Some models of this process include heterogenous preferences, but then assume that preferences are unchanging over time.[3]. Although this assumption is ...
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An Agent-based Model of Matching with Endogenous Preferences Katharine A. Anderson
Abstract— In this paper, I consider a set of heterogenous agents making group membership decisions, such as those made by an academics on the job market. The agents have an exogenous preference ranking over groups, which does not change and an endogenous preference ranking, which depends on group composition and is common to all individuals in the system. Using two different agent-based models, I find that when individuals are forced to weigh these two seperate preference rankings via a weighted preference ranking, tiers of groups emerge which have similar average ability levels and are thus treated as equivalent by individuals in making their decisions. I find that a dynamic model, in which individuals retire from groups randomly over time, exaggerates this tiering. The rate of rank-shifting also increases in this model, indicated in that lock-in of the rankings of lower-prestige groups may be mitigated by the retirement of high-ability individuals within those groups. Index Terms— Matching, Lock-in, Agent-based Models
The author acknowledges the support of the NSF in producing this work.
I. I NTRODUCTION There is a widely spread and diverse existing literature on the problem of how heterogenous individuals form into groups. Some models of this process include heterogenous preferences, but then assume that preferences are unchanging over time.[3] Although this assumption is very tractable analytically, and leads to some very powerful conclusions, it also ignores the fact that individuals often care about the composition of their group, which will change over time. Other models allow preferences to depend on group composition, but do not allow groups to have preferrences over individuals, effectively preventing groups from regulating their membership.[4] In this paper, I use an agent-based model to explore a problem that has proven difficult to address analytically– namely, how do individuals make their group membership decisions when 1) they are forced to weigh their personal, exogenous preferences against their preferences over groups of different compositions, and 2) they may be unable to join the group that they most prefer? This problem is interesting, in part because there are a number of real-world examples in which this is important, but also because it reflects a more fundimental tradeoff–between personal preferences, and lockin of current leaders via positive feedback. II. T HE M ODEL
can be a member of one group and each group can have up to m members. In this paper, I assume that M ≤ mN , so that there are always enough spots available for every individual to join a group.1 Each individual has a commonly-observed ability level, ai , drawn uniformly from the integers in the interval [0, 100]. I assume that the preferences of groups over individuals depend only on ability. That is, the groups have a complete, strict, and transitive preferences over members, denoted by PG where iPG j iff ai > aj . Since ability is commonly-observed, all groups have the same preference ordering over individuals. The average ability level of the group is commonly observed. I will call this average ability the prestige ofP a group 1 and denote group g’s prestige in period t by pg,t = m i∈g ai . Note that a group with an empty slot will suffer a decreased prestige level, because an empty slot is treated as having ability level 0. The preferences of individuals over groups are a little more complicated. As in most matching models, individuals have an exogenously-given comple, strict, and transitive preferenceordering over groups, which I will denote by PiX . This preference ordering does not depend on the current matching between individuals and groups thus it does not change over the course of the game. These preferences represent all of the qualities of a group that do not depend on the group’s composition. 2 However, in this model, individuals also care about the composition of their group–specifically, the average ability of the members, which I will call the prestige of the group. The prestige of the groups will change over the course of the game, as the matching between groups and individuals changes. However, since prestige is commonly observed, all individuals will have the same endogenous preferences over groups. I will denote this endogenous, prestige-based preference ordering over groups by PtN where gPtN h iff pg,t > ph,t . The exogenous and endogenous preferences of the individuals must somehow be combined into a single, overall preference ordering. There are many possible ways to do this. Dutta and Masso solve this problem by using lexicographic preferences.[1] Unfortunately, this approach gives one set of preferences (exogenous or endogenous) strict priority over the other, which seems like an unlikely way for a real-life individual to address the problem. Since the individuals are essentially facing a tradeoff between their exogenous preference for a particular group and
A. Setup and Notation There are two different types of agents in this model: M individuals and N groups. In the following, I denote the set of individuals by I and the set of groups by G. Every individual
1 I would conjecture that the results would be similar when there are more individuals than available spots, but I have not checked that possibility. 2 Depending on the particular problem being addressed, this might represent location, focus or mission, benefits offered, culture, administration, etc.
