Allocating Group Housing Justin Burkett, Francis X. Flanagan, and Amanda L. Griffith* August 18, 2017

Abstract We study mechanisms for allocating objects to pairs of agents when agents may have nontrivial preferences over objects and pairings. In this environment, the mechanism may distort agents’ preferences over pairings. Compared to certain distortive mechanisms, a non-distortive one always has a stable allocation in our model, and selects stable outcomes that are ex ante preferred by all students under a regularity condition on the distribution of pair values.

I

Introduction

We propose a novel theoretical model which combines two well known matching market models, the house allocation model and the roommates problem, in order to study the allocation of objects (rooms) to pairs of agents (roommates) in a setting where agents are free to form pairs as they choose, but objects are allocated to pairs via a centralized mechanism. Agents care about both their roommates and their rooms, thus the allocation mechanism can affect the agents’ decision over pair formation, and hence the quality of roommate matches. Our leading example is housing allocation on college campuses, thus we refer to the objects as housing/rooms and the agents as students/roommates, but the model could apply to other settings in which agents have preferences over both * Burkett: Department of Economics, Wake Forest University, Box 7505, Winston-Salem, NC 27109; [email protected]. Flanagan: Department of Economics, Wake Forest University, Box 7505, Winston-Salem, NC 27109; [email protected]. Griffith: Department of Economics, Wake Forest University, Box 7505, Winston-Salem, NC 27109; [email protected]. We thank Thayer Morrill and an anonymous referee for helpful feedback.

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the coalitions they form and the objects they are assigned.1 We focus on two particular random serial dictatorship mechanisms which differ in when they assign housing priorities to students. The first mechanism, which we denote φL , assigns priorities to individual students before roommate groups are formed. The second mechanism, φR , assigns priorities to groups after they are formed. More specifically, φL uses an underlying priority order over each individual student to assign priorities to roommate pairs, such that the priority of the pair is determined by the highest priority of any member; thus there may be significant incentive to trade roommate quality in exchange for room (priority) quality. Meanwhile, φR assigns priorities uniformly randomly to roommate groups after they form, which mitigates this incentive. We compare these mechanisms in regards to stability and students’ preferences over them. Despite the fact that these seem to be the most commonly used allocation schemes across U.S. colleges,2 and despite the large literature on the peer effects of college roommates, we are unaware of any existing formal examination of these mechanisms. We show that when students know which room they will receive as a function of which roommates they choose, which corresponds to the φL mechanism, it is less likely that a stable roommate grouping exists, and the mechanism, under a regularity condition, ex ante has a lower payoff for every student when students are symmetric ex ante. By ex ante, we are referring to a state in which every students’ preferences are uncertain. It is easy to construct examples in which after preferences are known students disagree on the mechanism they prefer, but we show that when all students’ preferences are uncertain they all prefer the φR mechanism. This result can be understood as a formal justification for the mechanism designer’s choice of one mechanism over the other, when the designer is uncertain about individual students’ preferences. These results show that different housing allocation mechanisms can have drastically different, yet predictable effects on roommate group formation. This 1 Some

examples may be business teams which are assigned projects/locations, or families with preferences over neighbors and schools. 2 A survey of the top 25 National Universities and the top 25 National Liberal Arts colleges shows that 23 of 50 determine the housing priority before students decide on their roommates, and 16 of 50 determine the priorities after roommate groups form. The rankings come from U.S. News & World Report. The remaining schools either do not make their rules available or follow some other scheme.

