characterize the instances for whom a unique stable fractional matching exists.

Envy-graphs are essential to our analysis and had previously been used largely only in work on fair division [chevaleyre2007allocating, chaudhury2019little]. Envy is defined very differently in matching settings. “Justified” envy has been studied for the many-to-one matching setting, such as that of school choice, and is a relaxation of stability in these settings [wu2018lattice, aziz2019random]. Other notions of fairness have also been studied for various matching applications [aziz2019matching, huang2016fair, nguyen2019stable]. Some work has also been done in designing incentive compatible mechanisms for finding fair matchings in constrained settings [yahiro2018strategyproof, zhang2018strategyproof]. However, the notions of fairness studied here are not the same as those studied in fair division literature. This is largely because fair division settings assume cardinal valuations. In contrast, traditional matching settings assume ordinal valuations.

3 Preliminaries

Having laid out the positioning of our work, we now discuss relevant preliminaries for our analysis. We shall refer to an instance of our problem of finding stable fractional matching as a stable matchings instance.

3.1 Definitions

We represent a stable matching instance as . Here, is the set of men and , is the set of women. The valuations of men and women are captured by matrices and respectively. In particular, is ’s valuation for being matched integrally to . Analogously, is ’s valuation for being matched integrally to . We assume that all entries of and are non-negative and that a linear order can be derived from the valuations of one agent. That is, for each man there do not exist two distinct women such that . Similarly, for each woman there do not exist two distinct men such that . We say that such preferences are strict.

Matching problems are traditionally studied as graph problems. Let us denote the induced bipartite graph for a stable matching instance as where and . Given , we shall use to denote that is incident on . Fractional matchings can be thought of in two equivalent ways.

Definition 1 (Fractional Matching).

is said to be a fractional matching on if such that .

An alternate way of looking at fractional matchings is to think of them as convex combinations of integral matchings, where integral matchings are defined as follows:

Definition 2 (Integral Matching).

Given a graph is said to be an integral matching if for each there is at most one edge in which is incident on .

By Birkhoff-von Neumann theorem, we given a fractional matching as defined in 1, we can decompose it into a convex combination of integral matchings. We will largely consider fractional matchings as defined in Definition 1. Let us clarify some notation with regards to matchings. For an integral matching and , denotes ’s partner under . We shall say that fractional matching is a subset of , when (i) they are defined for the same instance and (ii) for each such that if , then . Note that it is necessary for the underlying instance to be the same for this definition to make sense.

Before defining the stability of fractional matchings, we must define a blocking pair in the context of fractional matchings. We say that form a blocking pair under matching if both get strictly less utility from than they get by being matched integrally with each other. The utility of a woman under fractional matching is . Thus, it is essentially the weighted sum of the utility from each of the integral matchings in the support of . The utility of a man can be analogously defined. Hence, we can now define stability for fractional matchings.

Definition 3 (Stable Fractional Matchings).

A fractional matching is said to be stable is there does not exist a pair of agents such that and .

Recall that our model is identical to that of Gale and Shapley with the exception that the preferences are available in cardinal form. As a result, we can always derive an instance of the type studied by Gale and Shapley given . Consequently, a stable integral matching always exists. Further, no stable matching will have unmatched agents, as the number of men and women are equal. Also, it is easy to see that integral matchings that are stable under our definition are also stable under Gale and Shapley’s original definition. Hence, throughout our analysis, we will assume these facts, without explicitly stating them.

We explore the existence of incentive compatible mechanisms to find stable fractional matchings. This paper aims for what is generally known as Bayesian Incentive Compatibility. That is, we shall say that a mechanism is incentive compatible if truthful revelation of preferences by all agents is a Nash Equilibrium for all input instances. We show that there does not exist a mechanism to find a stable fractional matching which is incentive compatible for all agents. This clearly negates any possibility of a mechanism where truthful revelation is a dominant strategy even for general settings.

We identify a special class of stable matching instances having a unique stable fractional matching. We show that, in fact, any stable matching instance which has a unique stable fractional matching belongs to this class. Further, when the input instances are restricted to those that belong to this class, we have that any mechanism which finds a stable fractional matching is incentive compatible. Moreover, when all remaining agents are truthful, no coalition can strategically collude and misreport their valuations to increase their own utilities. We call this property being resistant to coalitional manipulations.

In defining this class we rely heavily on identifying pairs of nodes that satisfy the MFP property. We call a man woman pair to be MFP (mutual first preference) if they are each others’ first preferences. That is, and are said to be MFP if . Note that for any stable matching instance with strict preferences, if there exist a pair of nodes that are MFP, they must be matched under every stable matching.

