In this section, we build a framework for examining a certain class of preferences. Suppose that a given agent, , is looking to form relationships with other agents in . In choosing his matches, desires to be matched with agents he deems to be “closer” to him according to some measure of likeness. This can be thought of as representing some notion of similarity, compatibility, or any other trait that desires. For example, this could encompass where other agents hail from geographically or their political views. To capture this idea, we introduce the notion of subjective homophily.
The subjective homophily function represents the “difference” an agent perceives between himself and a fellow agent. The homophily function is called "subjective" because we do not require that
, that is, we do not require that
’s view of the difference between himself and
be the same as
’s view of the difference between himself and
. This relaxes the symmetry assumption of Bartholdi and Trick [
17]. We are interested in the case where agents would like to match with those who are perceived to be closest in terms of this notion of subjective homophily.
Given the subjective homophily function, we write that each agent in some sense represents his own ideal point. The greater the similarity between two agents (the smaller their differences according to the subjective homophily function), the more preferred they are.
We now define a particular restriction on the subjective homophily function that will prove useful in demonstrating our main results.
4.1. Existence of Pairwise-Stable Fixture Matchings
The following lemma will be useful in constructing our algorithm to prove the existence of pairwise-stable matchings for the Stable Fixtures Problem when agent preferences are consistent with subjective homophily and satisfy approximate symmetry.
Lemma 1 (Bartholdi and Trick). If among all available choices, agent most prefers agent , and agent most prefers agent , then in any pairwise-stable matching and must be matched.
Proof. If and are not matched, they form a blocking pair. ☐
We now demonstrate that the existence of a pairwise-stable fixtures matching is guaranteed when preferences are consistent with subjective homophily satisfying approximate symmetry. The following lemma can be viewed as a fairly straightforward extension of Abraham et al.’s [
18] results for the Stable Roommates problem to the case of the more general Stable Fixtures problem.
Lemma 2. If agent preferences are consistent with subjective homophily and satisfy approximate symmetry then there exist pairwise-stable fixture matchings.
Proof. We demonstrate a constructive algorithm for obtaining a pairwise-stable fixture matching:
Step 1: Begin with the first entry in . Let be this entry. Since neither nor currently has any matches, we match them. Remove from . By approximate symmetry, the next entry in is . Because and are matched, this entry can be removed as well. Reduce both and by 1. If either or has filled his capacity, remove any remaining subjective homophily function output corresponding to that agent from . If neither nor has filled his capacity, no entries are removed from . Define as the vector of subjective homophily function outputs remaining following Step 1.
Step k: If , the algorithm terminates. If , we match the agents corresponding to the first entry in . Remove the subjective homophily function outputs corresponding to a match between these two agents from , and reduce each agent’s capacity by 1. If either agent has filled his capacity, we remove all subjective homophily function outputs corresponding to that agent from and rename the resulting vector .
This algorithm will terminate after a finite number of steps (there are a finite number of agents, and therefore a finite number of subjective homophily function outputs to consider), when either all subjective homophily function outputs have been removed or there is a single agent remaining with excess capacity. We now demonstrate that the resulting matching is pairwise-stable. Assume for contradiction that the matching resulting at the termination of the above algorithm is not pairwise-stable. Then there exists a blocking pair, , such that
either for some that is matched with, or has not filled his quota and has excess capacity remaining, and
either for some that is matched with, or has not filled his quota and has excess capacity remaining.
Assume that for some that is matched with. This implies that . But since and are not matched, it must be the case that when the algorithm reached , ’s entries must have already been removed, meaning that his quota was filled. Therefore, there is no agent, , matched with such that . We now assume that has not filled his quota and therefore has excess capacity available. This implies that there is no one remaining to whom he can be matched as all other agents must have filled their quotas with agents closer to them than . Thus, can not be a blocking pair, and we have obtained the contradiction.
The matching is pairwise-stable. ☐
It is important to here note the relevance of the weak preference ordering. Given that the subjective homophily function may admit ties between and among agents, it is possible for there to be multiple pairwise-stable matchings. When preferences are strict, the pairwise-stable matching achieved at the termination of the above algorithm will be unique. However, in the case of weak preferences where there are ties among agents in terms of subjective homophily, the outcome of the above algorithm will depend on how the subjective homophily outputs are ordered. However, the assumption of approximate symmetry means that such an ordering is possible, but may depend on the order in which agents with equivalent subjective homophily function outputs are ordered. This generates strategic implications that will be explored in more detail in
Section 5.
4.2. Random Paths to Pairwise-Stable Fixture Matchings
We have demonstrated that when preferences are consistent with subjective homophily and satisfy approximate symmetry, there exists a pairwise-stable solution to the Stable Fixtures problem. A natural question is whether stable matchings can be obtained through a decentralized matching process as opposed to a centralized algorithmic mechanism.