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is an important consideration for the mechanism designer, as it is well documented that peer effects from college roommates can have a significant impact on a number of academic and social outcomes. For example, students’ GPAs may be affected if there is a shift in the ability level of students that are paired together, resulting in more (or fewer) high ability students rooming with lower ability stuGriffith and Rask, Rask 2014 2014; Carrell et al., al. 2009 2009; Foster Foster, 2006 2006; Sacerdote Sacerdote, 2001 2001; dents (Griffith Zimmerman, 2003 Zimmerman 2003). There may also be social impacts such as changes in drinking behavior, which ultimately can also significantly affect GPA (Kremer Kremer and Levy, Levy 2008; Sacerdote 2008 Sacerdote, 2001 2001). Finally, these effects may extend beyond the academic setting, and beyond graduation. Research has found links between roommate and suite groupings during college and race relations and political beliefs later Boisjoly et al., al. 2006 2006; Zimmerman et al., al. 2004 2004). Therefore, when evaluating in life (Boisjoly a housing allocation mechanism such as the ones described in this paper, which are used widely within the higher education sector in the U.S., it is important to think carefully about how the mechanism affects the formation of roommate groups. While we do not take a stand on which mechanism achieves the most desirable outcome in terms of peer effects, we view our results as a useful guide and starting point for housing allocation mechanism designers and future researchers. The remainder of the paper is organized as follows. In Section II we discuss related theoretical literature, to complement the discussion of the peer effects literature in the introduction. Next we introduce the model in Section III III. Section III also introduces the various mechanisms in question, as well as the stability notions we use. Section IV compares students’ preferences of each mechanism. Section V concludes. All proofs are in the Appendix Appendix.

II

Related Literature

The model we introduce in this paper unites two distinct subjects in the matching literature: housing allocation problems and roommate problems. The house allo1979), is characterized by cation problem, introduced by Hylland and Zeckhauser (1979 a set of n agents who each must be assigned exactly one object (house) from a set of n indivisible objects. This is closely related to the housing market problem, introduced by Shapley and Scarf (1974 1974), which is a house allocation problem in which each agent initially owns one object. Abdulkadiroglu and Sönmez (1999 1999) intro3

duce a hybrid model, the house allocation problem with existing tenants, in which some, but not all, objects are initially owned. Our model begins as a housing allocation market without existing tenants: initially the objects (rooms) are unassigned. The main difference between previous literature on house allocation and our model is that we wish to assign objects to pairs, rather than individuals. Dogan et al. (2011 2011) looks at the housing market problem with couples, however the authors assume that couples are determined exogenously. In the present paper we are explicitly interested in the coalition formation part of the game, which is why our model is also closely related to the roommates problem. 1962), is a one-sided The roommates problem, introduced by Gale and Shapley (1962 matching problem: there are 2n agents who must form n pairs. Gale and Shapley introduced this as an example of a market in which there is no guaranteed stable match. To avoid this issue students in our model have preferences based on the “complete rigidity in the division of output” preferences of Becker (1973 1973): students s1 and s2 in a roommate pair generate a symmetric “output” w(s1 , s2 ) = w(s2 , s1 ), and for each student si there exists a strictly increasing function f si such 2000) shows that these that f si (w(si , s j )) is si ’s payoff from living with s j . Chung (2000 preferences are part of a more general category of preferences, called no odd rings preferences, which always guarantee a stable match in the roommates problem. Likewise, Pycia (2012 2012) shows that if the set of feasible preference profiles is rich enough, then roommate preferences must be pairwise aligned to guarantee a stable coalition structure. For similar reasons we restrict our theoretical results to the situation in which students may only form pairs, rather than larger coalitions. Although coalitions may be larger than two in many “real world” room allocation mechanisms, Arkin et al. (2009 2009) show that even if preferences are generated by distances in a metric space stability is not guaranteed if the size of the roommate coalition is expanded beyond couples.3 Under one of the mechanisms we study, φR , the unique stable outcome corresponds to the positively-assortative student match. Following Becker (1973 1973), the literature on assortative matching often assumes that the “production function” between two matched agents exhibits a complementarity in agents’ attributes, which then implies that the efficient match is positively-assortative (ignoring costs 3 Preferences

generated by distance in a metric space, first proposed by Bartholdi and Trick (1986 1986), is another “no odd rings” type of preferences.

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associated with signaling). Our environment differs. Although we assume each pair receives a common match value, this does not correspond to any complementarities analogous to the ones in the assortative matching models.

