Matching Soulmates

Abstract

We study iterated matching of soulmates [IMS], a recursive process of forming coalitions that are mutually preferred by members to any other coalition containing individuals as yet unmatched by this process. If all players can be matched this way, preferences are IMS-complete. A mechanism is a soulmate mechanism if it allows the formation of all soulmate coalitions. Our model follows Banerjee, Konishi and Sönmez (2001), except reported preferences are strategic variables. We investigate the incentive and stability properties of soulmate mechanisms. In contrast to prior literature, we do not impose conditions that ensure IMS-completeness. A fundamental result is that, (1) any group of players who could change their reported preferences and mutually benefit does not contain any players who were matched as soulmates and reported their preferences truthfully. As corollaries, (2) for any IMS-complete profile, soulmate mechanisms have a truthful strong Nash equilibrium, and (3) as long as all players matched as soulmates report their preferences truthfully, there is no incentive for any to deviate. Moreover, (4) soulmate coalitions are invariant core coalitions – that is, any soulmate coalition will be a coalition in every outcome in the core. To accompany our theoretical results, we present real-world data analysis and simulations that highlight the prevalence of situations in which many, but not all, players can be matched as soulmates. In an Appendix we relate IMS to other well-known coalition formation processes.

xkcd comic #770 by Randall Munroe.

Republished under Creative Commons license.

1. Introduction

This paper studies a coalition formation procedure we call Matching Soulmates. To introduce the framework of our model and the results, let us begin with a simple 7-person example.

Alice would rather be with Alex than anyone else. Alex feels the same way about Alice. They are soulmates. Since they would be willing to leave any other partners to be together, they are a threat to the stability of all matching that does not pair them.

Bertie and Ben are not so smitten; Ben would rather be with Alice than anyone, and Bertie with Alex. As long as Alice and Alex are paired, Bertie and Ben are soulmates in a conditional sense. As long as Alice and Alex are paired, Ben and Bertie threaten the stability of any matching in which they are not paired.

Casey, Devin, and Eddie would rather be matched with any of the previously introduced players. But suppose that among the three, Casey would prefer to be with Devin, Devin would prefer to be with Eddie, and Eddie would prefer to be with Casey; no pair of these three players are (conditional) soulmates.

We refer to the process by which Alice and Alex and then Bertie and Ben are matched as iterative matching of soulmates [IMS] and, when all players can be matched by IMS, we say that preferences are IMS-complete. While the example considers only a simple model matching pairs of players, our work applies to coalition formation models in which players have preferences over arbitrary sets of coalitions of which they are members.

Since Banerjee, Konishi and Sönmez (2001), this process has been used in a number of papers albeit without being named or with a different name.¹ Our work differs from prior literature in that we do not impose conditions on the model to ensure IMS-completeness.

Another seminal paper is Pápai (2004) which asks, in a coalition formation model, whether the core is unique, consisting of only one partition of the set of players.² Papai demonstrates that a necessary and sufficient condition on the environment for uniqueness of the core under any preference profile in that the environment is such that any two admissible coalitions have at most one member in common and permissible coalitions do not form a cycle, the single-lapping property. This property ensures that any preference profile in that environment is, in our words, IMS-complete. Papai continues to show that in a strategic form of her model, the single lapping property ensures the existence of a strategy-proof equilibrium; players have no incentive to misrepresent their preferences.

The results of Banerjee, Konishi and Sönmez (2001), and Pápai (2004) are beautiful and important to our understanding of coalition formation. Their results, however, require strong restrictions. These restrictions rule out our example above. It is not hard to imagine situations where some, but not all, players can be matched as soulmates, such as marriage models or roommate models with cycles.

Our paper breaks from the prior literature in that we consider the properties of mechanisms that match soulmates, without imposing further restrictions on the model. We consider both situations where preferences are IMS-complete and ones where preferences do not necessarily satisfy IMS-completeness. We next list a number of our results. The statements are somewhat informal, but each will give the reader a rough idea of the result.

A fundamental result (Proposition 1) is that: Given reported preferences and a mechanism that matches soulmates, there does not exist a deviating coalition containing soulmates who have reported their true preferences.

This result has the following important corollaries. First, if preferences are IMS-complete, then the preference revelation game has a truthful strong Nash equilibrium (Corollary 1). Indeed, we can go further: for any soulmates coalition C and any player i ∈ C, if all other players in C have reported their true preferences, then player i’s best response is to report their true preferences (corollary 2).

Note that Proposition 1 and Corollary 2 above are positive results that do not require that all players can be matched as soulmates. However, for such situations, we also provide a negative result extending Rodrigues-Neto (2007). If true preferences contain a cycle of odd length, no mechanism that matches soulmates has a truthful strong Nash equilibrium (Proposition 2).

We also provide several results about the outcome properties of mechanisms that match soulmates. First, if the core of a coalition formation problem exists for a particular preference profile, then the coalitions formed by IMS are part of every core partition (Proposition 3 and Corollary 3). Further, for any profile, if there are blocking coalitions to a partition produced by a mechanism that matches soulmates, those coalitions do not contain soulmates (Proposition 5). Similarly, all those matched by IMS prefer their match to being alone (individual rationality, corollary 5) and if a partition exists that some subset of players mutually prefers, that subset contains no soulmates (partial Pareto efficiency, corollary 5). Again, these results pertain to preferences profiles that may be IMS-incomplete. On the other hand, if the profile is IMS-complete, then the set of coalitions produced by IMS is the unique core partition (Proposition 4), and is individually rational and Pareto optimal (corollary 4). The core result is also implied by a result in Banerjee, Konishi and Sönmez (2001) and generalized to settings with restrictions to the allowable set of coalitions by İnal (2019). We also provide a number of informative examples.

Since our paper establishes that IMS provides desirable properties for those players that it matches, even when it cannot match all players, we also study the commonality of soulmates and conditional soulmates in an empirical analysis of real-world datasets and computational experiments.

Our empirical analysis studies three settings: a roommates problem³ using data from a university social network; a similar problem involving building work teams using data on connections within a consulting firm; and a two-sided matching problem using data from a speed-dating experiment. By studying these environments, we can understand how mechanisms implementing IMS can impact productivity in work environments, and compatibility between matched groups in social settings. In fact, a surprising number of people can be matched by IMS; about 25% to 40% on average in the three environments and as many as 75% for particular instances of work-teams, indicating that IMS may be particularly powerful in producing efficient teams in work settings.⁴

To study how the structure of preferences affects IMS, our computational experiments analyze how likely players are to be matched by IMS under various preference patterns. We consider unconstrained preference profiles as well as profiles that are a relaxation of reciprocal preferences,⁵ and profiles that are a relaxation of common-ranking. While IMS-complete preferences are rare among unconstrained preferences, they are quite common when preferences exhibit strong reciprocity or are close to commonly ranked.