An important contribution of our work is to give a polynomial-time algorithm to find a stable fractional matching, which is not integral, whenever one exists. This algorithm is instrumental in establishing that whenever there are no MFP pairs, under strict preferences, a stable fractional matching which is not integral can be found. This, in turn, characterizes the instances which have unique stable fractional matchings. Envy-graphs are critical in establishing these results. Before defining the construction of an envy-graph, we define envy.

Definition 4 (Envy).

Given stable matching instance and fractional matching , for any , is said to envy under if

Similarly, for any , is said to envy under if

For integral matching , envies under if . We can further define, given a matching, the envy-graph under that matching as follows.

Definition 5 (Envy-graph).

Given stable matching instance and fractional matching , the envy-graph on women under is a directed graph where envies under . The envy-graph on men under , , can be analogously defined.

When there are no MFP agents, we can use envy-graphs to find stable fractional matchings which are not integral.

3.2 Structural Observations

(a) Non-convexity
(b) Unstable Support
(c) Unstable Support
Figure 1: Counterexamples

Before we present our analysis, it is important to demonstrate that it is non-trivial to compute stable fractional matchings which are not integral. We now list some structural observations regarding the space of stable fractional matchings.

Observation.

The convex combination of stable fractional matching need not always be stable.

Caragiannis et al. [caragiannis2019stable] illustrate this with an example which does not have strict preferences.Figure 0(a) demonstrates that this holds even with strict preferences. There are exactly two stable integral matchings. These are: , , and , , . Fractional matching is not stable for all , as form a blocking pair. Thus, we have that the space of stable fractional matchings is not easy to iterate over.

Note that for any instance, stable integral matchings are a subset of stable fractional matchings. However, knowing the space of stable integral matchings does not give information about the space of stable fractional matchings. We shall now demonstrate this through two observations.

Observation.

A matching instance may have a unique stable integral matching but multiple stable fractional matchings.

Consider the stable matching instance represented in Figure 0(b). The unique stable integral matching is , , . Consider matching , , . Fractional matching is stable for all .

This observation is consequential in conjunction with the observation that there may be stable fractional matchings with only unstable integral matchings in the support.

Observation.

There exist stable fractional matchings whose support consists solely of unstable integral matchings.

This can be illustrated by the stable matching instance described in Figure 0(c). The unique stable integral matching is . The matchings , and are all unstable. However, the fractional matching is in fact stable.

These observations demonstrate that the space of stable fractional matchings can be unintuitive, even under strict preferences. Finding a stable fractional matching which is not integral is not straightforward. Given any integral matching, it is not clear how to see whether it is present in the support of a stable fractional matching. Further, if so, what should be the weight on this matching. Our analysis of the incentive compatibility of stable fractional matching procedures proposes a way to overcome these hurdles.

4 Results on Incentive Compatibility

Real-world applications often would want the agents involved to act according to their true preferences. Roth [roth1982economics] showed that it is not possible to have any incentive compatible mechanism to find stable integral matchings. However, stable integral matchings form a mere subset of stable fractional matchings, and it is not explicit whether incentive compatible mechanisms exist in this case. We now resolve this question by demonstrating that this, in fact, is not possible.

4.1 Impossibility in General Settings

We first consider the algorithm given by Caragiannis et al.[caragiannis2019stable]. The algorithm essentially proceeds as follows: first find a stable matching, considering only heavy edges444These are edges where both agents have positive valuations for each other. Edges that are not heavy are said to be light. This matching is then completed by finding a maximum weight matching for the unmatched agents by only considering the light edges.

They do not explicitly state the algorithm used to compute stable matchings. However, as a consequence of [roth1982economics], the algorithm will be incentive compatible for at most one side. Simple counterexamples suffice to show that neither of the algorithms are incentive compatible for any set of agents. Consider the example in Figure 1 where there are two men and two women.

Figure 2: Algorithms in [caragiannis2019stable] are not incentive compatible

Here, there are only two integral matchings possible, both are stable and have the same social welfare. For this input, the algorithm given may return either of the two integral matchings. In both the integral matchings only one of the two sides will be happy and the agents on the other side will benefit by misreporting their preferences. Suppose the matching is . If even one of the women reports her value for the man she likes as any value greater than 1, say the algorithm will then yield the matching . For the other matching men can analogously misreport their preferences.

In fact, we can show that there is no incentive compatible mechanism to find a stable fractional matching. chooses a value of for the matching .