Our main result shows that when preferences are consistent with subjective homophily and satisfy approximate symmetry, a pairwise-stable matching can always be attained from a pairwise-unstable matching by satisfying a finite sequence of blocking pairs. Starting from an arbitrary matching , if form a blocking pair and it is true that both and , then we can simply match both agents to generate a new matching, . In this case, no other agents are affected by the match. However, it is possible that an agent may have his entire capacity filled under , that is, . If form a blocking pair for matching in this case, this means that must “dump" one of his current matches in order to match with . In this case, will dump his least preferred match among his current matches, that is, will dump such that for all . In words, will dump the current match of his who is farthest away from him, according to the perceived subjective homophily, of all his current matches under in favor of matching with . The dumped agent will then gain one unit of excess capacity.
Definition 11 (Satisfying the Blocking Pair). Let be a set of agents with preferences consistent with subjective homophily that satisfy approximate symmetry. Let be a matching. Let be a blocking pair for μ. A new matching, , is obtained from μ by satisfying the blocking pair if:
- 1.
and .
- 2.
If , then s.t. and dumps in favor of matching with .
- 3.
If , then s.t. and dumps in favor of matching with .
- 4.
If , then and if , then and .
- 5.
Condition (1) states that after satisfying the blocking pair, and must now be matched with each other. Conditions (2) and (3) state that if or is currently matched at full capacity under the original matching , they must dump their least preferred current match to satisfy the blocking pair. Condition (4) states that under the new matching , dumped agents remain matched to the same set of agents that they were matched with under minus the members of the satisfied blocking pair. Condition (5) states that all other agents not affected by the blocking pair have the same matches under the new matching as under the old matching .
Remark 3. Any individually irrational matching can be transformed into an individually rational matching by having agents dump any unacceptable matches.
We now demonstrate that starting from an arbitrary matching we can achieve a stable matching by sequentially satisfying a finite number of blocking pairs.
Lemma 3. When agent preferences are consistent with subjective homophily and satisfy approximate symmetry, for any matching μ, there exists a finite sequence of matchings , such that , is pairwise-stable, and for each , there is a blocking pair for such that is obtained from by satisfying that blocking pair.
Proof. We provide a constructive algorithm that will transform the current matching into a stable matching in a finite number of steps:
Step 1: Let . Consider the first entry of that corresponds to a pair of agents that are not matched under . This represents the first potential blocking pair. If these agents do not form a blocking pair, then either or must be matched to capacity and does not wish to dump any of his current matches. As subjective homophily outputs are increasing, no future blocking pair will arise involving these two agents together. Remove any subjective homophily outputs corresponding to this agent from . Define . If, however, does constitute a blocking pair, we have two possible cases:
Case 1.If and , we match and . Since neither nor is currently matched at full capacity, no other agents are affected. Call the resulting matching .
Case 2.If or , then either or are at full capacity under the current matching, and must dump their least preferred agent to satisfy the blocking pair. We satisfy the blocking pair and any dumped agents gain one unit of excess capacity. Call the resulting matching .
Define as the vector of subjective homophily function outputs remaining following Step 1.
Step k: Consider the first entry of corresponding to a pair of agents that are not matched according to the matching . This represents a potential blocking pair. If these agents do not form a blocking pair, then either or must be matched to capacity and does not wish to dump any of his current matches. As subjective homophily function outputs are increasing, no future blocking pairs involving this agent can arise. Remove any subjective homophily function outputs corresponding to this agent from . Define If, however, does constitute a blocking pair, we have the same two possible cases as above and proceed accordingly.
This algorithm terminates in a finite number of iterations resulting in the matching, . The proof of stability follows the same argument as given in the proof of Lemma 2. ☐
The critical step in the above proof is that blocking pairs can be satisfied sequentially based on subjective homophily. Any time an agent is dumped when a blocking pair is satisfied, the dumped agent has a greater subjective homophily function output value than the newly matched agent. This means that a dumped agent will not create any new instability among the matches that have been generated in previous steps of the algorithm.
Having proved the above lemma, the random paths to pairwise-stability result is an immediate consequence of the standard Markov-chain argument, summarized as follows: starting from an arbitrary matching , a random process can generate a sequence of matchings by satisfying a single randomly chosen blocking pair. The probability of any one blocking pair being chosen is positive for all such blocking pairs for a given matching. The following proposition results from the fact that for any matching, every blocking pair has a positive probability of being chosen.
Proposition 1. If agent preferences are consistent with subjective homophily and satisfy approximate symmetry, then a decentralized process of allowing randomly chosen blocking pairs to match will converge to a pairwise-stable fixtures matching with probability one.