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their endogenous preferences over the group composition, I will use an approach that allows individuals to weigh their exogenous and endogenous preferences. Let RiX (g) be the rank of group g in individual i’s exogenous preference ranking, where RiX (g) = M if g is ranked first in the preference ratings and RiX (g) = 1 if it is ranked last (ie: RiX (g) = r if the set h | gPiX h has cardinality r). Similarly, let RtN (g) be the rank of group g in the endogenous (prestige) rankings at time t (ie: RtN (g) = r N if the set h | gPt h has cardinality r). These two rankings are combined via a linear combination with weight α on the T exogenous preference ranking: Ri,t (g) = αRiX (g) + (1 − N α)Rt (g). I define individual i’s overall preference rankings T T T in time t as PitT where gPi,t h iff Ri,t (g) ≥ Ri,t (g).3 B. The Process At the start of a run, the individuals are sorted into the order of movement.4 This order of play remains constant throughout the run. On each step, a single individual, i, is allowed to move. She first sorts the groups according to her overall preferences ordering in that period, P T (i). She then applies to the groups in order, starting with her most preferred group. A group decides whether to accept an applicant in the following way. If the group is not yet full, the individual is automatically accepted. If the group already has m members, it keeps the top m individuals according to its own (abilitybased) preferences, PG . That is, if the applicant is of higher ability than the lowest ability member of the group, the lowest ability member is rejected and the new applicant is accepted– otherwise, the applicant is rejected. If the applicant is rejected from agroup, she applies to the next group on her overall preference list. If she is accepted, she stops applying and joins that group. If she was unmatched, she becomes matched. If she was already a member of another group, she leaves her old group and joins the new one. I will call the set of groups that would accept an individual in a given period, that individual’s feasible set. C. Discussion of this Game There are several important aspects of this model, which should make the results interesting. Most importantly, this formulation provides a parameter, α, which can be used to tune the preferences of the individual–by using a process that resembles that in the Gale-Shapely matching literature, I can observe a whole range of behaviors. When α = 1, the individuals care only about their own exogenous preferences, and the resulting matching is a basic Gale-Shapley matching. On the other hand, when α = 0, the individuals care only about the prestige of the group. In this case, the departments that are the first to gain members will attract all subsequent 3 Note that this preference ordering is no longer strict (though it is complete and transitive). This should not cause any difficulty in the following and brings this framework closer to that in the existing literature. 4 Note that the order of play had no effect on the qualitative results presented, although it did have an effect on the time to convergence.
members, and the resulting matching will dominated by lockin. As a result, the individuals sort entirely by ability. In between is a region where the sort is neither Gale-Shapley or a simple ability sort. This is the region that I hope to characterize in this work. Appendix A places some rough 1 , the analytic bounds on this “interesting” region. For α < M M−1 individuals will sort perfectly by ability. For α > M , the individuals will sort as in the Gale-Shapley algorithm. The interesting region will lie between these values. In that region, the individual will have to weigh the prestige of an institution against her personal preferences. There are several things that make this intermediate region interesting. Firstly, although at the end of her turn the individual will be matched with the group that she most prefers in her feasible set, there is no reason to suspect that she will continue to be pleased with her match after the overall matching has changed. Thus, the individuals have the potential to not only join groups, but also leave them, in switching from one group to another. Whereas in the Gale-Shapley model, the average ability level within a particular group could only increase, in the current model, it could also decrease. This means that it is possible for groups to change rankings during the game– potentially reversing early lock-in. III. R ESULTS On a gross scale, the behavior of this model is much as one would expect. Extreme values of α yield the expected results, and the intermediate region yields more complex behavior. When α is near 0, the groups that gain members at the beginning of the game gain all subsequant members, locking in the gains for these early winners. When α is near 1, the Gale-Shapley matching occurs. The region in between is much more interesting. For intermediate values of α, there is some rank-shifting early on, because although individuals care about the ranking of the groups according to ability, they are willing to ignore small differences in prestige if there are large differences in their exogenous preferences. However, lock-in of the highestranked groups still occurs fairly early in the game–well before all of the individuals have moved once. The rest of the rank ordering also solidifies very early in the game. Eventually, all individuals match with a group. When all of the individuals find that they are matched with the group they most prefer within their feasible sets, the matching is stable and remains the same forever. As would be expected, when individuals put more weight on their exogenous preferences, the groups have a more even distribution of ability levels. The results for N = 100, M = 10, and m = 10 are shown in Fig. 1 . Individuals match closer to the top of their overall rankings and are, on average, happier with their match when α is high, because more weight is put on personal preferences and less on common preferences. This is a fairly straightforward conclusion, but it does indicate that the computational model is working correctly. Additionally, it is possible to see in Fig. 1 the regions of behavior predicted by the analysis above. = .9, the individuals behave consistantly– For α > M−1 M
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Fig. 1.