III

Model

Let R be the finite set of n ≥ 2 rooms, and S be the set of 2n students. A (roommate) pair is a subset of S of size two, denoted (si , s j ) with i 6= j. A match µ is a partition of S into pairs. We refer to student i’s roommate in match µ as µ(si ), and we have the usual requirement that s j = µ(si ) ⇐⇒ si = µ(s j ). We refer to the set of feasible matches as M. Student payoffs are a combination of payoffs over roommates and rooms. Student s’s payoff from living with s0 is given by f s (w(s, s0 )). The function w : S2 → R+ assigns a symmetric “index” to each roommate pair, and for each s ∈ S the strictly increasing function f s : R+ → R+ translates the index into student s’s payoff from the roommate pairing.4 Thus, for any s, s0 ∈ S, w(s, s0 ) = w(s0 , s) ≥ 0. We impose no further restrictions on these functions, which allow for nontrivial differences between students in their preference orderings. For example, student i may rank j first, while j ranks several other students above i. For each student s, room values are set according to vs : R → R+ . We assume throughout that students have common ordinal preferences over rooms. That is, for any s, s0 ∈ S, r, r 0 ∈ R, vs (r ) > vs (r 0 ) ⇐⇒ vs0 (r ) > vs0 (r 0 ). We also assume that when n ≥ 3 there exist at least three rooms that are distinct from the others. That is, |{r |vs (r ) 6= vs (r 0 ), r 0 ∈ R \ r }| ≥ 3 when n ≥ 3. The complete payoff function for student s for being matched with roommate s0 and room r is us (r, s0 ) = vs (r ) + f s (w(s, s0 )).5 We assume all feasible roommate-room pairs are acceptable to all students: for any (r, s0 ) ∈ R × (S \ {s}), us (r, s0 ) > max{us (∅, s0 ), us (r, ∅), us (∅, ∅)}.6 4 This

model borrows from Becker Becker’s (1973 1973) model of marriage. the common order assumption is strong, it allows us to avoid specifying a method for how roommates jointly decide on a room, given their possibly distinct preferences over rooms. We leave such a model as an avenue for further research. 6 All of the results would follow if, rather than all roommates being acceptable and all students having common ordinal preferences over rooms, S could be partitioned into subsets such that for each subset S0 , for any student s ∈ S0 , S0 contains all acceptable roommates for s, and within these sets students share ordinal preferences over rooms. For example, athletes may all prefer to live near the athletic facilities on campus, in contrast to other students. We require then that they also 5 Although

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Formally, an allocation A = (µ, Mµ ) is a match, µ, and a bijection, Mµ : µ → R, where Mµ (s, s0 ) = r is the room assigned to pair (s, s0 ). The primary object of interest for our paper is not the allocation, but rather the mechanism through which the allocation is created. The following section introduces the class of mechanisms we study and introduces stability.

III.1

Mechanisms and Stability

Anecdotal evidence suggests that the most common way colleges distribute rooms is via some sort of random serial dictatorship: a priority order is created over the set of roommate pairs, and pairs choose rooms in order of this priority order. Formally, for any µ ∈ M, a priority order is a total order µ over roommate pairs in µ. For any feasible match µ, let Σµ be the set of all priority orders. A mechanism is a mapping φ which takes feasible matches µ ∈ M and assigns a probability distribution over priority orders, ∆(Σµ ). Therefore the room allocation procedure is as follows: students form a feasible match µ, a priority order is chosen according to φ(µ), and student pairs choose rooms in order of the priority, with the top priority pair choosing first, and so on. Throughout the paper we assume the mechanism, the reported match, the priority order realized, and all preferences are common knowledge when pairs form or when rooms are chosen. One type of mechanism we are interested in we refer to as φL . Given any match µ ∈ M, a φL mechanism assigns a deterministic probability distribution over Σµ based on an underlying order over individual students. Formally, a mechanism is a φL mechanism if there exists a total order >S over the set of students, s1 >S s2 >S . . . >S s2n , such that for any µ ∈ M, for any (si , s j ), (sk , sl ) ∈ µ, φL (µ) puts probability one on priority order µ such that (si , s j ) µ (sk , sl ) if and only if min{i, j} < min{k, l }. That is, a φL mechanism assigns all probability to a priority order which puts a higher priority to a roommate pair based entirely on the rank order of the two highest ranked students between the pairs according to > S .7 Generalizing, one can think of φL mechanisms as a subset of an even larger class of mechanisms which, given any match µ ∈ M, put probability one on some prefer to live with each other. 7 In our examples > is often given, in which case we refer to “mechanism” φ , with the L S understanding that this is the φL mechanism with respect to >S .