These results shed additional light on settings where IMS can be particularly effective. For instance, in work settings, strong complementarities in worker strengths and expertise may be reflected in highly reciprocal preferences that would lead many teams to be matched by IMS. This would provide both strong incentive properties to the mechanism and produce highly productive teams.

The structure of our paper is as follows. In Section 2 we present the general matching environment as well as several definitions used throughout the paper. In Section 3 we define IMS formally and provide several examples. We turn to our key results on incentive compatibility and stability in Sections 4 and 5 respectively. Section 6 contains the results of our computational and empirical analysis of soulmates.

2. Environment

We closely follow the model of Banerjee, Konishi and Sönmez (2001), but do not assume that preferences are known to the mechanism designer. In contrast to Pápai (2004) we do not impose restrictions on the set of permissible coalitions. Instead, we consider unrestricted environments and focus on situations where preferences are such that some, possibly all, players can be matched as soulmates.

The total player set is given by N = {1, …, n}. A coalition is a non-empty set of players C ∈ 2^N\{∅}. Each player i ∈ N has a complete and transitive preference ≻_i over the collection of coalitions to which she may belong, denoted ${\mathcal{C}}_i$ for player i. Coalition formation problems with this feature are often called “hedonic”, and this property may be understood as requiring that there are no externalities in the group-formation process. As the notation ≻_i suggests, we assume that preferences over coalitions are strict.

The set of i’s possible preferences is ${\mathcal{D}}_i$. A coalition $C\in {\mathcal{C}}_i$ is acceptable for i if C = {i} or C ≻_i {i}.

A profile (of preferences) $\succ \in \mathcal{D}={\times }_{i\in N}{\mathcal{D}}_i$ is a list of preferences, one for each player in N. Given profile $\succ \in \mathcal{D}$ and any subset S ⊆ N, the subprofile (of preferences) for players in S is denoted by ${\succ }_S\in {\mathcal{D}}_S={\times }_{i\in S}{\mathcal{D}}_i$. As is customary, we let ≻_−i = ≻_N\{i}.

The domain of possible true profiles is $\mathcal{R}\subseteq \mathcal{D}$. In general, the domain of true profiles $\mathcal{R}$ need not be equal to $\mathcal{D}$. Although a preference ${\succ }_i\in {\mathcal{D}}_i$ may be a “conceivable” preference for i, ≻_i need not be player i’s preference in any true profile $\succ \in \mathcal{R}$.⁶ It is also possible that, although all preferences ${\succ }_i\in {\mathcal{D}}_i$ are i’s true preference for some profile $\succ \in \mathcal{R}$, some profiles $({\succ }_i,{\succ }_{-i}^{\prime })$ with ${\succ }_{-i}^{\prime }\ne {\succ }_{-i}$ are not elements of $\mathcal{R}$ because true preferences are interdependent. (That is, ${\succ }_{-i}^{\prime }$ cannot be the subprofile for players in N\{i} when i’s preference is ≻_i. See, for example, the domains of k-reciprocal profiles in Section 6.3).

A coalition structure π is a partition of N. For any coalition structure π and any player i ∈ N, let π_i denote i’s coalition of membership, that is, the coalition in π which contains i.

A (direct coalition formation) mechanism is a game form M that associates every reported profile ${\succ }^{\prime }\in \mathcal{D}$ with a coalition structure π ∈ Π (Π is the set of all coalition structures). For every i ∈ N, the set of preferences ${\mathcal{D}}_i$ is i’s strategy space for the mechanism M. Because mechanisms are simultaneous game forms, $\mathcal{D}$ must be the Cartesian product of the sets ${\mathcal{D}}_i$; simultaneity makes it impossible for any player or group of players to condition their reports on the report of other players.⁷

Together, a pair (M, ≻), where $\succ \in \mathcal{R}$ is a profile of true preferences, determines a preference revelation game. Again, the strategy space of a player i in this game is the set of her reported preferences ${\mathcal{D}}_i$. Once profile ${\succ }^{\prime }\in \mathcal{D}$ is reported, the mechanism M determines a coalition structure, denoted M(≻′). The coalition containing player i is denoted M_i(≻′); we say that M matches i with M_i(≻′). Each player i evaluates their assigned coalition according to their true preference ≻_i.

This model of direct coalition formation mechanisms generalizes many common matching environments. When mechanisms are individually rational, restrictions on the collection of feasible coalitions can often be translated into restrictions on the domain of preferences by forcing infeasible coalitions that contain i to be “unacceptable” for i (i.e., ordered strictly below i given ≻_i). As an example, in our environment, the roommates problem is obtained by restricting $\mathcal{D}$ to the set of profiles in which only singletons and pairs of players are acceptable. In the marriage problem (Gale and Shapley, 1962), only singletons and pairs of players with a player from each side of the market are acceptable. In the college admission problem (Roth, 1985), $\mathcal{D}$ is such that only coalitions containing a single college and some (or no) students are acceptable (and that students are indifferent between any two coalitions with the same college).

3. Iterated Matching of Soulmates

Given $\succ \in \mathcal{D}$, a coalition C ∈ 2^N is a 1^st order soulmates coalition if

$$C{\succ }_i{C}^{\prime }\text{for all}i\in C\text{and all}{C}^{\prime }\in {\mathcal{C}}_i\setminus C.$$ (1)

The process of iterated matching of soulmates [IMS] involves repeatedly forming (1^st order) soulmates coalitions from player sets decreasing in size as coalitions of soulmates are formed.

While no ordering is required in the formation of soulmates coalitions, it is easiest to describe IMS as if it were a dynamic process. In the first round, given a profile of reported preferences, the mechanism forms soulmates coalitions. The members of these coalitions prefer their assigned coalition to all others. In the second round, the mechanism forms soulmates coalitions among the players who are not assigned to coalitions in the first round, etc. Formally, the process of iterated matching of soulmates is defined as follows.

Round 1: Form 1^st order soulmates coalitions (i.e., coalitions satisfying 1). Denote the collection of these coalitions by ${\mathcal{S}}_1(\succ )$. The set of players who belong to a coalition in ${\mathcal{S}}_1(\succ )$ is denoted by N₁(≻). These players are called 1^st order soulmates.

⋮

Round r: Form coalitions of 1^st order soulmates among the players who are not part of a coalition that forms in any round preceding round r. Call these coalitions r^th order soulmates coalitions and denote the collection of these coalitions by ${\mathcal{S}}_r(\succ )$. The set of players who belong to a coalition in ${\mathcal{S}}_r(\succ )$ is denoted by N_r(≻). These players are called r^th order soulmates.