Average, Minimum, and Maximum Prestige Levels 100 individuals and 10 groups with maximum size 10 100
Maximum Prestige Average Prestige Minimum Prestige
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constrain their decisions. However, despite being able to join whichever group they wish, the highest ability individuals do not all end up in the group with the highest prestige. For example, in the case shown above, the highest ability individual was matched with the group with the second-highest prestige level. This provides some insight into why the tiering occurs–if the schools are “close enough” in prestige rankings, then the exogenous preference rankings of the individuals take over in the decision-making process. In the case where individuals care only about rankings, I conjecture that the size of these tiers is well-defined with respect to α.
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specifically, the match is the same as it is under the Gale1 = .1, the individuals sort Shapley algorithm. For α < M perfectly according to ability. What is somewhat surprising is the large range in which individuals sort more-or-less according to ability level. For levels of α up one half, the individuals sort entirely by ability. At this point, it is unclear what is driving that result, but it seems likely that there is an underlying mathematical reason for this behavior.5 Most interesting, however, is the “tiering” that emerges from the model. Over the course of the run, the groups tend to fall into tiers–collections of groups that have similar prestige levels and are thus treated more or less the same by individual players. Table X shows the prestige levels of different groups at the end of a typical run with N = 100, M = 10, m = 10, and α = .7. Note that the groups tend to cluster into sets that have similar ability levels. TABLE I Department 1 5 9 8 7 6 2 1 10 4
Average Ability Level 90 88 62 60 56 41 32 25 17 5
Another way of observing ths tiering is to examine where the highest ability individuals choose to settle. These individuals are interesting because the group preferences do not 5 These results change predictably with group size/number. Fewer, larger groups provide more individuals with the opportunity to join their top-ranked group. Thus, the average utility of the individuals (as measured by the ranking of their current group in their overall preference ranking) is higher when groups are larger. Obviously, this increase in utility is most pronounced for the lowest-ability individuals, because high-ability individuals are already able to join the group of their choice. The threshold value of α for which behavior is insensitive to exogenous preferences also varies predictably. Fewer, larger groups push this threshold down, and a larger number of small groups push it up.
IV. T HE DYNAMIC M ODEL There is one obvious question implicit in the above analysis–namely, where do the initial prestige values come from? In the above model, I assumed that all of the groups were empty to start with, and thus began the game on equal footing. In this section, I present a more dynamic model that allows the groups to evolve organically over time.Now, each individual retires from a group with probability p in every period. Every time an individual retires, his spot opens up and a new, unmatched individual is created. The results of this model are obviously less clear-cut than in the more static model. The system obviously never reaches a stable matching, in the sense that the last model did. However, there are still a few clear trends. First, tiering is much more pronounced in this model than it was in the last. This is somewhat surprising, since the loss of individuals over time seems as though it should limit the effects of tiering. However, I attribute most of this effect to the fact that many of the lower tier groups remain empty in this more dynamic model. Rank-shifting also occurs much more frequently in this model. As higher-ability individuals retire, the prestige level of their group drops significantly, which occassionally prompts other high-ability individuals to leave that group in favor of another, more prestigious group. However, the rank-shifting seldom takes place among the higher-ranked groups. The lower-ranked groups, which are less often full in this model, are more suseptible to loss of even low-abiltiy members. Further results from this model are forthcoming. V. E XTENSIONS There are a number of potential extensions to this model that would potentially be interesting. First, while the rankings measure used above does provide some analytic results, and most closely resembles the measures used in other similar work, it also tends to compress large preference differences and expand small ones. A similar measurePwhich weights a preference ranking (eg, {vi1 ...viM } where j vij = 100) and the prestige level would be a potentially interesting variation. Secondly, I would like to consider the change in behavior when the distribution of abilities is more similar to that found in real-life matching problems. This may be especially pertinent in the dynamic model, where the loss of a highability individual is felt more heavily if the individual is unusual in the department. If ability were distibuted according