(a) true valuations
(b) misreports
(c) misreports
Figure 3: Counterexample for DSIC mechanism
Theorem 1.

There is no incentive compatible mechanism to find stable fractional matchings which gives incentives for truthful revelation of preferences to all agents on all inputs.

Proof.

Consider the stable matching instance described in Figure 2(a). There are exactly two stable integral matching , , and , , . Further fractional matching is stable for all . No other fractional matching is stable for this instance. Thus any stable fractional matching algorithm when run on this instance essentially chooses a value of for the matching .

It is easy to see that would like to be matched to integrally, i.e. for the given stable fractional matching algorithm to choose value . Let the algorithm choose . can now misreport his preferences as shown in Figure 2(b) with . Under this instance, the only fractional matchings stable are for . Thus, if a given algorithm to chooses some , has an incentive to misreport his preferences as shown in Figure 2b with , giving him strictly higher utility. Similarly, would like to be matched integrally to . Thus, she can misreport her preferences as in Figure 2(c) to ensure that she receives higher utility whenever the algorithm chooses a value of .

Thus no mechanism resulting in a stable fractional matching can be incentive compatible for all the agents when there are no restrictions on the input instances. ∎

4.2 Incentive Compatibility under Restricted Settings

We know from Roth [roth1982economics] that there is no incentive compatible mechanism for finding stable integral matchings in general. However, when there is a unique stable integral matching we have an exception. Under the mechanism implementing the Gale Shapley algorithm, truthful revelation of preferences by all agents forms a Nash equilibrium. This is a simple consequence of prior work. Roth [roth1982economics] showed that when the Gale Shapley algorithm is run with men as proposers, it is a dominant strategy for men to be honest. By [teo2001gale] women can find their optimal manipulation for a given instance in polynomial-time. This optimal manipulation matches her with her best possible partner under Gale Shapley, when men propose. Vaish and Garg [vaish2017manipulating] show that the resultant matching is also stable under the true preferences. Thus, each woman’s partner, even after the optimal manipulation, will remain the same.

This motivates us to look at the class of instances where there is a unique stable fractional matching and see if there are incentive compatible mechanisms for this class. While there has been much work on finding stable integral matchings, so far there is no work which aims at finding a stable matching which is not integral. For some instances, such a matching need not exist. Previous work establishes that there is always a stable integral matching, and thus one stable fractional matching always exists. It is of interest to have an algorithm, which when given a stable matching instance, finds a stable fractional matching which is not integral whenever one exists. To this end, we first try to categorize the class of instances where there exists a unique stable fractional matching. Note that there is a unique stable fractional matching in the following settings.

Consider the idealistic “soulmate” setting where if man ’s first preference is , then ’s first preference is . Recall that we call such pairs of nodes as MFP pairs (mutual first preference). Here it is easy to see that the unique stable matching is the one where everyone is matched to their first preference.

Another setting where there is a unique stable fractional matching is where all men having identical preferences and all women have identical preferences. We shall call this setting the “popularity” setting. Here the only stable matching is where the most popular man is matched with the most popular woman for all . This is because the most popular man and woman must be matched as they are each other’s first preference. Given this, now the second most popular man and woman become each other’s first preference as the most popular man and women are now unavailable. In fact any combination of these two settings will have a unique stable fractional matching.

Note that if any stable matching instance has MFP pairs, any stable matching must match them. Based on this, we give the following polynomial-time algorithm which returns a matching which must be a subset of any stable matching.

Input:
Output:
;
;
;
;
while  s.t.  do
       ;
       is with and removed;
       ;
      
;
Algorithm 1 CMFPmatching

We use this algorithm to define a special class of stable matching instances called Conditioned Mutual First Preference (CMFP).

Definition 6.

We say that a stable matching instance is in class CMFP if and only if Algorithm 1 returns a perfect matching when is given as input.

Lemma 1.

Given any stable matching instance , Algorithm 1 returns a matching that is a subset of any stable integral matching on . Any stable fractional matching for instance I must set a weight of 1 each pair contained in the matching returned by Algorithm 1.

Proof.

We prove this by contradiction. Let us assume that this is not true. That is, there is a stable matching instance where is the matching returned by Algorithm 1 on , and there is a stable matching on such that is not a subset of . Let555If then is the empty matching, and as a result is a subset of every matching on . . Let and such that , , , where are matched in the round in Algorithm 1.

Let be the lowest index in such that . As is stable at least one of and must have higher utility for than from matching integrally with each other. Without loss of generality, let this be . By construction of Algorithm 1, this is only possible when for some . Thus, , . This is a contradiction as we assumed to be the lowest index for which this happens.

