Unique Stable Matchings

In this paper we consider the issue of a unique prediction in one to one two sided matching markets, as defined by Gale and Shapley (1962), and we prove the following. Theorem. Let P be a one-to-one two-sided matching market and let P be its associated normal form, a (weakly) smaller matching market with the same set of stable matchings, that can be obtained using procedures introduced in Irving and Leather (1986) and Balinski and Ratier (1997). The following three statements are equivalent (a) P has a unique stable matching. (b) Preferences on P* are acyclic, as defined by Chung (2000). (c) In P* every market participant's preference list is a singleton.


Introduction
We consider the classic model of one-to-one two-sided matching with nontransferable utility, the so-called stable matching problem. The framework, introduced in Gale and Shapley (1962), has been applied widely to settings ranging from college admissions, labour markets, the market for kidney donors, refugee resettlement, and others (see Roth (2008) for a survey). Yet despite all the attention that the model has received, a very basic question has remained unanswered: under what conditions is there exactly one stable matching? In this paper we provide necessary and sufficient conditions on the preferences of market participants that guarantee a unique stable matching.
Determining when a matching problem has a unique stable matching is of great practical importance. First of all, uniqueness is typically viewed as a desirable property in any economic model as it takes care of the issue of prediction and is useful for comparative statics. Second, in the context of allocation problems, uniqueness has the additional benefit that it ensures Pareto efficiency and guarantees that the truthful reporting of preferences is strategy proof (Sönmez, 1999). Third, there is a literature that highlights the role of incomplete information in matching environments (Roth, 1989;Liu et al., 2014), and the recent paper of Fernandez et al. (2021) shows that matching problems with a unique stable matching are more resistant to the infusion of uncertainty. Finally, and perhaps most importantly, unique stable matchings appear often in real-world matching markets, so a better understanding of what assures uniqueness might lead to both greater efficiency and greater transparency in practice. 1 Due to the problem's importance, a host of conditions on preferences that are sufficient for uniqueness have been proposed. 2 We show that imposing 1 Unique stable matchings have been observed in a wide range of settings including the National Resident Matching Program (Roth and Peranson, 1999), Boston school choice (Pathak and Sönmez, 2008), online dating (Hitsch et al., 2010), and the Indian marriage market (Banerjee et al., 2013). Ashlagi et al. (2017) are "not aware of any evidence of a large core in a matching market", and they shed some light on why it is that uniqueness is more likely to occur in "unbalanced markets" (those with strictly more participants on one side).
conditions on the full preference lists is, in a sense to be made precise later, excessive. Rather what matters is how preferences operate on the essential part of any matching problem, the subproblem that we term the normal form. We show, in Theorem 1, that a unique stable matching is equivalent to preferences being acyclic on the normal form. We emphasise that neither the definition of acyclic preferences (Chung, 2000), nor the concept of a matching problem's normal form (Balinski and Ratier, 1997), are original to this paper. 3 Rather our contribution is to combine the two. We explain both in detail now. (The working example we use, and the one that we carry throughout this paper, is that of a labour market with workers on one side of the market and firms on the other.) In our setting of workers and firms, the shortest possible cycle has length 4 and is described as follows: the 1 st worker prefers the 2 nd firm to the 1 st firm, the 2 nd firm prefers the 2 nd worker to the 1 st worker, the 2 nd worker prefers the 1 st firm to the 2 nd firm, and the 1 st firm prefers the 1 st worker to the 2 nd worker. Acyclic preferences are simply those that do not possess a cycle of the kind described above. In fact this very condition was already applied to the uniqueness question in Romero-Medina and Triossi (2013) (and considered briefly in Eeckhout (2000)). However, requiring that preferences are acyclic is by itself only sufficient for a unique stable matching. To add necessity, one need only require the weaker condition that preferences are acyclic on the normal form. In Example 2 we present a matching problem with preferences that are not acyclic, and thus do not satisfy the aforementioned sufficient conditions, and yet there is a unique stable matching since the preferences are acyclic on the normal form.
So what then is the normal form of a matching problem? The normal form is arrived at by stripping away parts of the preference lists that are not relevant to the set of stable matchings. 4 We term the procedure that discards the irrelevant information the iterated deletion of unattractive alternatives (IDUA). The IDUA procedure works by repeatedly pivoting around a particular kind of blocking pair. Suppose that firm f is worker w's most preferred firm. Then there cannot be a stable matching that matches f to a worker that it prefers less than w, since w and f would form a blocking pair. We say that all the workers less preferred to w by f are unattractive to f. Since f can guarantee doing better than these unattractive workers in every stable matching, we can delete these workers from f's preference list. Similarly, all the workers deleted by f will realise that a match with f ain't happening and so will delete f from their preference lists -unattractiveness is reciprocated. 5 Importantly, further rounds of deletion may be possible with the simplified preference lists because market participants that were not initially unattractive can become so. Eventually the deletion procedure can go no more, and what remains is the normal form.
The IDUA procedure that reduces a matching problem to its normal form parallels closely the iterated deletion of dominated strategies (IDDS) applied to strategic games. Once IDDS stops, the set of surviving strategies are the only "rational" way to play a game. That is, IDDS strips a game of strategies that no rational individual who fully understands the environment could ever justify choosing; the resulting game is smaller and yet the set of solutions remains unchanged. The IDUA procedure has a similar effect on matching problems in that while it discards information, no "important" information is discarded in the sense that it only deletes pairs that cannot be part of any stable matching; the resulting matching problem is smaller and yet the set of solutions remains unchanged. 6 In Section 2.3 we compare and contrast these two procedures with particular focus on the sorts of higher order reasoning required for each to be operationalised. Just as an ultra sophisticated player can determine when a game is dominance solvable, so too could an ultra sophisticated market participant determine when a matching problem has a unique stable matching.