Formally, given any integer r, the r^th order soulmates coalitions are the coalitions C that contain no players from ${\cup }_{j=1}^{r-1}{N}_j(\succ )$ and are such that

$$C{\succ }_i{C}^{\prime }\text{for all}i\in C\text{and all}{C}^{\prime }\in {\mathcal{C}}_i\setminus C\text{with}{C}^{\prime }\cap ({\cup }_{j=1}^{r-1}{N}_j(\succ ))=\varnothing .$$

End: The process ends when no coalition forms in some round r*, i.e., ${\mathcal{S}}_{r^{\ast }}(\succ )=\varnothing$.

For convenience, we will denote the collection of coalitions ${\cup }_{j=1}^{r^{\ast }-1}{\mathcal{S}}_j(\succ )$ formed by this process as IMS(≻). We refer to any player who is matched by IMS as a soulmate, and to every coalition that forms under IMS as a soulmates coalition (or coalition of soulmates).

A mechanism M is a 1^st order soulmate mechanism if for every $\succ \in \mathcal{D}$, every 1^st order soulmates coalition forms under M (i.e., ${\mathcal{S}}_1(\succ )\subseteq M(\succ )$). Similarly, a mechanism M is a soulmate mechanism if for every $\succ \in \mathcal{D}$, the coalitions that form under IMS form under M (i.e., IMS(≻) ⊆ M(≻)).

3.1. Examples

Our next examples illustrate the process of iterated matching of soulmates for particular profiles. Though preferences in our environment are over coalitions to which a player might belong, to save space in examples below we will often express these as the player’s preferences over partners.

Example 1 (Formation of parliamentary groups) A Left (L), a Center (C), a Right (R) and a Green (G) party have to form parliamentary groups. Their preferences form a roommates profile : (i) every party prefers a coalition of two to being alone, and (ii) every party prefers being alone to being in a coalition of more than two players. The parties have the following preferences over partners

$$\begin{array}{c}\hfill R:C{\succ }_{R}G{\succ }_{R}L\hfill \\ \hfill C:R{\succ }_{C}G{\succ }_{C}L\hfill \\ \hfill G:L{\succ }_{G}C{\succ }_{G}R\hfill \\ \hfill L:C{\succ }_{L}G{\succ }_{L}R\hfill \end{array}$$

If we apply IMS to this profile, coalition {R, C} forms in the first round, and coalition {G, L} forms in the second round.

Example 2 (Marriage, aligned women and cyclic men) In a marriage profile (i) N = M ∪ W, (ii) every woman w ∈ W (resp. man m ∈ M) prefers being in a pair with a man m (resp. woman w) to being alone, and (iii) every woman w ∈ W (resp. man m ∈ M) prefers being alone to being in any coalition different from a pair with a man m (resp. woman w ). Consider the following profile of preferences over partners

$$\begin{array}{c}\hfill w_1:m_1{\succ }_1^w{m}_2{\succ }_1^w{m}_3\hfill \\ \hfill w_2:m_1{\succ }_2^w{m}_2{\succ }_2^w{m}_3\hfill \\ \hfill w_3:m_1{\succ }_3^w{m}_2{\succ }_3^w{m}_3\hfill \end{array}\begin{array}{c}\hfill m_1:w_1{\succ }_1^m{w}_2{\succ }_1^m{w}_3\hfill \\ \hfill m_2:w_2{\succ }_2^m{w}_3{\succ }_2^m{w}_1\hfill \\ \hfill m_3:w_3{\succ }_3^m{w}_1{\succ }_3^m{w}_2\hfill \end{array}$$

In the first round of IMS, {m₁, w₁} forms. In the second round, given that m₁ has already been matched, {m₂, w₂} is a coalition of soulmates and forms. In the third round, given that m₁ and m₂ have already been matched {m₃, w₃} is a coalition of soulmates and forms. It is easy to see how this example extends to larger sets of players.

Some famous coalition formation mechanisms are soulmate mechanisms. This is the case, for example, for the deferred acceptance [DA] mechanism in two-sided matching. As we show in Section 5, this follows from the well-known fact that DA always select a core partition. In contrast, despite being a 1^st order soulmate mechanism, the immediate acceptance [IA] mechanism (or Boston mechansim Abdulkadiroğlu and Sönmez, 2003) also used in two-sided matching is not a soulmate mechanism.

Example 3 (IA is not a soulmate mechanism) Consider the following profile of preferences over partners in a marriage profile.

$$\begin{array}{c}\hfill w_1:m_1{\succ }_1^w{m}_2\hfill \\ \hfill w_2:m_1{\succ }_2^w{m}_2\hfill \\ \hfill w_3:m_2{\succ }_3^w{m}_1\hfill \end{array}\begin{array}{c}\hfill m_1:w_1{\succ }_1^m{w}_2{\succ }_1^m{w}_3\hfill \\ \hfill m_2:w_1{\succ }_2^m{w}_2{\succ }_1^m{w}_3\hfill \end{array}$$

In the first round of IA, women propose to their favorite man, and men immediately form a coalition with the woman they like best among the women from whom they receive a proposal (hence the name “immediate acceptance”). Thus, at the end of the first round, coalitions {w₁, m₁} and {w₃, m₂} have formed. In the subsequent round, there are no more men available to form a coalition with w₂. Thus, the coalition structure selected by IA is {{w₁, m₁}, {w₃, m₂}, {w₂}}.

In the first round of IMS, only coalition {w₁, m₁} forms. Coalition {w₂, m₂} forms in the second round followed by coalition {w₃} in the last round. Hence, the coalition structure selected by IMS is {{w₁, m₁}, {w₂, m₂}, {w₃}} which differs from that selected by IA.

3.2. IMS-complete Profiles

A profile $\succ \in \mathcal{D}$ is IMS-complete if all players match through IMS. All the profiles in Examples 1 to 3 are IMS-complete, although it is not hard to construct IMS-incomplete profiles (see Example 5). In the literature, two important classes of IMS-complete profiles are : (1) profiles satisfying the common ranking property (Farrell and Scotchmer, 1988)⁸; and (2) profiles satisfying the top-coalition property (Banerjee, Konishi and Sönmez, 2001). A profile satisfies the common ranking property if, for any two coalitions, the players in the two coalitions have the same preferences over these two coalitions.⁹ A profile satisfies the top-coalition property if for every subset S ⊆ N, there exists a coalition C* ⊆ S which is preferred by all its members to any other coalition made of players from S.¹⁰ The common ranking property implies the top-coalition property, which itself implies IMS-completeness. We illustrate the relationship between the three conditions in Figure 1. See Appendix A.1 for a more complete analysis of the relationship between IMS-completeness and other profile conditions in the literature.¹¹

Figure 1:

Venn diagram of the profile conditions. A dot indicates that the section of the Venn diagram is non-empty. The inclusion relationship is trivial. For examples of profiles of type A see Banerjee, Konishi and Sönmez (2001, Section 6). For an example of a profile of type B, see Banerjee, Konishi and Sönmez (2001, Game 4). Example 1 in this paper is a profile of type C. Example 5 in this paper is a profile of type D (any other profile for which the core is empty would also be an example).