Thus we have the result. ∎

As a result, we have a polynomial-time algorithm to tell when instance is CMFP. Algorithm 1 helps us fix certain edges which must be included in any stable matching. Further, if the matching returned is a perfect matching then it is the unique stable fractional matching for the given instance. This is a consequence of Lemma 1. The class CMFP is also special in that incentive compatibility can be achieved for this class.

Theorem 2.

Given any matching instance belonging to CMFP, under any mechanism to find a stable fractional matching, truthful revelation of preferences forms a Nash Equilibrium.

Proof.

Given a stable matching instance , every mechanism resulting in a stable fractional matching will return the same matching . We use Algorithm 1 to give us labellings and such that , , , where are matched in the round in Algorithm 1.

Let all other agents be truthful. Clearly, have no incentive to misreport their preferences as they are already matched to their first preference. As long as and stay truthful, they will continue to be matched to each other, irrespective of how other agents are behaving. Now for for neither can gain by increasing or decreasing their value for any agent matched earlier. This is because the agents who are matched before round are truthful and will not become MFP pairs with or . Thus, those pairings will not change. Of the remaining agents, and have highest value for each other and cannot benefit from misreporting their preferences. Consequently, when all other agents are truthful, no agent has an incentive to misreport their preferences. ∎

In fact, the same reasoning also shows that the stable matching in a CMFP instance is also robust to coalitional manipulation.

Corollary 1.

For each stable matching instance in , no coalition can collude to strategically misreport their preferences and improve their utilities under any mechanism to find a stable fractional matching.

Consequently, when the matchings instances are restricted to those in , we have incentive compatible mechanisms which result in stable fractional matchings. What we now show is that in fact, this is the only set of instance where there is a unique stable fractional matching.

4.3 Matching Instances with Unique Stable Fractional Matchings

We shall now design a polynomial-time algorithm to find a stable fractional matching which is not integral, whenever . There is no algorithm (irrespective of the time complexity) to find a stable fractional matching which is not integral in prior work. The various hurdles to this were illustrated in Section 3.2. However, when the preferences are strict, we can use envy to help find stable fractional matchings given a stable integral matching. The assumption of strict preferences is very important to establish our results.

Note that by Lemma 1, the matching returned by Algorithm 1 must be a subset of any stable matching. As a result, it suffices to have an algorithm which works on instances without MFP pairs. We can first run Algorithm 1 and then use the aforementioned algorithm on the reduced instance returned by Algorithm 1.

The key idea is that under strict preferences, when there are MFP pairs, then it is not possible to reduce the weight on the edge between them and still be stable. However, whenever there are no MFP pairs, each edge matched under a stable matching will have at least one node who prefers another agent to their current partner. As a result, we can use envy-graphs to construct other matchings which improve the utility of some of the agents. As the preferences are strict, there will always exist a small enough weight to place on these matchings such that there are no blocking pairs with other nodes. While it is not necessary to explicitly construct envy-graphs to help find such matchings, envy-graphs help in expressing these ideas in an intuitive manner.

Cycles in Envy-Graphs

Let be a stable integral matching and assume contains a cycle. We can produce an alternate matching where all women not in are matched as in and all women in are matched to their successor’s666The successor of a node on a directed path or a cycle is the vertex to which it has an outgoing edge in the path/cycle. There can only be one such vertex. partner under .This resultant matching need not be stable, even if the original was stable. However, it will increase the utility of the women in the cycle. Define .

Let us now discuss the stability of . In this case, no woman sees a decrease in utility. As a result, it suffices to ensure that no man experiences a large enough drop in utility for a blocking pair to form. Let us first consider men whose partners are unchanged. They do not form any blocking pairs. This is because any woman such a man prefers more to his current partner either has the same or higher utility. Now, consider the men whose partners do change. Note that as is stable, these men prefer their partners under over those under . Let us define . In order for to be stable we need that for each such that , . As we have strict preferences, a value of can be found in polynomial-time. Clearly a fractional matching can also analogously be found when there is a cycle in .

Acyclic Envy-Graphs

The approach of using envy-graphs can be extended when the given instance has no MFP pairs and both envy-graphs are acyclic. Let be such a stable integral matching on an instance which has no MFP pairs. Consequently, at least one of and will be non-empty. Without loss of generality, let this be . If this is acyclic, every path on the graph will have a sink. Note that a node in an envy-graph is a sink if and only if the corresponding agent matched to their first preference. Further, as there are no MFP pairs, the woman matched to this node under will have an outgoing edge in . The reason we are able to attain stability by resolving envy along cycles is that all agents in the cycle get higher utility. When there is a path, the sink, let us call this , will be matched to the partner of the first node on the path, whom she clearly prefers less. Further prefers to her predecessor as is a stable matching. As a result, to ensure stability, we need to increase the utility of .