We stated above that Theorem 1 shows the equivalence between a unique stable matching and preferences being acyclic on the normal form. But in fact we show what appears, on first inspection, to be a much stronger equivalence. That is, whenever there is a unique stable matching, the IDUA procedure whittles down every market participant's preference list to a singleton, where all that remains in the normal form is the unique stable matching and nothing else. (Given the similarity between IDDS for strategic games and IDUA 5 That is, to be clear, if a pair of this kind is ever blocked then it is always blocked. 6 So from a practical perspective running IDUA should be the first port of call when considering a matching problem. It reduces the size of the input, rendering it more tractable, without affecting the desired output (the set of stable matchings). In much the same way, one typically begins the analysis of a strategic game by looking for dominated strategies.
for matching problems, this seemingly stronger equivalence result is, in a sense, the matching problem analog of a strategic game being "dominance solvable".) While the IDUA procedure can be justified by inferences based on higher ordering reasoning, it also admits another, more practical, interpretation that we believe is interesting. The deferred acceptance algorithm of Gale and Shapley (1962) can be imagined as an interactive forum in which one side of the market are assigned as active proposers and the other side as passive responders. Given this perspective, the IDUA procedure can be interpreted as a dynamic marketplace wherein, perhaps more realistically, all participants are simultaneously both proposing and responding to proposals. While Theorem 1 confirms that a matching market operating in such a fashion will not fully "clear" unless there is a unique stable matching, the market will become smaller and easier to parse. 7 Interpreted in this way, IDUA has the flavour of a bargaining situation or price discovery mechanism typically more associated with the traditional "markets with prices", in the sense that it closes in on the upper and lower boundaries of the core. 8 Given any matching, our techniques allow an analyst to identify the full set of preference lists with the property that the given matching is the unique stable matching for each of the preference lists. This is done by effectively running IDUA "in reverse". This works because a matching can be viewed as a collection of preference lists with only one participant per list. One can then augment these preference lists by adding participants in such a way that the IDUA procedure would have deleted the additional entries. While this does not yield a succinct condition on preferences like acyclicity (or any of the conditions provided by the papers listed in Footnote 2), and the procedure may be computationally intensive, it is exhaustive. That is, for every matching one can construct the entire set of equivalent instances, where equivalence is defined as possessing the same unique stable matching.
7 The market will only not become smaller in the statistically rare and economically unusual case that every participant's favourite partner views them as least desirable.
8 For a more classical setting with this kind of feature, imagine a seller who owns an object and values it at e5, and a buyer who values the same object at e10. In a real-world bargaining situation the buyer may start out with a bid far short of e5 while the seller may first make a demand far in excess of e10; only when one party crosses into the interval [e5, e10] does the real bargaining begin. Viewed like this, the deferred acceptance algorithm bestows upon the proposing side in a two-sided market the sort of extreme bargaining power afforded the proposer in the ultimatum game (Harsanyi, 1961;Güth et al., 1982).
Ever-shortening preference lists can be cumbersome to manage. Fortunately the directed graph (digraph) approach to matching introduced by Maffray (1992) allows all the information of a matching problem to be encoded on one digraph instead of 2n preference lists, allowing reductions to be handled easily. This equivalent formulation is simple to explain. Instead of "lining up" the participants on opposite sides of the market, one constructs a grid wherein each row corresponds to a specific worker, each column corresponds to a specific firm, and every vertex in the grid corresponds to a pair. Preferences are depicted by horizontal arcs (directed edges) for workers and vertical arcs for firms. A matching is depicted by a subset of vertices no two of which are in the same row nor the same column. The acyclic condition corresponds to no cycle in the matching digraph. In this formulation, the IDUA procedure involves deleting vertices and arcs, and if at some point the reduced digraph becomes acyclic, one can immediately conclude that there is a unique stable matching.
The paper proceeds as follows. Section 2 introduces the matching problem, the IDUA procedure, and the normal form. 9 Section 3 presents our result classifying uniqueness. In Section 4 we introduce the directed graph approach to stable matchings, illustrate how the IDUA procedure works in this setting, and finally we sketch how an analyst can recover, and thereby classify, the full collection of preference lists for which a given matching is the unique stable matching. Section 5 concludes.
2 Matching Problems and their Normal Form

Matching problems
Let W be a set of n workers and let F be a set of n firms. Each worker w ∈ W has a strict preference relation, w , over the set of firms, and each firm f ∈ F has a strict preference relation, f , over the set of workers. That is, if worker w prefers firm f to firm f then we will write f w f , with an analogous statement for firms. A preference relation w is said to be 9 Many of the properties of IDUA used in our proof of the main result, Theorem 1, were already obtained by Balinski and Ratier (1997) using matching digraphs. To make our paper self-contained, we prove every property that we use and whenever a result of ours is similar to one from Balinski and Ratier (1997) we explicitly state it as such. Our main result, Theorem 1, is completely new.
complete if all firms are in the relation; similarly for f .
We have the following definition.
Definition 1. An instance of the stable matching problem, P , is defined as the pair ({ w } w∈W , { f } f∈F ), where { w } w∈W and { f } f∈F are the collection of complete preference relations, one for each worker and each firm.
Note that we are assuming that all preferences are complete. We do so purely for convenience. All results in the paper can be shown to hold for incomplete preference lists, with the caveat that in such instances there can exist stable matchings in which not all market participants are assured to be matched. In fact, one of the main points we wish to make with this paper is that assuming that all preference relations are complete is typically, except in a small fraction of cases, excessive. To emphasise this point as strongly as possible it makes sense to begin with complete preference lists.
A matching is a mapping µ from W ∪ F to itself such that for every worker w ∈ W , µ(w) ∈ F ; for every firm f ∈ F , µ(f) ∈ W ; and for every The following is the key definition first introduced by Gale and Shapley (1962).
Definition 2. Worker w and firm f form a blocking pair with respect to matching µ, if they prefer each other over their partners in µ. That is, f w µ(w) and w f µ(f).
In words, (w, f) form a blocking pair with respect to µ when both would prefer to break away from their current partners and pair up themselves. The notion of a matching being stable is defined by the absence of blocking pair.
Definition 3. A matching µ with no blocking pairs is called a stable matching. Gale and Shapley (1962) introduced the deferred acceptance algorithm and used it to prove the following result.
Theorem (Gale and Shapley (1962)). Every instance of the matching problem has a stable matching.
The following example illustrates the above definitions.
Example 1. Let P 1 be an instance of the stable matching problem in which there are three workers and three firms. That is, let W = {w 1 , w 2 , w 3 } and let F = {f 1 , f 2 , f 3 }. Preferences are given as follows.
It can be checked that P 1 possesses two stable matchings, µ 1 and µ 2 . They are, While Example 1 above is simple, in the next subsection we will use this example to highlight how an instance of the stable matching problem may contain more information than is required in order to compute the set of stable matchings.