While profiles satisfying the top-coalition property are IMS-complete the converse is not necessarily true.¹² To gain intuition why, consider a profile in which IMS completes in two rounds. This implies that there is a coalition C₁ and a coalition C₂ such that (i) C₁ is a coalition of 1^st order soulmates in N, (ii) C₂ is a coalition of 1^st order soulmates in N\C₁, and (iii) C₁ ∪ C₂ = N. The top-coalition property is much stronger as it requires that, for any coalition C, there be a coalition of 1^st order soulmates in N\C. This is illustrated more concretely in Example 1, where the profile is IMS-complete but does not satisfy the top-coalition property because there is no coalition of 1^st order soulmates in {C, G, L}.¹³

4. Incentive Properties of soulmate mechanisms

4.1. Introduction

In this section, we introduce the desirable properties of IMS with respect to players’ incentive to report preferences truthfully. There are at least two reasons to favor mechanisms that provide incentives to report preferences truthfully. First, if players do not report preferences truthfully, then desirable properties that the mechanism satisfies with respect to the reported preferences might not be satisfied with respect to the true preferences.¹⁴ Second, mechanisms with good incentives to report preferences truthfully “level the playing field” (Pathak and Sönmez, 2008) by protecting naive players who report preferences truthfully against the manipulations of more strategically skilled players.¹⁵

4.2. Main Result

Any player who reports her preference truthfully and is matched by IMS can find no alternative preference report that makes her better off. In fact, as the next Proposition shows, no group of players containing a truth-telling soulmate can collude to simultaneously make themselves better off. Moreover, it follows that the reduced game with player set consisting only of players that are matched as soulmates has a truthful strong Nash equilibrium.

Proposition 1 (No Soulmates Among Deviators) Suppose that M is a soulmate mechanism. For any reported preference profile $\succ \in \mathcal{D}$ and set of players W that contains some soulmates, all of whom report their true preferences, there does not exist a joint deviation ${\succ }_W^{\prime }$ by the members of W that makes every player in W better-off than reporting ≻_W.

Proof. Consider a player matched in the first round of IMS. That is, i ∈ N₁ (≻). If i is in W then ≻_i is the true preference of i and they cannot be made better off since N₁ (≻) must contain their favorite coalition.

Now suppose i is matched in some other round of IMS. i ∈ N_k (≻) for k > 1. If i is in W then ≻_i is their true preference and they can only be made better off being matched with a coalition that contains at least some players matched earlier in the process of IMS.

Thus, to make i better off, there must be some player i′ matched in an earlier round of IMS under reported preferences ≻ that is also in W. But, this requires that ≻_i′ is also the true preference of i′ and that i′ be made better off. Either i′ ∈ N₁ (≻), in which case i′ cannot be made better off (by the logic above), or i′ ∈ N_k′ (≻) for k′ < k in which case this logic applies recursively- there must be an i″ matched even earlier in IMS who is in W and who has reported truthfully.

Since the number of players (and thus the number of possible rounds of IMS) is finite, induction leads to the requirement that for players in W matched in some round k of IMS for k > 1 to be made better off, W must contain a first-order soulmate who, by the truthful reporting requirement, cannot be made better off. Thus, either W contains no players who have truthfully reported and are matched in some round of IMS, or not every player in W is made better off. ■

4.3. Corollaries of Proposition 1 and Strong Nash Equilibrium

Proposition 1 implies some remarkable incentive properties for IMS-complete profiles. Perhaps more significantly, soulmate mechanisms retain these properties in general for the players matched by IMS, even in profiles that are not IMS-complete. These properties are presented more formally in Corollaries 1 and 2 below.

Before presenting these corollaries, we provide the formal definition of strong Nash equilibrium. Given a true profile $\succ \in \mathcal{R}$, game (M, ≻) has a truthful strong Nash equilibrium if there exists no group of players S ⊆ N and no reported subprofile ${\succ }_S^{\prime }\in {\mathcal{D}}_S$ different from ≻_S such that

$$M_i({\succ }_S^{\prime },{\succ }_{N\setminus S}){\succ }_i{M}_i({\succ }_S,{\succ }_{N\setminus S})\text{for all}i\in S.$$¹⁶ (2)

Often, we care about whether a mechanism M induces games that have a truthful strong Nash equilibrium for every profile in the domain of true profiles $\mathcal{R}$. Mechanism M is said to have a truthful strong Nash equilibrium on domain $\mathcal{R}$ if (M, ≻) has a truthful strong Nash equilibrium for all $\succ \in \mathcal{R}$.

For IMS-complete profiles, the following is a corollary of Proposition 1.

Corollary 1 (Truthful Strong Nash Equilibrium) Suppose that M is a soulmate mechanism. For any IMS-complete profile ≻, the preference revelation game (M, ≻) has a truthful strong Nash equilibrium.

In particular, a soulmate mechanism M always has a truthful strong Nash equilibrium on a domain $\mathcal{R}$ containing only IMS-complete profiles. However, the incentive compatibility properties extend beyond IMS-complete profiles. Proposition 1 implies that even if a profile is not IMS-complete, soulmate mechanisms are incentive compatible for all soulmates under the mutual belief that every soulmate tells the truth.

Corollary 2 (Mutual Truthfulness Among Soulmates) Suppose that M is a soulmate mechanism. Let S be the set of players who are part of a soulmates coalition in ≻. For all i ∈ S there is no profitable deviation from truth-telling under the condition that every other j ∈ S (j ≠ i) tells the truth.