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priority order µ ∈ Σµ . We refer to this class as the Φ L class of mechanisms. Although φL mechanisms are our leading example, all of the theoretical results apply to any φ ∈ Φ L . A natural alternative to mechanisms in Φ L that we consider is a mechanism that assigns a priority order uniformly randomly to a match after the students commit to that match, which we refer to as φR . Formally, for any match µ ∈ M, φR assigns probability 1/|Σµ | = 1/(n!) to each priority order in Σµ . In some sense, φR and the mechanisms in Φ L represent two extremes. The mechanisms in Φ L are deterministic in that the mapping from matches to priority orders is known when the students form matches, while the mapping is unknown at that time under φR . Note that at this point we make no assumptions on how the priority orders are assigned for any φ ∈ Φ L . In either case, the mechanism specifies when roommate pairs form and how rooms are allocated to roommate pairs, but it does not specify how pairs form. Roommate pairs are formed in a decentralized process which we do not model explicitly. Instead, we assume that the match resulting from this decentralized process satisfies a stability criterion which we give in Definition 1. That is, stability is both an equilibrium concept and a desideratum. Stability in this environment is defined with respect to the room allocation mechanism, because a student’s preferences over roommates are affected by how the rooms are allocated, and hence roommate preferences are endogenous to the mechanism. The stability notion we use is similar to pairwise stability in other matching models. Informally, a match is not stable if there exist two students who would rather be roommates with each other than their current roommates. Such a pair is called a blocking pair or is said to block the original match. This corresponds to the usual notion of pairwise stability; however, it is necessary in this model to specify what rooms the students expect to occupy after forming a blocking pair. First, to make the blocking decision a feasible calculation, we assume that if two students block µ, then the roommates who are “left” by the blocking pair will pair together in the subsequent match. That is, for the roommate pairs (s, s0 ), (t, t0 ) ∈ µ, if (s, t) blocks µ, then the new roommate match after the block will be µ0 = (µ \ ((s, s0 ) ∪ (t, t0 ))) ∪ ((s, t) ∪ (s0 , t0 )). Because of this assumption it is often useful to look at matches which differ in this manner: we say two matches µ and µ0 are adjacent, written µ ↔ µ0 , if exactly two pairs are different between 7