To give an example of what we mean by this, consider Example 1 from the perspectives of worker w 1 and firm f 2 . Since w 1 's most preferred firm is f 2 and f 2 's most preferred worker is w 1 , it must be that the pair (w 1 , f 2 ) are matched in any stable matching since otherwise (w 1 , f 2 ) would form a blocking pair. This is confirmed by the two matchings, µ 1 and µ 2 , listed in (1). But we can build on this. Given that w 1 will certainly be matched with their most favourite f 2 in every stable matching, the fact that w 1 has a relative preference for f 1 over f 3 is irrelevant. By this we mean that if w 1 's preferences were altered so that his relative preference for f 1 over f 3 was swapped the set of stable matchings would remain unchanged. In fact, we observe that if worker w 1 's preferences were incomplete, such that they preferred to be unmatched over being matched with either f 1 or f 3 , the set of stable matchings would still remain unchanged. This motivates the notion of an unattractive alternative which is the subject of the next section.

Unattractive Alternatives
The informal discussion following Example 1 above highlights that for some instances of the matching problem, the input supplied may exceed that which is required to compute the set of stable matchings. That is, we saw that since worker w 1 was highly sought-after, nothing changed if they have a very incomplete preference list. But then the preference lists of those who are deemed unattractive by sought after participants often contain too much information as well. We now formalise this.
Definition 4 (Unattractive alternatives). We say that (iii) f is an unattractive alternative to worker w whenever w is an unattractive alternative to firm f (and vice versa).
In words, Part (i) of Definition 4 says the following: worker w deems firm f unattractive if w will definitely do better than f in any stable matching. This will occur when w is the most preferred firm of some firm f that w prefers to f, because then w and f form a blocking pair to any matching that matches w and f. Part (ii) of Definition 4 is the analog to part (i) but for firms instead of workers. Part (iii) stipulates that unattractiveness is reciprocated.
Let us now revisit Example 1 using this new terminology. We begin by considering condition (i). As noted in the previous section, firm f 2 is worker w 1 's most preferred firm. Given that w 1 's most desired match is firm f 2 , the other two firms f 1 and f 3 are unattractive to w 1 . We then have that f 2 w 1 ∼ w 1 {f 1 , f 3 }, where we have gathered the collection of w 1 's unattractive alternatives in a set in which the order that they are listed is immaterial. By similar reasoning, condition (ii) yields If we now pause and consider how the above impacts instance P 1 of Example 1, we can see that P 1 is, from the perspective of the market participants, identical to the instance P 1 , where P 1 is defined as, W = {w 1 , w 2 , w 3 } and F = {f 1 , f 2 , f 3 }, and preferences are as follows: 11 Let us now consider condition (iii) from Definition 4. Since worker w 1 has deemed both firms f 1 and f 3 unattractive, condition (iii) requires that both f 1 and f 3 reciprocate. The reason for this is that a match with w 1 is not happening for either of these firms, so maintaining w 1 in one's preference list serves no purpose. A similar statement holds for workers w 2 and w 3 , both of whom reciprocate unattractiveness to firm f 2 . This means that instance P 1 above is also identical to instance P 1 , where P 1 is defined as, W = {w 1 , w 2 , w 3 } and F = {f 1 , f 2 , f 3 }, and preferences are as follows: Let us now make some observations. The first, and it is easily verified, is that the set of stable matchings for instance P 1 coincides precisely with that of P 1 . The second is that the set of stable matchings coincide despite the fact that P 1 is, in a very precise sense, strictly smaller than instance P 1 . To see this we note two features: (i) for instance P 1 , every market participants' preference list is no longer than for instance P 1 (in fact each is strictly shorter), and (ii) the relative ordering of any pair in a preference list of P 1 is the same as for P 1 . So in essence P 1 contains all the relevant information of P 1 and yet is simpler to parse.
The above hints that (mutually) unattractive alternatives play no role in the set of stable matchings of any instance. In many ways, unattractive alternatives have much the same effect on stable matchings as strictly dominated strategies have on strategic games. We recall that deleting strictly dominated strategies from a strategic environment reduces the size of the game and yet does not change the set of rationalisable outcomes (the set of possible predictions). One might wonder whether deleting unattractive alternatives has the effect of reducing the input to a matching problem and yet does not affect the set of stable matchings (the set of possible predictions). The answer turns out to be yes.
Another natural question to ask is whether deleting unattractive alternatives from an instance of the matching problem can only be performed once. When deleting dominated strategies, the deletion operation can be applied on the reduced version of the game since strategies that were not initially strictly dominated can become so. The same occurs with unattractive alternatives, and the reason is that there may be participants who were not initially unattractive but are unattractive in the reduced environment. We address this in the next section wherein we define the iterated deletion of unattractive alternatives (IDUA), a procedure that parallels closely the iterated deletion of dominated strategies (IDDS) due to Moulin (1979).

IDUA and a Matching Problem's Normal Form
We now define the iterated deletion of unattractive alternatives (IDUA), a procedure that repeatedly prunes redundant information from the preference lists. Technically it just repeatedly deletes unattractive participants from preference lists until there is no market participant who views any participant on their preference list as unattractive.
Definition 5 (The iterated deletion of unattractive alternatives (IDUA)). Given an instance of the matching problem P = ({ w } w∈W , { f } f∈F , we define 0 w := w and 0 f := f , and for each k ≥ 1, form the matching (sub)problem P k = ( k w w∈W , k f f∈F ) where for every worker w and every firm f, ( Finally, define the normal form of matching problem P , P * , as P k with minimum k such that P k+1 = P k . That is, the normal form is what remains when no further deletions are possible for some P k . The iterative part of Definition 5, given in (2), says that if worker w and firm f do not find each other mutually unattractive at some round of the iteration procedure, then neither deletes the other from their preference list during that round. That is, worker w carries firm f forward to the next round of the procedure and vice versa.
The IDUA procedure has parallels with the IDDS procedure that are quite striking. Once the IDDS procedure stops (and it must), the set of surviving strategies are the only "rational" way to play the game. The following result, whose proof is in the Appendix, confirms a similar feature of the IDUA procedure. Precisely, it shows that while the IDUA procedure may discard information from a matching problem, no "important" information is discarded in the sense that the set of stable matchings for P can be computed using only P * . That is, only pairs that are not part of any stable matching are deleted.
Lemma 1 (Balinski and Ratier (1997)). The iterated deletion of unattractive alternatives does not change the set of stable matchings. That is, P and its associated normal form, P * , contain exactly the same set of stable matchings.