4.4. Impossibilities

The incentive compatibility implied by Proposition 1 does not generally extend to players who are not matched by IMS, as we illustrate in the following modification of Example 1:

Example 4 (Potential Deviation for Non-Soulmates) A Left (L), a Right (R) and a Center (C) party have to form parliamentary groups. All parties prefer being in a coalition to being alone. The parties have the following preferences over partners:

$$\begin{array}{c}\hfill R:C{\succ }_{R}L\hfill \\ \hfill C:L{\succ }_{C}C\hfill \\ \hfill L:R{\succ }_{L}C\hfill \end{array}$$

Consider for instance IMS-serial dictatorship [IMS^SD] which consists in first applying IMS and then using the serial dictatorship mechanism to determine the assignment of the players who do not match under IMS (any such mechanism is therefore a soulmate mechanism). Suppose that the series of dictators is R, C, L. Because IMS(≻) = ∅, we have IMS^SD(≻) = SD(≻), the outcome of the serial dictatorship mechanism given profile ≻. Thus, IMS^SD(≻) = {{R, C}, {L}}. L can deviate by reporting C as their most preferred partner. This forces IMS^SD to form {C, L} as a coalition of soulmates, and both C and L prefer to their match in IMS^SD(≻).

Our next result shows that the kind of deviation in Example 4 implies that the remarkable incentive properties identified in Proposition 1 cannot generally be strengthened much further. Proposition 2 below shows that if $\mathcal{R}$ is sufficiently rich, with the possibility of containing the kind of preference structure in the example above, any soulmates mechanism M will fail to induce a truthful Nash equilibrium in game (M, ≻) for some IMS-incomplete $\succ \in \mathcal{R}$.

A similar incompatibility can be found in Takamiya (2012, Proposition 3), which shows that, without further restrictions, no two-sided matching mechanism that (in our terminology) is also a 1^st order soulmate mechanism is strategy-proof. As we demonstrate below, this impossibility extends to many settings outside of two-sided matching.

Notice that there is a cycle of preferences among the three parties in Example 4. Suppose that M is a 1^st order soulmate mechanism. Mechanism M can form at most one of the three pairs {L, R}, {L, C}, {R, C}. But then, any party who is not in one of these pairs can manipulate by reporting that they are the soulmate of one of the others. Hence, no 1^st order soulmate mechanism M has a truthful Nash equilibrium on a domain that includes the type of cyclic profile in Example 4 (and that allows the aforementioned deviations).

The reason the above profile prevents 1^st order soulmate mechanisms to have a truthful Nash equilibrium is that it contains a cycle of the type: 1 likes 2 best, 2 likes 3 best, and 3 likes 1 best. In roommates problems, this impossibility can only occur in profiles featuring such cycles. As Rodrigues-Neto (2007) argued, roommates profiles that do not feature cycles of any length are IMS-complete and therefore have a truthful strong Nash equilibrium by Corollary 1.

It is possible to extend the cycle condition from Rodrigues-Neto (2007) to general coalition formation environments. In a general coalition formation environment, soulmate mechanisms may fail to have a truthful Nash equilibrium for reasons other than cycles, but the presence of a cycle of odd size is sufficient to induce the impossibility.

Our generalized cycle conditions are defined formally in Appendix A.2. Intuitively, individually cyclic domains have cycles in which coalitions {1, 2}, {2, 3} and {3,1} are replaced by coalitions of the form {1, 2} ∪ O₁₂, {2, 3} ∪ O₂₃ and {3, 1} ∪ O₃₁, where the O_jh are set of players that are allowed to rank coalition {j, h, O_jh} as their best coalition. In Appendix A.2, we also define cyclic domains in which 1, 2 and 3 are replaced by groups of players N₁, N₂, and N₃ that can jointly deviate. Individually odd-cyclic and odd-cyclic domains have cycles of the corresponding type that involve an odd number of coalitions.

Proposition 2 (Impossibilities) (i) If $\mathcal{R}$ is an odd-cyclic domain, no 1^st order soulmate mechanism M has a truthful strong Nash equilibrium on $\mathcal{R}$. (ii) If $\mathcal{R}$ is an individually odd-cyclic domain, no 1^st order soulmate mechanism M has a truthful Nash equilibrium on $\mathcal{R}$.

The proof of Proposition 2 generalizes the logic of roommates problem cycle discussed above and can be found in Appendix A.2.

5. Outcome Properties of soulmate mechanisms

We now introduce properties of the outcomes of a mechanism, where these properties are evaluated with respect to the reported preferences. As discussed above, properties with respect to reported preferences will only reflect properties with respect to true preferences when and for players who have the incentive to report preferences truthfully. In each case below, either our results pertain to IMS-complete profiles, in which case truth-telling is a strong Nash equilibrium by Corollary 1, or the properties are focused on the outcomes of those players who match as soulmates. By proposition 1, there are strong incentives for truth-telling among those who would match as soulmates when reporting truthfully. Thus, for soulmates, the properties with respect to reported preferences can be seen as reliable with respect to true preferences.

Properties with respect to reported preferences may also be relevant per se to the mechanism designer. For example, in school choice, a mechanism that selects a core matching with respect to the reported preferences provides a protection against challenges of the matching in court.¹⁷

Before presenting the results, we first define some relevant concepts.

5.1. Definitions

Given any profile $\succ \in \mathcal{D}$, a core partition is a coalition structure π* in which no subset of players strictly prefers matching with each other rather than matching with their respective coalitions in π*. Formally, π* is a core partition if there does not exist a blocking coalition to π*, that is, a coalition C ∈ 2^N such that $C{\succ }_i{\pi }_i^{\ast }$ for all i ∈ C.

A particular kind of blocking coalitions are singletons coalitions {i}, where i prefers being alone to being in the coalition to which she is matched by the mechanism. Given any profile $\succ \in \mathcal{D}$, a partition π* is individually rational if, for all i ∈ N, ${\pi }_i^{\ast }=\{i\}$ or ${\pi }_i^{\ast }{\succ }_i\{i\}$. Mechanism M is individually rational if M(≻) is individually rational for all $\succ \in \mathcal{D}$.

Blocking coalitions also exist if the outcome of a mechanism fails to be Pareto optimal. Given any profile $\succ \in \mathcal{D}$, a partition π* is Pareto optimal if there exists no other coalition structure that is preferred by every player to π* . Clearly, a core partition π is Pareto optimal, because any coalition in a partition π′ that is preferred by every player to π is a blocking coalition to π. Mechanism M is Pareto optimal if M(≻) is Pareto optimal for all $\succ \in \mathcal{D}$.

5.2. Soulmates and the Core

Our first result pertains to the relationship between coalitions formed by IMS and the core of a matching problem. Soulmates coalitions are coalitions in any core partition. The logic of the proof is similar to that of Proposition 1.

Proposition 3 (Soulmate Coalitions are Core Coalitions) Given any $\succ \in \mathcal{D}$, let π* be a partition in the core and let C be a coalition formed by IMS(≻). Then C ∈ π*.