them: µ ↔ µ0 ⇐⇒ |µ \ µ0 | = 2. Next, define student s’s expected payoff given match µ ∈ M and mechanism φ ∈ {φR } ∪ Φ L as Us (φ, µ) = Vs (φ, µ) + f s (w(s, µ(s))), where Vs (φ, µ) is the expected room value and f s (w(s, µ(s))) is the known roommate value, which is independent of the mechanism. Since all students have common ordinal preferences over rooms, the probability that a pair receives the ith best room is the probability that the pair receives the ith best priority: let Pi (φ, µ, (s, µ(s))) be the probability that (s, µ(s)) receives the ith highest priority given µ and φ, and let rooms be ordered r1 , . . . , rn such that vs0 (r1 ) ≥ vs0 (r2 ) ≥ . . . ≥ vs0 (rn ) for all s0 ∈ S, then Vs (φ, µ) = ∑in=1 vs (ri ) Pi (φ, µ, (s, µ(s))). Since the expected payoffs depend on the mechanism, the stability of a match will also depend on the mechanism, as the following definition explains. Definition 1. Given mechanism φ ∈ {φR } ∪ Φ L , a match µ is stable with respect to φ if, for any two pairs (s1 , s2 ) and (t1 , t2 ) in µ and any adjacent µ0 with (s1 , t1 ) ∈ µ0 , we have that Usi (φ, µ0 ) > Usi (φ, µ) implies Uti (φ, µ0 ) ≤ Uti (φ, µ) for i ∈ {1, 2}. This stability criterion applies at an interim stage, in the sense that it applies before rooms are allocated, but after preferences are known. We define stability at an interim stage because we use it as the solution concept in the decentralized pair formation game, and we define it with respect to the mechanism because the mechanism influences students’ preferences over roommates. Ex post, following the assignment of rooms, one could define stability independently of the room 2010) argues that allocation mechanism. Once rooms have been assigned, Morrill (2010 if a student cannot unilaterally evict their roommate, stability should incorporate the constraint that for a blocking pair to form all students involved in a swap should agree to the swap (i.e., blocking pairs must induce Pareto improvements). We define an ex post stability criterion next that does not rely on the nature of the room assignment mechanism. In the analysis that follows we use the phrase “stable with respect to the mechanism” (or simply “stable” if the mechanism is implied) to refer to the interim concept in Definition 1, and the phrase “ex post stable” to refer to the ex post concept in Definition 2. Definition 2. Let the allocation A = (µ, Mµ ) be such that (s1 , s2 ), (t1 , t2 ) ∈ µ, Mµ (s1 , t1 ) = r1 , and Mµ (s2 , t2 ) = r2 , and let A0 = (µ0 , Mµ0 ) be any alternative allocation in which all students in S \ {s1 , s2 , t1 , t2 } are assigned to the same 8

roommate and room in A and A0 . The allocation A is ex post stable if at least one of {s1 , s2 , t1 , t2 } strictly prefers their allocation in A to their allocation in A0 or all of {s1 , s2 , t1 , t2 } are indifferent between A and A0 . We refer to both the match and the resulting allocation as being stable. Theorem 1 shows that under our assumptions a stable match always exists under φR ; however, Theorem 3 states that under any φ ∈ Φ L a stable match is not guaranteed to exist, as the following example helps illustrate. s2

s4

s3

s1

s5

s6

0

2

3

5

8

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Figure 1: Example with No Stable Match Example 1. Let roommate indices equal the roommate values and be represented by some constant minus the distance between students on the number line in Figure 1, and let the mechanism be φL , where >S is given by the students’ indices, with s1 the highest ranked. Let each student have the same room value function, with v(r1 ) = 1.5, v(r2 ) = 1.25 and v(r3 ) = 0. First look at the match µ = {(s1 , s5 ), (s2 , s6 ), (s3 , s4 )}, listed in order of each pair’s priority. This match is the unique stable match under φR . However, since v1 (r1 ) + f s1 (w(s1 , s3 )) > v1 (r1 ) + f s1 (w(s1 , s5 )) and v3 (r1 ) + f s3 (w(s1 , s3 )) > v3 (r3 ) + f s3 (w(s3 , s4 )), the pair (s1 , s3 ) blocks this match under φL . The resulting match, µ0 = {(s1 , s3 ), (s2 , s6 ), (s4 , s5 )}, is itself blocked by (s2 , s4 ), and one can check that all other possible matches are also blocked.8 The essential problem leading to non-existence of a stable match is that a pair may receive different rooms in different matches, leading that pair’s utility to depend on the arrangement of the other students. Under φR however, the expected utility a student assigns to a given pair is always the same. As a consequence, it is straightforward to prove the following. Theorem 1. There exists a match which is stable with respect to φR . This result follows from Becker (1973 1973) and Chung (2000 2000): under φR , 0 0 Vs (φR , µ) = Vs (φR , µ ) when µ(s) = µ (s). Therefore if w(s, µ(s)) ≥ w(s, s0 ) for 8 Blocking

pairs are one of (s1 , s2 ), (s1 , s3 ), (s1 , s4 ), (s1 , s5 ), (s2 , s4 ) and (s3 , s4 ).