In the discussion of unattractive alternatives following Example 1, statements about unattractiveness were made in a particular order. For an analyst who is deleting unattractive alternatives, a natural concern is whether the order in which the unattractive alternatives are deleted might matter. Fortunately, the following lemma, whose proof is also in the Appendix, shows that there is no issue with this as the resulting normal form is arrived at independently of the order in which unattractive alternatives are deleted. 12 Lemma 2 (Balinski and Ratier (1997)). Let P be an instance of the matching problem. Then the normal form of P , P * , is uniquely defined. That is, no matter in which order we repeatedly delete unattractive alternatives from preference lists, we always end up with the same P * .
Given that strictly smaller instances are by definition computationally easier, together Lemmas 1 and 2 imply that performing the IDUA procedure should be the first port of call for an analyst when considering a stable matching problem.
We now document some further connections between the IDDS procedure and the IDUA procedure. The first point of note concerns effectiveness. There are many games in which IDDS does nothing. IDUA on the other hand almost always has some bite except for the rare cases in which every market participant's favourite partner ranks them last.
Moreover, just like IDDS and the solution concept of rationalizability (Bernheim, 1984;Pearce, 1984;Tan and Werlang, 1988), IDUA can be justified by appealing to a form of "higher order reasoning" in recognising how other participants view the environment. However, the higher order reasoning invoked is different for IDUA. Both IDDS and rationalizability work by assuming "rationality" and "sophistication" on the part of individuals: rational individuals avoid dominated strategies and sophisticated individuals expect their rational opponents to do the same. And so on. With IDUA there is a "moment" at which both participants simultaneously recognise each other's unattractiveness and delete each other from their preference lists. It is then required that third parties are capable of recognising this and processing it. They do so as their relative placing in the preference lists of others can have changed. While this might seem unreasonable at first, the solution concept of stability is coalitional in nature, so perhaps it is not unreasonable that the sort of higher order reasoning in which participants engage should be too.
Another way to highlight how the higher order reasoning differs between IDUA and IDDS can be seen by considering how each procedure works in the first round of deletion. In a strategic game, a player can delete his own strictly dominated strategies without forming any view as to how his opponents will behave. In fact, a player can delete a strictly dominated strategy even without any knowledge of the other players payoffs. The same is not true of IDUA. Worker i can only decide that firm j is unattractive if there is some other firm, say k, who worker i prefers to firm j and for whom worker i is the most preferred worker. But for worker i to be able to do this, in addition to knowing his own preferences, importantly he must also have knowledge of firm k's preferences. (We note however that worker i need not know firm j's preferences.) We conclude this section with a useful observation about the normal form. If any two instances of the matching problem possess the same normal form, then they must have the same set of stable matchings. But one can show by example that the reverse implication does not hold. That is, it is not the case that if two instances of the matching problem have the same set of stable matchings then they must have the same normal form. However, this reverse implication does hold for instances that possess the same unique stable matching. This can be exploited as follows: for a given matching one can generate the full class of instances for which that matching is the unique stable matching; simply start out with the matching in question and run all possible variants of the IDUA procedure "in reverse". We will return to this issue in Section 4.3.

Classifying Unique Stable Matchings
Checking for uniqueness in an instance of the matching problem can be done as follows. Run the deferred acceptance algorithm twice, once with workers in the role of proposers and once with firms in the role of proposers, and check if the stable matching found for each run is the same. If yes, then there is a unique stable matching. The reason for this is that the set of stable matchings forms a distributive lattice of which the worker-proposing stable matching is the top element and the firm-proposing stable matching is the bottom element (Knuth, 1996). Since the deferred acceptance algorithm runs in polynomial time, the algorithmic approach above provides an efficient solution from the perspective of computational complexity.
But while computationally efficient, the algorithmic approach above sheds no light on the structure of instances possessing a unique stable matching. It is for this reason that a host of sufficient conditions on preferences ensuring uniqueness have been proposed (see Footnote 2). Perhaps the reason a necessary condition had not been found was that, as per Lemma 1, what really matters is how preferences operate on the normal form. It turns out that the barrier to uniqueness are preference lists that possess cycles, first defined by Chung (2000), but cycles on the normal form. A cycle is formally defined below.
Definition 6. Let P = ({ w } w∈W , { f } f∈F ) be an instance of the matching problem with n workers and n firms. We say that the preference lists of P possess a cycle, if there exists a subset of workers of size k and a subset of firms of size k (with k ≥ 2), and an enumeration of and ordering of the participants {f 1 , w 1 , f 2 , w 2 , f 3 , . . . , w k−1 , f k , w k } such that f j+1 w j f j for all j = 1, . . . , k (modulo k), and w j f j w j−1 for all j = 1, . . . , k (modulo k) We say that an instance P of the matching problem is acyclic if its preference lists do not possess a cycle.
Each participant's preference list is generated by a binary relation that is antisymmetric and negatively transitive. Together these imply transitivity, which effectively translates as "individually acyclic". But while each individual preference relation is acyclic, cycles can materialise in the system as a whole due to the interconnectedness of the entire market. We defer further discussion of this until Section 4 because matching digraphs allow the representation of cycles in a very intuitive way.
We now state our main result whose proof is found in the Appendix.
Theorem 1. Let P be an instance of the matching problem and let P * be its associated normal form. Then the following three statements are equivalent.
(a) P has a unique perfect stable matching.
(b) The normal form of P , P * , is acyclic.
(c) In the normal form, P * , every market participant's preference list a singleton.
The equivalence of (a) and (b) confirms that it is cycles in preferences on the normal form that prevent uniqueness. To see why, consider a stable match, µ, and consider a subset of market participants, S, whose preferences possess a cycle. Well it turns our that we can "shuffle around" participants in S, by assigning them different partners, also in S, and arrive at another stable matching. To illustrate this, we return again to Example 1 that possessed two stable matchings. We restrict attention to P 1 that is the normal form of P 1 (in that no further deletions are possible).
Let S be the subset of market participants, {w 2 , w 3 , f 1 , f 3 }. Restricted to this subset, there are two stable (sub)matchings. They are (w 2 , f 1 ), (w 3 , f 3 ) and (w 3 , f 1 ), (w 2 , f 3 ). Note that the first of these stable (sub)matchings is firm-optimal, as evidenced by the fact that both firms are with their most preferred partner available in the normal form (while worker w 1 is the favourite worker of all firms in the original instance, P 1 , worker w 1 was deleted from the preference lists of all firms bar firm f 2 ). Likewise, the second stable (sub)matching is worker-optimal.