Proof. Again, we use the notation in the definition of IMS. Let π* be a partition in the core and suppose that $C^1\in {\mathcal{S}}_1(\succ )$, the set of 1^st order soulmates coalitions under IMS(≻). Suppose that C¹ ∉ π*. Then, from the definition of 1^st order soulmates, since each member of C¹ would strictly prefer to be in C¹ rather than in any other coalition, C¹ can improve upon π*; that is, $C^1{\succ }_i{\pi }_i^{\ast }$ for all i ∈ C¹. But since π* is in the core, this is a contradiction; therefore C¹ ∈ π*. Now consider the player set N\C¹ and suppose that $C^2\in {\mathcal{S}}_2(\succ )$. The coalition C² can improve upon any partition of N\C¹ that does not contain C². This process can be continued until no more players can be matched by IMS. ■

Proposition 3 does not rule out the possibility that the core is empty. If the core is empty, then there are no core partitions π* and Proposition 3 is trivially true. When the core is non-empty, however, the following is a corollary of Proposition 3. Let I(≻) be the invariant portion of the core, i.e., the collection of coalitions that belong to every core partition given ≻. Then, because every player belongs to at most one coalition in IMS(≻), we have the next result.

Corollary 3 (Soulmate Coalitions are Invariant Core Coalitions) Given any $\succ \in \mathcal{D}$, if a core partition exists, then the collection of coalitions IMS(≻) ⊆ I(≻).

In this sense, IMS(≻) captures a part of the invariant portion of the core. Observe that, by Corollary 3, if there are multiple core partitions but I(≻) = ∅ (i.e., no player matches with the same coalition in every core partition), then IMS(≻) = ∅.¹⁸ Also observe that, if mechanism M always selects a core outcome when one exists, I(≻) ⊆ M(≻) for all $\succ \in \mathcal{D}$. Thus, by Corollary 3, IMS(≻) ⊆ M(≻) for any such mechanism M (examples include the famous deferred acceptance mechanism in two-sided matching).

Our next proposition can be viewed as a consequence of Corollary 3: If IMS successfully matches all players, the entire match must be a part of any core coalition structure, implying that it is the unique core coalition structure. Obviously, this implies that any soulmate mechanism M selects the unique core coalition structure for every IMS-complete profile. This also implies that, for any profile $\succ \in \mathcal{D}$, mechanism M selects the unique core coalition in the reduced game in which the player set is shrunk to IMS(≻) itself.

This result is also implied by the proof of Theorem 2 in Banerjee, Konishi and Sönmez (2001) which shows that the top-coalition property is sufficient to guarantee the existence of a unique core coalition structure. As noted by Banerjee, Konishi and Sönmez (2001), their proof can be generalized to IMS-complete profiles (in our terminology).¹⁹ The result is also similar to Theorem 1 of İnal (2019), which strengthens the implied extension of Theorem 2 of Banerjee, Konishi and Sönmez (2001) by allowing restrictions on the set of allowable coalitions while preferences are defined over all coalitions.

Proposition 4 (Unique Core under IMS-complete Profiles) For every IMS-complete profile $\succ \in \mathcal{D}$, the coalition structure IMS(≻) is the unique core coalition structure.

Example 1 together with Proposition 4 proves that the top-coalition condition is sufficient but not necessary for the core to be non-empty. The same is true of IMS-completeness. Although, by Proposition 4, IMS-completeness guarantees that the core is non-empty and unique, IMS-completeness is not a necessary condition for the core to be non-empty or unique, as the next example shows.

Example 5 Consider the following profile of preferences over partners in a roommates profile.

$$\begin{array}{c}\hfill 1:3{\succ }_{1}2{\succ }_{1}4\hfill \\ \hfill 2:4{\succ }_{2}1{\succ }_{2}3\hfill \\ \hfill 3:2{\succ }_{3}1{\succ }_{3}4\hfill \\ \hfill 4:1{\succ }_{4}2{\succ }_{4}3\hfill \end{array}$$

If 1 matches with 2 then 3 matches with 4. In this case 2 and 4 form a blocking coalition. If 1 matches with 4 then 2 matches with 3. In this case 1 and 2 form a blocking coalition. Thus, 1 must match with 3 in any core partition. It is easy to check that {{1, 3}, {2, 4}} is indeed the unique core coalition structure (because 1 and 2 match with their favorite coalition, they cannot be part of a blocking coalition, and 3 and 4 can only benefit by joining a coalition including 2 or 1, respectively). Clearly, IMS does not match all the players under this profile as there are no soulmates in N.

5.3. Soulmates, Blockers, and Pareto Efficiency

As for incentives to misrepresent preferences, soulmate mechanisms retain part of their stability on IMS-incomplete profiles. Although coalitions can block the outcome of a soulmate mechanism in IMS-incomplete profiles, these coalitions can only consist of players that do not match in IMS.

Proposition 5 (No Soulmates Among Blockers) Suppose that M is a soulmate mechanism. For any profile $\succ \in \mathcal{D}$, any blocking coalition to M(≻) contains only players who are not soulmates.

Proof. Again, we use the notation in the definition of IMS. Clearly, no coalition W₁ that blocks M(≻) contains any players from N₁(≻), the set of 1^st order soulmates.

Now suppose that a coalition W₂ that blocks M(≻) contains a player from N₂(≻), but contains no player from N₁(≻). Then, for any player i* ∈ W₂ ∩ N₂(≻), there must exist a coalition $C^{\ast }\in {\mathcal{C}}_{i^{\ast }}$ such that (i) C* ⊆ W₂, and (ii) C* ≻_i* M_i* (≻). However, by definition of IMS, coalition M_i* (≻) is i*’s most preferred coalition among the coalitions made of players in N\N₁(≻). Hence, by (ii), C* must contain at least one player from N₁(≻), which contradicts (i). The same logic extends by induction to soulmates of any order. ■

The following are corollaries of Propositions 5.

Corollary 4 (Individual Rationality And Pareto Optimal) Given any IMS-complete profile ≻, the coalition structure IMS(≻) is individually rational and Pareto optimal.

Corollary 5 (Partial Individual Rationality and Pareto Efficiency) Suppose that M is a soulmate mechanism. Given any profile $\succ \in \mathcal{D}$,

1. any player i who is matched with a coalition that she likes less than {i} is not a soulmate, and

2. for any subset of players S ⊆ N such that there exists a partition π^S of S with ${\pi }_i^S{\succ }_i{M}_i(\succ )$ for all i ∈ S, subset S contains no soulmates.

6. How Common are Soulmates?

How common are IMS-complete profiles? How likely are players to be matched into a coalition by IMS? In this section, we study these questions using empirical analysis of real-world data and computational experiments. We focus primarily on the roommates environment (with an even number of players) where every player prefers being in any coalition of two better to being alone.