We resolve this as follows. Let the path found be in with source and sink . Define matching where each man not in is matched as in . is matched to . The remaining nodes along are matched to their successor’s partner under . As there are no MFP pairs, we have that has an outgoing edge in . Find a similar path to a sink, say in and analogously define . If lies on , then we are done. Else, we repeat the same procedure for . We continue till the partner of the sink of the path considered lies on a previous path considered. For each new path encountered, we define a new integral matching.

These matchings form the support of the stable fractional matching. To achieve stability, we need to set weights on the matchings which ensure that the utility of each agent is greater than their utility from the fractional matching from matching with agents who prefers them to their current allocation. This is possible as the preferences are strict. As in the case of when cycles are present, a linear program can be solved to find the weights on and all newly defined matchings to find a stable fractional matching.

Input:
Output:
CMFPmatching();
;
if  is non-empty then
        ;
        GaleShapley();
        Generate Envy-Graphs and ;
        if Either or contain a cycle  then
               if  cycle  then
                      ;
                     
              else
                      ;
                      Find cycle ;
                     
              Define as follows;
               if  and  then
                      ;
                      ;
                     
              else
                      ;
                      ;
                     
              
       else
               Set of agents whose utilities increase;
               Find sink node ;
               );
               ;
               while   do
                      ;
                      Find path in ;
                      Define as follows;
                      if  and  then
                             ;
                             ;
                            
                     else
                             ;
                             ;
                             if  then
                                    ;
                                   
                            
                     ;
                      if  then
                            ;
                     else
                            ;
                     
              
       Find such that and
        , and
        , ;
        ;
       
Algorithm 2 Find a stable fractional matching

Before detailing the algorithm, we first set some notation. and are the envy-graphs of men and women under as defined. For each if else . For each if else . For cycle and node , denotes ’s successor under . For path and node , if is the sink of the path, then denotes the source of the path, else denotes ’s successor under .

Time Complexity of Algorithm 2: A cycle or a path in an envy-graph can always be found in polynomial-time. Thus, computing a new matching can be done in polynomial-time. As each matching is defined to improve the utility of a previously unimproved agent, there can be at most matchings. Further, the required values can be found by solving a linear program with the appropriate constraints. The number of variables are , which is the number of matchings. Thus, this linear program can be solved efficiently. Consequently, the algorithm detailed is a polynomial-time algorithm. It finds a stable fractional matching which is not integral whenever one exists.

Theorem 3.

A stable matching instance has a unique stable fractional matching if and only if it is in CMFP.

Proof.

Let be stable matchings instance. We have already established that if , has a unique stable fractional matching. We now prove the other direction by establishing the contrapositive. Let , and and be the matching and instance returned by Algorithm 1 on . Clearly, is not a perfect matching. Any matching on which is stable, can be combined with to obtain a stable matching for .

Let be the stable matching returned by Gale Shapley on with men proposing. Note that, as is the instance returned by Algorithm 1, there is are no MFP pairs. Consequently, at least one of and will be non-empty. Algorithm 2 will give a stable fractional matching which is not integral. ∎

A simple corollary of Theorem 3 is that under strict preferences, a stable matching instance has either a unique stable fractional matching, or uncountably many.

5 Conclusions and Future Work

This paper looks into the design of incentive compatible mechanisms for finding stable fractional matchings. We first showed that this is not possible under general settings. We then discovered a class of stable matchings instances which have a unique stable fractional matching, namely those satisfying the Conditioned Mutual First Preference () property. We showed that every mechanism that finds a stable fractional matching is incentive compatible if and only if the input instances are in . We presented the first algorithm to compute stable fractional matchings which are not integral. The algorithm makes intelligent use of envy-graphs, hitherto unused in the stable matchings literature.

Our work suggests several interesting directions of future work. Firstly, the hardness of finding an optimal manipulation for our algorithm is not clear. Another relevant direction of future work would be to find algorithms to find stable fractional matchings and are hard to manipulate. Another possibility would be investigating if it is possible to achieve incentive compatibility by relaxing the stability constraint. Envy-graphs are an essential tool for the analysis done in this paper. An interesting problem is whether the use of envy-graphs can give better approximation algorithms for finding social welfare maximizing stable fractional matchings. Finally, it would be important to see if the same results hold without strict preferences.

References