When we "zoom in" further on the subset of market participants S in the normal form P 1 , we note that there is a cyclic structure to the collection of these individuals' preferences. To see this, we construct a sequence that begins with an arbitrarily chosen participant from this subset, and every subsequent element in the sequence is the most preferred participant of the participant listed before. As an example, let us begin the sequence with w 2 .
f 3 w 2 f 1 and w 3 f 3 w 2 and f 1 w 3 f 3 and w 2 f 1 w 3 (4) Now if we relabel w 2 by w 1 , f 1 by f 1 , w 3 by w 2 , and f 3 by f 2 , then the expressions in (4) read as f 2 w 1 f 1 and w 2 f 2 w 1 and f 1 w 2 f 2 and w 1 f 1 w 2 (5) where we note that the expressions in (5) above provide an example, with k = 2, of the conditions in (4) of Definition 6 for a cycle.
Let us now consider parts (b) and (c). That (c) implies (b) is immediate, since for preference lists to possess a cycle at least four participants must each have preference lists of length at least 2. That (b) implies (c) is far from obvious and is proved in the Appendix. If we consider now the results on uniqueness that assume that the full preference lists are acyclic, we can see how this is a sufficient condition. If the full preference lists are acyclic to begin with, then clearly reducing the preference lists cannot generate a cycle that was not there before. That is, applying IDUA yields the normal form which, by part (c) leaves everyone with a single available match, must be acyclic too.
In the next section we formally introduce the directed graph approach to matching problems due to Maffray (1992). While the reader not interested in the proofs of our results can skip this section, we believe that reformulating stable matching problems as directed graphs is very useful for tackling certain problems. To illustrate how visually appealing this equivalent formulation is, Figure 1 depicts the normal form of instance P 1 , P 1 , from Example 1. With three workers and three firms there is a 3 × 3 grid, where each vertex in the grid is a pair with vertex (i, j) corresponding to the pair (w i , f j ). A (full) matching is a subset of three vertices no two of which are in the same row nor the same column The vertices that are hollow are pairs that cannot be part of any stable matching as both participants in that pair deemed the other unattractive at some point in the deletion procedure. So, for example, the fact that vertex (2, 2) is hollow connotes that at some point during the IDUA procedure, worker w 2 deemed firm f 2 unattractive and firm f 2 did likewise to worker w 2 . The directed edges between pairs that remain capture preferences. For example, the directed edge from vertex (2, 1) to (2, 3) indicates that worker w 2 prefers firm f 3 to firm f 1 (i.e., f 3 w 2 f 1 ). The clockwise cycle in the normal form is easily identified by simple eyeballing, and the reader can verify that this cycle corresponds precisely to that given in (4). (In the directed graph representing the original instance P 1 , there were many more directed edges, but when a vertex in the directed graph (a pair) is deleted so too are all directed edges that were incident to that vertex.)

Digraph terminology and notation
A directed graph (or just digraph) D consists of a non-empty finite set V (D) of elements called vertices and a finite set A(D) of ordered pairs of distinct vertices called arcs. We shall call V (D) the vertex set and A(D) the arc set of D and write D = (V (D), A(D)). For an arc xy the first vertex x is its tail and the second vertex y is its head. Moreover, x is called an in-neighbour of y and y an out-neighbour of x. We also say that the arc xy leaves x and enters y. We say that a vertex x is incident to an arc a if x is the head or tail of a. For a vertex v ∈ V (D), the out-degree of in a digraph D is a sequence of vertices x 1 , x 2 , . . . , x p for which there is an arc from each vertex in the sequence to its successor in the sequence. Such a walk is written as W = x 1 x 2 . . . x p . Special cases of walks are paths and cycles. A walk W is a path if the vertices of W are distinct. If the vertices x 1 , x 2 , . . . , x p−1 are distinct, for p ≥ 2 and x 1 = x k , then W is a cycle.
For a digraph D = (V, A) and an arc xy ∈ A, deletion of xy from D results in the digraph

Matching digraphs
Given an instance of the matching problem, P , we define the associated matching digraph, D(P ) = (V, A), where V is the vertex set and A is the arc set. The vertex set V is defined as V := W × F . The arc set A is defined as follows.
Furthermore let D W (P ) = (V, A W ) and let D F (P ) = (V, A F ). A matching in P is depicted in D(P ) by a set of vertices, M , such that for every w ∈ W there exists exactly one vertex, (w, f ), in M containing w and for every f ∈ F there exists exactly one vertex, (w , f), in M containing f. (Going forward we will abuse terminology and refer to such a collection of vertices in D(P ) as a matching.) A stable matching in D(P ) is a matching M such that for every vertex (w, f) ∈ V (D(P )) either (w, f) ∈ M or (w, f) has an out-neighbour that belongs to M . In the language of directed graphs, M is a kernel. 13 The following example serves three purposes. First, we will use it to introduce matching digraphs in a more rigorous manner than at the end of Section 3. Second, the preference lists possess a cycle (in fact more than one), and hence none of the existing sufficient conditions available in the literature would immediately conclude that there is a unique stable matching. And yet there is a unique stable matching. Third, given that there is a unique stable matching, part (c) of Theorem 1 confirms that the IDUA will collapse each individuals preference list to a singleton. We will use this fact to illustrate how IDUA operates, and to show how visually intuitive the procedure is when a matching problem is reformulated using digraphs.
Example 2. Let P 2 be an instance of the matching problem in which there are three workers and three firms. That is, W = {w 1 , w 2 , w 3 } and F = {f 1 , f 2 , f 3 }. Preferences are as follows: It can be checked that P 2 possesses a unique stable matching, µ * , given by, We emphasise that preferences of P 2 are not acyclic. To see this, consider the subpopulation {w 1 , w 2 , f 1 , f 3 }. The cycle here is given by, f 3 w 1 f 1 and w 2 f 3 w 1 and f 1 w 2 f 3 and w 1 f 1 w 2 (7) A relabelling of participants in (7), as was done in going from (4) to (5), confirms the cycle. Figure 2 illustrates the matching digraph for P 2 . The complete digraph D(P 2 ) is displayed in the left hand panel, while the digraph in the right hand panel is a "condensed" version where the arcs implied by transitivity have been suppressed for readability.
The vertex (i, j) in each digraph of Figure 2 denotes the pair (w i , f j ). That is, rows are indexed by workers and columns are indexed by firms. The preferences of workers are depicted by horizontal arcs, the arc set A W , and the preferences of firms are depicted by the vertical arcs, the arc set A F . A matching in P 2 corresponds to a set of vertices, M , no two of which are in the same row nor the same column.