While IMS-complete profiles are relatively rare (in large player sets) when the set of profiles is unconstrained, there is often some structure to preference profiles encountered in real-world problems. Two particularly natural properties of preference profiles are reciprocity, where individual preferences for others are mutually correlated, and common ranking, in which individual preferences over coalitions are correlated.

We demonstrate that IMS matches many players in three real-world datasets concerning environments where reciprocal or commonly-ranked preferences are at least intuitively likely. Our computational experiments indicate that when preferences are highly reciprocal, or close to commonly ranked, IMS matches many players on average and profiles are often IMS-complete.

6.1. Soulmates in the Field

Here we present empirical results on soulmates in applied problems using real-world datasets in three different environments.

We first consider a roommates problem using social-network data from 1350 students at a university. We next consider a similar problem involving matching coalitions of no more than two in a work setting using data from 44 consultants. Finally, we consider a two-sided matching problem using data from 551 people attending “speed dating” events.

Each data set includes information we use as a surrogate for preferences.²⁰ In each case, ties occur in the preferences we derive from the data. Because of this, for each set of data, we have run IMS 1,000 times with random tie-breaking each time and recorded the proportion of players matched in each case. More details about the data sets and the precise assumptions used in deriving preferences are given below.

In each environment IMS was able to match about a third of players on average and sometimes substantially more depending on the tie-breaking- up to three-quarters in one instance of the work teams data. This is far more than would be predicted by our results on unstructured preferences in section 6.2.

To illustrate better how IMS is operating in these environments, the table below details the number of individuals left after each round of IMS (for a single random tie-breaking of preferences). This demonstrates that the iterated nature of IMS is far from trivial in practice. In each case, IMS is able to match at least 6^th order soulmates and, for example, in the roommates data there are 34 5^th order soulmates for this particular tie-breaking of preferences.

6.1.1. University Roommates

Our roommates dataset comes from a network of 1,350 users²¹ of a “Facebook-Like” social network at the University of California Irvine. The data is provided by and described in Panzarasa, Opsahl and Carley (2009). The data includes, for each user, the number of characters sent in private messages to each other user. We use this information as a surrogate for the preference data of each user, assuming that if a user sends more characters to i than to j than the user would prefer to be matched with i over j. We assume that a user would prefer to be matched with any random partner than to remain alone.

Over 1,000 trials, an average of 39.0% of users are matched by IMS. The maximum was 39.4%. Since the data measures characters sent- a relatively fine-grained measure, most of the ties occur where the users have sent zero characters to each other. This has the effect of randomizing the “bottom” of each users’ preference list and does not substantially affect IMS. The success of IMS in this case is likely due to the reciprocity in the derived preferences. If i sends many messages to j, then it is likely that j sends many to i.

6.1.2. Work Teams

The work coalitions data set comes from a study of 44 consultants within a single company. The data is provided by and described in Cross et al. (2004). The consultants responded on a 1-5 scale for each of the other consultants to the question “In general, this person has expertise in areas that are important in the kind of work I do.” Here, we assume that if a consultant gave a higher score to i than to j, the consultant would rather be on a coalition with i. (Again, ties in preferences are broken randomly.) We again assume that a consultant would prefer to be matched with any random partner than to remain alone.

Over 1,000 trials, an average of 31.3% of consultants are matched by IMS. The maximum was 77.3%. Since the preference measure is less fine-grained in this case, tie-breaking randomization has a stronger effect. Here, it is likely that elements of reciprocity and common-ranking are present in the data. Expertise is a relatively objective measure, while the fact that the question asks about “work I do” makes two consultants with the same focus likely to give each other higher scores.

6.1.3. Speed Dating

The speed dating dataset comes from a study of 551 students at Columbia University invited to participate in a speed dating experiment. The data is provided by and described in Fisman et al. (2006). Each participant had a four-minute conversation with roughly 10-20 partners and was then asked to rate each partner on a 1-10 scale in various aspects. Here, we focus on two ratings: “Overall, how much do you like this person?” and a shared interest rating. For both, we assume that if a participant gave a higher ranking to i than to j this participant would rather be matched with i than with j.

The proportion matched depends on the question. On average 26.2% can be matched by IMS based on the overall ranking but 31.0% can be matched using the shared interest rating. The maximum for the overall rating was 36.6% and 43.2% for shared interest. The improvement of using shared interest is likely due to the additional reciprocal structure in shared interest data.

6.2. Unconstrained Preferences

In this section, we compare the proportion of IMS-complete profiles to the proportions of other types of profiles that imply IMS-completeness, some of which are discussed in the literature: top-coalition, common-ranking, and reciprocal profiles.

Recall that common ranking profiles are those in which if player i prefers to be matched with j over k then every other player prefers to be matched with j over k.²²

Reciprocal preferences are preferences such that, for any player i, if i prefers coalition T to all other coalitions, then for all j ∈ T, j also prefers coalition T to all other coalitions.²³

To do our analysis, we randomly generated 10,000 preference profiles for total player set sizes n = 4, 6, 8, and 10 and tested the IMS-completeness and the top-coalition property. We have limited the size of player sets since it is computationally hard to test every subset of players to determine whether even a particular profile has the top-coalition property. Also, even the simple problem of counting preference profiles that contain at least some soulmates is a complex problem, not conducive to standard counting techniques. For some analytical results on this problem see Appendix A.3.

The proportion of common-ranking and reciprocal roommates profiles are much easier to count than IMS-complete and top-coalition profiles. The number of common-ranking profiles is n!. Hence, the proportion $\frac{n!}{(n-1){!}^n}$ of common-ranking profiles is very small. Even for total player sets of size 6 the proportion is 2.41 × 10⁻¹⁰. There are $n!((n-2)!{)}^n∕2^{\frac{n}{2}}$ reciprocal profiles. In comparison to common-ranking, reciprocal profiles are far more abundant. When n = 6 about a half of one percent of all profiles are reciprocal. However, each of these accounts for only a small portion of the IMS-complete profiles.

Table 2 compares the approximate proportions of IMS-complete, top-coalition, common-ranking, and reciprocal profiles. Top-coalition profiles are only a small portion of IMS-complete profiles as well. In fact, when the number of players is greater than or equal to 8, there were no top-coalition profiles among the 10,000 test cases.

Table 2:

Approximate proportions of unconstrained profiles meeting each of the conditions from 10,000 randomly generated test-cases.

	IMS-complete	Top-Coalition	Common-Ranking	Reciprocal
n=4	0.624924	0.3442	0.0185	0.0741
6	0.3064	0.0087	0.0000	0.0057
8	0.1219	0.0000	0.0000	0.0004
10	0.0460	0.0000	0.0000	0.0000

Even though IMS-complete profiles are far more abundant, they still form a vanishing subset of profiles under unconstrained preferences. In the next two subsections, we consider preference domains endowed with natural structure.