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to a normal distribution, then the loss of one star may have a larger effect. If ability were distributed according to a power law, the effect might be even more dramatic. Finally, and most interestingly, I would like consider other ways that individual preferences might depend on group composition. For example, there is some evidence that individuals on the job market care more about the average ability of individuals higher ability individuals than about the overall average ability in a working group. This might have interesting ramifications, because the highest ability individuals would tend to have weaker endogenous preferences than lower ability individuals, adding another layer of heterogeneity. Alternatively, individuals may be most comfortable if their ability level is similar to the average ability level, which would almost certainly consolidate ability levels. Finally, if individuals are connected in a social network, they may care about how many individuals they know within a group, in addition to the overall ability level. ACKNOWLEDGMENT Thanks to Michael Aylward, Christopher Cameron, Chris Fowler, Sera Linardi, Tobias Lorenz, Vincent Matossian, Jolie Martin, John Miller, Ryan Muldoon, Emre Ozdenoren, Scott Page and Robi Ragan A PPENDIX A Recall that M is the number of groups in the system.6 For a linear weighting of preference ranks, the following two theorems hold. 1 Theorem 1: For α ∈ 0, M , individuals will sort perfectly according to ability. Proof: Suppose RtN (a) = M for a ∈ G. That is, suppose A ranks first in the prestige ranking, PtX . T X • Find the α st aPi b ∀ b ∈ G. That is, α st αRi (b) + (1 − N X N α)Rt (b) < αRi (a) − (1 − α)Rt (a) ∀ b ∈ G • To prove the sufficient condition take the following, worst case scenario. – Suppose RiX (a) = 1, That is, suppose that a is ranked last in the individual’s exogenous ranking. – Suppose further that ∃ b ∈ G such that RiX (b) = M and RtN (b) = M − 1. That is, consider a group that is ranked first in the individual’s exogenous preference rankings and second in prestige. • We want α st αM + (1 − α)(M − 1) < α + (1 − α)M 1 => α < M Theorem 2: For α ∈ M−1 M , 1 , individuals will sort as they would in the Gale-Shapley algorithm with the individuals as the proposers. Proof: Suppose RiX (a) = M for a ∈ G. That is, suppose A ranks first in the individual’s exogenous preference ranking, PiX . T X • Find the α st aPi b ∀ b ∈ G. That is, α st αRi (b) + (1 − α)RtN (b) < αRiX (a) − (1 − α)RtN (a) ∀ b ∈ G 6 Actually, the following can be generalized to a case where individuals have exogenous preference orderings over disjoint sets of groups, as long as all individuals partition the schools into the same sets. In that case, the result holds by replacing M with the number of sets of schools in the partition.
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To prove the sufficient condition take the following, worst case scenario. – Suppose RtN (a) = 1, That is, suppose that a is ranked last in prestige. – Suppose further that ∃ b ∈ G such that RtN (b) = M and RiX (b) = M −1. That is, consider a group that is ranked first in prestige and second in the exogenous preference rankings. We want α st α(M − 1) + (1 − α)M < αM + (1 − α) => α > M−1 M
Note that both of the above results are sufficient conditions, meaning that they provide upper and lower bounds on the regions specified. In between those values, the behavior of the agents is not particularly well characterized in this way. R EFERENCES [1] Bhasker Dutta and Jordi Masso, “Stability of Matchings when Individuals Have Preferences over Colleagues,” Journal of Economic Theory, (1997) [2] Frederico Echenique and Mehmet B. Yenmez, “A Solution to Matching With Preferences over Colleagues,” WP (June, 2005) [3] D. Gale and L.S. Shapley, “College Admissions and the Stability of Marriage,” The American Mathematical Monthly, (January, 1962) [4] Ken Kollman, John H. Miller, and Scott E. Page, “Political Institutions and Sorting in a Tiebout Model,” American Economic Review, (December, 1997)