Every participant's preference is complete and transitive, so there cannot be a cycle in any row (corresponding to a worker's preferences) nor in any D(P 2 ) with arcs implied by transitivity suppressed Figure 2: The matching digraph D(P 2 ) for the instance P 2 of Example 2.
column (corresponding to a firm's preferences) of a matching digraph. Note however that there can be cycles in the digraph as a whole. In Figure 2, we have labelled one such preference cycle by colouring the arcs that comprise it in blue, and we note that this is precisely the preference cycle identified in (7). So the acyclic condition due to Chung (2000), given in Definition 6, requires that preferences are intertwined in such a way that the transitivity property of individual preferences is inherited at the population level appearing in collection of preference lists and not just individual preference lists. Given a matching problem P and its associated matching digraph D(P ), we now introduce a reduction, R, that "prunes" the matching digraph of extraneous information. Specifically, it identifies vertices in D(P ) that represent pairs that view each other as mutually unattractive and deletes them.
For every vertex v ∈ V (D(P )), it will be useful to decompose d + That is, the out-degree of a vertex is split into the horizontal out-degree and the vertical out-degree. (Note that for for a given pair (w, f), the horizontal out-degree corresponds to the number of firms that w prefers to f and the vertical out-degree corresponds to the number of workers that f prefers to w.) We then have the following.
Definition 7. Given a matching digraph D(P ), we define R D(P ) as the result of the following procedure.
Choose (w, f) ∈ V (D(P )) with either d + 14 The reduction procedure, R, operates as follows. If d + F (v) = 0, then all vertices that are the tail of an arc with head v in A W are to be deleted. This is because if v = (w, f), and d + F (v) = 0, then worker w is firm f's most preferred worker and hence w can not be matched with any firm that they prefer less than f in any stable matching. When d + W (v) = 0, analogous vertex deletions are performed.
A version of IDUA, which we call IDUA R , repeats R after setting D(P ) := R D(P ) until further reductions are no longer possible. When IDUA R stops we obtain the normal form of the initial D(P ) denoted by D * (P ). By definition, D * (P ) = D(P * ).
We emphasise that the reduction procedure IDUA R , generated by repeated applications of R, differs slightly from the IDUA procedure of Definition 5. The difference is as follows. The way we defined IDUA is more in line with the way the IDDS procedure for games is typically motivated in the sense that multiple deletions happen simultaneously. 15 Specifically, in iteration k, IDUA deletes all vertices that cannot be part of some stable matching. It does this by pivoting around all blocking pairs simultaneously. This should be contrasted with the IDUA R , that in iteration k finds one blocking pair and pivots only around it. But while there is a formal difference between IDUA and IDUA R , from a practical perspective the difference is immaterial as Lemma 2 confirms that the order in which unattractive alternatives are deleted does not affect the final output.
We now illustrate how IDUA R operates, using instance P 2 from Example 2. (Let us recall that these preferences are not acyclic and yet this instance does possess a unique stable matching.) Figure 3 below contains six panels that show repeated applications of R to the matching digraph of P 2 , D(P 2 ). As in Figure 2, vertex (i, j) denotes the pair (w i , f j ). In each panel the red arrow indicates the row or column where the R is being applied. The black vertex is the vertex with no arcs out of it in the row or column that is being considered (i.e., it is the blocking pair that R pivots around).
14 If more than one vertex (w, f) ∈ V (D(P )) satisfies the condition above, then R(D(P )) will depend on which of them is chosen. However, we can ignore this fact since our interest lies in repeated application of R (see below) and, as per Lemma 2, the end result of repeated application does not depend on the intermediate values.
15 While algorithmically more care must be taken with operation that delete multiple objects simultaneously, simultaneous deletion procedures are a closer fit to the higher ordering reasoning systems that game theorists assume of rational agents.
The dotted vertices and arcs are the vertices and arcs that get deleted in that iteration (the dotted vertices have arcs into the black vertex, and these arcs are perpendicular to the row/column that determined the choice of the black vertex). Once the repeated application can go no further, we have the normal form of D(P 2 ), D * (P 2 ), which in this case consists of the matching in (6). Let us now be slightly more concrete about Figure 3. In the first panel, we consider the second row that captures worker w 2 's preferences. The (horizontal) arcs in this row indicate that f 1 is the most preferred firm to w 2 , and so we pivot around vertex (w 2 , f 1 ). We indicate this by colouring this vertex black. Firm f 1 recognises that worker w 2 is, in effect, a lower bound on who they can pair with. (Since if firm f 1 were matched with any worker that they prefer less than worker w 2 , then (w 2 , f 1 ) would constitute a blocking pair since w 2 will leave any firm for f 1 .) Worker w 3 is one such worker. So firm f 1 deems w 3 unattractive and by the reciprocal nature of this relation, w 3 deems f 1 unattractive. Hence by considering the second row, we can delete the vertex (w 3 , f 1 ) and any arcs incident on it.
In the second panel, we consider the third row that captures worker w 3 's preferences. At the onset, f 1 was worker w 3 's most preferred firm. But the first application of R, depicted in the first panel, showed that this pair can never be part of a stable matching as vertex (w 3 , f 1 ) was deleted. As such, w 3 's most preferred feasible firm is f 3 , which we depict by colouring the vertex (w 3 , f 3 ) black. This means that firm f 3 can at worst match with worker w 3 . Thus we can delete all vertices with arcs in A F that have head (w 3 , f 3 ). (In fact, given that w 3 is in fact f 3 's most preferred feasible partner, we can now conclude that they will certainly be paired in all stable matchings.) Note that this deletes vertices (w 1 , f 3 ) and (w 2 , f 3 ), and doing so this deletes three of the four arcs comprised the cycle in the preference lists, i.e., the arcs that were coloured blue in the left hand panel of Figure 2. This highlights that cycles are not always a barrier to uniqueness.
The remaining panels should now be easily understood.

Equivalent Instances
We now sketch how an analyst can recover the full set of preference lists for which a given matching is the unique stable matching.
While Lemma 1 ensures that if two instances of the matching problem possess the same normal form then they must possess the same set of stable matchings, the reverse implication need not hold. However, parts (a) and (c) of Theorem 1 confirm that when an instance of the matching problem possesses a unique stable matching, the IDUA procedure collapses each participant's preference list to a singleton. And as such the reverse implication of Lemma 1 does hold for instances with a unique stable matching.