6.3. Soulmates Under Relaxed Reciprocity

Relaxed forms of reciprocity are natural in some environments: if Alice prefers Alex to all others, it may be that Alex ranks Alice highly as well. If preferences are purely reciprocal then the preference profile is IMS-complete, but what happens under relaxed reciprocity?

To study IMS under different degrees of reciprocity, we now introduce a generalization of reciprocal preferences, k-reciprocal preferences. For the roommates problem, a profile ≻ is k-reciprocal if for any two players i and j, j is in i’s top k most preferred players if and only if i is in j’s top k most preferred players.

Example 6 Consider the following profile of preferences over partners in a roommates profile.

$$\begin{array}{c}\hfill 1:3{\succ }_{1}2{\succ }_{1}4\hfill \\ \hfill 2:1{\succ }_{2}4{\succ }_{2}3\hfill \\ \hfill 3:4{\succ }_{3}1{\succ }_{3}2\hfill \\ \hfill 4:2{\succ }_{4}3{\succ }_{4}1\hfill \end{array}$$

The profile is trivially 3-reciprocal, as there are only 3 possible partners for each player. In general, n − 1 reciprocal preferences are unconstrained preferences.

The profile is also 2-reciprocal. For example, player 1’s two most preferred partners are players 3 and 2, and 1 is the most preferred partner for player 2 and the second most preferred for player 3.

As player 3 is player 1’s most preferred partner and 1 is not 3’s most preferred partner, the profile is not 1-reciprocal.

We randomly generated 10,000 k-reciprocal profiles for each combination of total number of players n ∈ {4, 6, 8,10} and k ∈ {2, 4, 6, n − 1} (recall that k = n − 1 are unconstrained profiles), tested each profile for IMS-completeness and recorded the number of players matched by IMS. Results for N = 10 are shown in figure 2 and full results are reported in the online appendix table 4.

Figure 2:

Proportion of Players Matched and IMS-Complete Profiles by Level of Commonality (D) for N = 10

Most 2-reciprocal profiles are IMS-complete. While, as expected, the percentage of IMS-complete profiles decreases k, a large portion of players can be matched by IMS even with moderate amounts of commonality. For instance, on average in groups of ten, about half of players can be matched by IMS. For comparison, on average only about 16% can be matched in unconstrained profiles. Interestingly, for k = 2,4, and 6, the average proportion of players that can be matched by IMS is nearly constant as n increases.

6.4. Soulmates Under Relaxed Common-Ranking

We now consider a relaxed form of common-ranking where players’ preferences are partially correlated. Unlike in the previous section, where we study a class of preference profiles, here we will instead study distributions over preference profiles that can be thought of as “noisy” common rankings.

In this experiment, we randomly generated 10,000 noisy common-ranking profiles for N = 6, 8, 10 using the following procedure. Each player’s preference starts with an underlying common cardinal vector of utilities over partners. Players’ utilities are then perturbed via a normally distributed “noise” term. A player’s new ordinal ranking becomes the player’s preference. The larger the variance of the noise, the larger the deviation of the expected perturbed preference from the original common-ranking profile.

To make the variance of the noise term informative of the extent of commonality of rankings, we calibrated variances to target specific amounts of ordinal distance between the rankings. We measure the distance between any two players’ ordinal rankings using the Kemeny distance (swap-distance, Kemeny 1959). The Kemeny distance counts the number of pairwise components which have a different ordering in two preference lists.

In the experiment, we chose noise amounts that correspond to an average Kemeny distance between any two players ranging from 0.2 to 2 (in 0.2 increments). For instance, d = 1 indicates that, on average, it takes a “swap” of a single pairwise preference component to transform one player’s preference list into another.²⁵

Figure 3 below reports the results for N = 10 and the full results are reported in appendix table 5. Even with a moderate commonality, a large proportion of profiles are IMS-complete. For example, with N = 10, approximately two-thirds of profiles are IMS-complete when the variance in common ranking is such that player’s preference lists difference by on-average one swap (d = 1). Approximately one-half of profiles are IMS-complete for d = 2. Interestingly, the likelihood of a profile being IMS-complete does not decrease much with the size of the total player set, holding the average distance constant.

Figure 3:

Proportion of Players Matched and IMS-Complete Profiles for by Level of Commonality (D) for N = 10

7. Conclusion

In this paper, we have provided the first detailed study of iterated matching of soulmates (IMS). Our paper demonstrates that IMS is a relevant property regardless of whether IMS can match all players, and even when the core is empty, as is the case in many environments studied in the literature. For instance, in a recent paper, Graziano, Meo and Yannelis (2020) study housing market scenarios where the core may be empty, Liu (2019) studies the potential emptiness of the core in matching problems with participation constraints, and Choi (2020) studies international matching markets between workers and entrepreneurs. What sort of results can be obtained if soulmate coalitions are formed before applying other solution concepts in such models?

It may also be fruitful to extend results about IMS beyond the hedonic matching environments studied in this paper. For instance, how can IMS be applied to matching environments with externalities such as those studied in Stamatopoulos (2021) and Gonzalez, Marciano and Solal (2019)?

Beyond coalition formation, IMS may also have implications for concepts that can be mapped into problems about the existence or structure of the core in a cooperative game. For instance, Jackson and Van den Nouweland (2005) study the concept of “strongly stable networks” through the use of cooperative game theory and the core. How can the invariant core properties of IMS be used in studying similar concepts?²⁶

Club economies seem like another situation in which IMS may provide interesting, new results. For instance, an innovative paper by Windsteiger (2021) introduces the idea of a sorting technology, such as clubs, that may facilitate segregation. Do exclusive clubs, for example, London social clubs (once called “gentlemen’s clubs”) match soulmates in contexts for which there may be discrimination both by price and by personal characteristics of club members?

While this is all speculative, we conjecture that there will be future research advancing the application of IMS. We conclude by noting two classic contributions on coalition formation which may be inspiring: see Gamson (1961) for a discussion of coalitions from a sociological perspective, and Ray (2007) who provides a broad more recent, game-theoretic approach to coalition formation problems.

Table 1:

Group Size Remaining After Each Round of IMS. Bold numbers indicate the final size at the end of IMS.

	Roommates	Work Teams	Dating (Overall)	Dating (Shared Interest)
Start	1350	44	551	551
1	1116	36	469	471
2	1000	34	431	435
3	936	32	407	413
4	896	30	395	397
5	862	28	387	385
6	840	26	385	375
7	826		383	371
8	824			369
9	822