The fact above is useful as it allows the analyst to reconstruct the full set of preference lists for which a given matching is the unique stable matching. Let us elaborate on this. Consider Figure 3 and observe what happened in going from the second to last panel to the last panel. By the rules of the deletion procedure, the vertex (3, 2) was deleted and hence so were the arcs that made up the path (2, 2) → (3, 2), (3, 2) → (3, 3) (interpreted as f 3 w 3 f 2 and w 3 w 2 w 2 ). Note however, that the matching digraph in the final panel would have been the same if the path in the second to last panel had been the reverse, (3, 3) → (3, 2), (3, 2) → (2, 2) (f 2 w 3 f 3 and w 2 w 2 w 3 ), or even if the path had instead been from vertex (2, 2) to (3, 3) via the vertex (2, 3) (f 3 w 2 f 2 and w 3 w 3 w 2 ).
This insight can be built upon. While R reduces a matching problem without altering the set of stable matchings, one can define an operation, call it R −1 , that expands a stable matching subproblem in such a way that the set of stable matchings is not altered. The inverse operation R −1 simply adds participants to preference lists in such a way that they would be deleted by an application of R, though we note that R −1 is a relation and hence R −1 (D) denotes the set of all appropriate digraphs. Just as R can be applied repeatedly, the analyst can repeatedly apply R −1 until all 2n preference lists are complete, at which point the analyst has generated an instance of the matching problem for which the identified matching is the unique stable matching.
We will illustrate this reverse procedure for a specific example. When there are n workers and n firms, the number of possible instances is enormous, (n!) 2n , so for reasons of space we consider an example with n = 2. We let W = {w 1 , w 2 } and F = {f 1 , f 2 }, and we consider the matching µ * = (w 1 , f 1 ), (w 2 , f 2 ). We then ask: what is the full set of instances for which µ * is the unique stable matching? Let us now show how to reverse engineer the set of preferences for which µ * is the unique stable matching.
When n = 2, there are a total of (2!) 2×2 = 16 possible instances. Since the matching digraph for an instance with n = 2 is a 2 × 2 grid, and a minimum cycle has 4 vertices, there are only two possible instances that possess a cycle (one going "clockwise" and the other "anti-clockwise"). This means that for n = 2 there are 14 instances with a unique stable matching. Since there are only two possible matchings, we can be sure that there are exactly 7 instances for which µ * = (w 1 , f 1 ), (w 2 , f 2 ) is the unique stable matching. These are given in Figure 4 below, where each panel presents one of the instances using a digraph and in terms of preferences. In every digraph the vertices that comprise the original matching, µ * , are shaded grey while the vertices that are added back are hollow. The arcs represent preferences.
The above discussion illustrates how an analyst can "build up from scratch" the full set of equivalent instances of the matching problem. We conclude by noting that while the above procedure is computationally intensive, it is exhaustive.

Conclusion
The matching framework of Gale and Shapley (1962) is one of the classic models of economic theory. It has been applied to a wide array of real-world settings like matching students to schools, matching doctors to hospitals, matching kidney donors to those in need of transplants, online dating websites, and others (see Roth (2008) for a survey). Yet despite all the attention (1,1) that the model has received, a classification of what structural properties of a given matching problem guarantee a unique stable matching remained an open question. In this paper we have resolved this open question.
The key to answering the question is to first reduce a given matching problem to its essential constituents. We do this by applying a reduction procedure, the iterated deletion of unattractive alternatives (IDUA) that strips away the parts of the preference lists that are not relevant to the set of stable matchings. We term the resultant subproblem the normal form. Our main result, Theorem 1, shows that a matching problem has a unique stable matching if and only if preferences on the normal form are acyclic. (In fact we show the stronger result that IDUA reduces every market participant's preference list to a singleton.) Given a particular matching, the complete collection of preferences for which this matching is the unique stable matching can be generated by running the IDUA procedure "in reverse", thereby allowing the analyst to identify those preference lists that ensure uniqueness.
We conclude by noting that the environment we considered in this paper (Definition 1) is that of the original Gale and Shapley (1962) paper: a oneto-one matching environment wherein both sides of the market are the same size and every market participant has a complete preference ordering over those of the other side. There are many ways in which this environment has since been extended. For example, some worker may find it preferable to be unmatched over being paired with a particular firm (not requiring that preferences are complete, or, equivalently, beginning with some participants already unacceptable), firms may have positions for more than one worker (many-to-one matchings), there may be external constraints that need to be satisfied (see for example, Kamada and Kojima (2015)), and so on. While our results immediately carry over to the case wherein preference lists need not be complete we leave the study of how IDUA operates in other richer environments to future work.

APPENDIX
Lemma 1, Lemma 2, and Theorem 1 are all stated in the text in terms of the preferences. Since all of our results are proved using the equivalent matching digraph formulation, below we state each result as it appeared in the main text and then state the equivalent result in terms of matching digraphs. That is, Lemma 1 is restated as Lemma 1 , Lemma 2 as 2, and Theorem 1 as Theorem 1 .
A Proofs omitted from the main text Lemma 1. The iterated deletion of unattractive alternatives does not change the set of stable matchings. That is, P and its associated normal form, P * , contain exactly the same set of stable matchings.
Lemma 1 . The iterated deletion of unattractive alternatives does not change the set of perfect stable matchings. That is, D(P ) and D * (P ) contain exactly the same perfect stable matchings.
Proof of Lemma 1 and 1 .
Proof. We will show that one iteration of R does not change the set of stable perfect matchings, which will imply that the lemma holds. Assume that there is no arc in A F leaving (w, f) ∈ V (D(P )) and we have therefore deleted all vertices (w, f i ) such that (w, f i )(w, f) ∈ A W (see (i) in Definition 7). Let D 1 be the matching digraph before the reduction and let D 2 denote the matching digraph after the operation.
For the sake of contradiction assume that one of the deleted vertices, say (w, f i ), lies in a stable perfect matching M 1 of D 1 . Then for some j, we have (w j , f) is also in M 1 . Now we will show that (w, f) is a blocking edge in M 1 . Since (w, f i ) and (w j , f) are in M 1 , (w, f) ∈ M 1 . Moreover, f prefers w to w j as (w j , f)(w, f) ∈ A F and w prefers f to f i as (w, f i )(w, f) ∈ A W . Therefore, M 1 does not exist (as there is a blocking edge). So no deleted vertex can belong to a stable perfect matching of D 1 . Therefore, if M 1 is a stable perfect matching in D 1 , then M 1 is also a stable perfect matching in D 2 (as D 2 is an induced subdigraph of D 1 and M 1 ⊆ V (D 2 )). Conversely assume that M 2 is a stable perfect matching in D 2 . Clearly no vertex in D 2 is a blocking edge for M 2 . For the sake of contradiction assume that (w, f i ) is a blocking edge for M 2 in D 1 (where (w, f i ) is deleted when constructing D 2 ). Recall that (w, f i )(w, f) ∈ A W and there are no arcs in A F leaving (w, f) ∈ V (D(P )). As (w, f) is not a blocking edge for M 2 in D 2 , we note that (w, f) either belongs to M 2 or there is a vertex (w, is not a blocking edge for M 2 in D 1 . This implies that there is no blocking edges for M 2 in D 1 and therefore M 2 is a stable perfect matching in D 1 . This completes the proof.
Lemma 2. Let P be an instance of the matching problem. Then the normal form of P , P * , is uniquely defined. That is, no matter in which order we repeatedly delete unattractive alternatives from preference lists, we always end up with the same P * .
Lemma 2 . Let P be an instance of the matching problem. Then the matching digraph of the normal form, D * (P ), is uniquely defined. That is, no matter in which order we repeatedly apply R to perform IDUA, we always end up with the same D * (P ).

Proof of Lemma 2 and 2 .
Proof. Let P be an instance that contains a perfect stable matching. Let M ⊆ V (D(P )) be the vertices of D(P ) that belong to some perfect stable matching of P . Let M * denote the set of vertices (w, f ) ∈ D(P ) where either (w, f ) ∈ M or there are vertices (w, f i ) ∈ M and (w j , f ) ∈ M such that (w, f i )(w, f ) ∈ A W and (w j , f )(w, f ) ∈ A F . We will show that no matter in what order we perform the reductions we always obtain V (D * (P )) = M * and A(D * (P )) contain the arcs of D(P ) with both tail and head in M * .
Let m * be an arbitrary vertex in M * . If m * would be deleted by some Reduction R, then either all vertices with arcs in A W with head m * will also be deleted or all vertices with arcs in A F with head m * will be deleted (by the definition of Reduction R). In both cases at least one vertex from M will be deleted, a contradiction to Lemma 1 . So, if m * ∈ M * then m * is not deleted by any Reduction R and m * ∈ V (D * (P )). This implies that M * ⊆ V (D * (P )).
For the sake of contradiction assume that there exists a (w, f) ∈ V (D * (P ))\ M * . As (w, f) ∈ M * we note that (w, f) ∈ M and without loss of generality we may assume that there is no arc a ∈ A W such that (w, f) is the head of a and the tail of a lies in M . Since M F (see the definition in Lemma 3) is a perfect stable matching it contains a vertex (w, f i ) ∈ M F . As (w, f) has no in-neighbour of the form (w, f j ) belonging to M , and therefore also not belonging to M F , we have that (w, f i ) is an out-neighbour of (w, f). However as there are no arc in A F leaving (w, f i ) and (w, f)(w, f i ) ∈ A W , we note that (w, f) will be deleted by Reduction R, a contradiction to (w, f) ∈ V (D * (P )).
Theorem 1. Let P be an instance of the matching problem and let P * be its associated normal form. Then the following three statements are equivalent.
(a) P has a unique perfect stable matching.
(b) The normal form of P , P * , is acyclic.
(c) In the normal form, P * , every market participant's preference list a singleton.
Theorem 1 . Let P be an instance of the matching problem, and let D(P ) be the associated matching digraph, and let D * (P ) be the matching digraph of the normal form. Then the following three statements are equivalent.
(a) P has a unique perfect stable matching.
The proof of Theorem 1 requires the following two additional lemmas, Lemma 3 and Lemma 4. Before stating and proving these Lemmas, we introduce the following notation. Let A * W denote the arcs from A W with both endpoints still in D * (P ). Analogously let A * F denote the arcs from A F with both endpoints still in D * (P ). That is A(D * (P )) = A * W ∪ A * F . Let d * ,+ W (x) denote the number of arcs out of x ∈ V (D * (P )) in A * W and analogously let d * ,+ F (x) denote the number of arcs out of x ∈ V (D * (P )) in A * F . Lemma 3. Let P be an instance of the matching problem. Then the following two sets, M W and M F , are both perfect stable matchings in P .
Proof. By Lemma 1, we note that there does exist some perfect stable matching in D * (P ). So for every w ∈ W some vertex (w, f) belongs to V (D * (P )). Let M W be defined as above and note that for every w ∈ W there exists exactly one vertex (w, f w ) in M W . We first show that if w i , w j ∈ W are distinct then f w i and f w j are distinct. For the sake of contradiction, assume that this is not the case, and f w i = f w j . Let f = f w i = f w j and without loss of generality assume that (w i , f)(w j , f) ∈ A(D * (P )). However this implies that we could perform Reduction R on (w j , f), which would imply that we would have deleted (w i , f), a contradiction. Therefore f w i = f w j for all distinct w i and w j . Thus M W is a perfect matching. By the proof of Lemma 2, every (w, f) ∈ V (D * (P )) either belongs to M W or is an in-neighbour of (w, f w ) which belongs M W . Thus, there are no blocking edges in M W , which implies that M W is indeed a perfect stable matching in D * (P ). By Lemma 1 M W is therefore also a perfect stable matching in P .
The fact that M F is a perfect stable matching in P can be shown analogously.
Lemma 4. Let P be an instance of the matching problem. If A(D * (P )) is nonempty, then P contains at least two distinct perfect stable matchings.

Proof of Lemma 4.
Proof. Let P be an instance that contains a perfect stable matching, such that A(D * (P )) = ∅. Without loss of generality assume that (w, f i )(w, f j ) is an arc in A(D * (P )). Let (w, f k ) ∈ V (D * (P )) be chosen such that d * ,+ W ((w, f k )) = 0 (j = k is possible) and note that i = k. This implies that (w, f k ) ∈ M W (defined in Lemma 3).
If M W = M F then P contains at least two distinct perfect stable matchings (namely M W and M F ). So, we may assume that M W = M F . Hence, (w, f k ) ∈ M F . However this implies that (w, f i ) will be deleted by Reduction R, as (w, f i ) is an in-neighbour of (w, f k ) and there is no arc a ∈ A F such that (w, f k ) is the tail of a. This is a contradiction to the fact that we cannot perform Reduction R on D * (P ).
Proof of Theorem 1 .