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all s0 6= µ(s), then neither s nor µ(s) can be part of a pair which blocks µ. Therefore we can construct a match which is stable under φR by successively identifying the highest value pairs not already formed and placing them in the match. Although at the time the roommate match forms under φR the allocation of rooms is not yet known, the allocation of rooms does not influence the ex post stability of the match. Note that we assumed that students are able to commit to their match before rooms are selected in φR . One interpretation of this next result is that this commitment is self-enforcing. Theorem 2. The stable match with respect to φR is ex post stable. The possible non-existence of a stable match is not unique to φL . Our next result, a kind of impossibility result, shows that a stable match is not guaranteed for any φ ∈ Φ L . Theorem 3. If φ ∈ Φ L and n ≥ 3, it is possible to construct roommate indices so that no stable match exists under φ. Therefore φR dominates any mechanism in Φ L with respect to interim stability. Furthermore, the stability of a match with respect to a mechanism in Φ L does not imply the ex post stability of that match. In other words, even if preferences are such that a stable match can be found at the interim stage, that same match need not be stable ex post. Intuitively, a distinction may arise between the two concepts, because there may be a blocking pair which, if formed, would lead to different rooms changing hands at the interim and ex post stages. Therefore ruling out that the blocking pair does not form at the interim stage does not imply that the same blocking pair does not form at the ex post stage — even after requiring that the ex post blocking pair must generate a Pareto improvement. Theorem 4. Given a match µ, when n ≥ 4 there exist preferences and a φ ∈ Φ L such that µ is stable with respect to φ but not ex post stable.

IV

Students’ Ex Ante Preferences

We next analyze students’ preferences over the two mechanisms, and provide a formal justification for why all students may prefer to participate in φR instead 10

of φL . In our model the roommate pairs in φR are formed independently of the decision about which room to select, while the priorities over rooms may directly influence the students’ roommate preferences in φL . The fact that room preferences influence roommate preferences in φL , but not φR , suggests that the pairs that form in φR might be preferable — only considering the students’ roommate preferences — to those that form under φL . However, an individual student’s preferences are heavily influenced by the priority received in φL . It is easy to construct an example in which a student prefers participating in φL to participating in φR if she receives a high priority in φL , but prefers φR if she receives a low priority in φL . There are two connected reasons. First, a high priority in φL makes it likely that she will end up in a preferred room, and second, receiving a high priority in φL makes that student more attractive to other students, making it more likely that she will be able to pair with a roommate higher up her roommate preference list. It is therefore ambiguous whether an individual student would prefer φL or φR once their preferences and random priorities have been determined. We are able to rank the mechanisms for an individual, however, if we consider a student’s preferences ex ante, whereby ex ante we mean before roommate preferences or random priorities have been determined. While this ex ante perspective might be an odd one to take for an individual student, this perspective is more appropriate for a school administrator choosing between mechanisms, particularly if there is little information available about students’ preferences. We extend our model to incorporate uncertainty over student preferences by making the roommate indices, w(s, s0 ), random variables. We assume they are determined by i.d.d. draws from an absolutely continuous distribution function G : [0, ∞) → [0, 1] with density g. In other words, roommate indices are symmetrically distributed across students. We make the stronger assumption that students are identical ex ante. In other words, f s ( x ) = x and vs (r ) = vs0 (r ) for all s, s0 ∈ S and r ∈ R. Finally, our results apply for any realization of room values, so we do not place any additional restrictions on these. With this structure in place we can be more specific about the stable matches under φR . For each realization of roommate indices, the students with the highest roommate index must be paired, then the students with the next highest of the remaining indices must be paired, and so on. The roommate index of the “first” 11

student pair is therefore the first order statistic of the i.i.d. draws from G. The φR mechanism is “greedy” because it always pairs the students with the highest index at each stage. The φL mechanism is also greedy, in the sense that it will tend to pair students with high indices, but the stable matches do not coincide between the two mechanisms. It is therefore possible that the stable match under φL does not include the pair with the highest index. It could be, for example, that one member of the pair with the highest index prefers to live with a less preferred roommate who received a high room priority and therefore has access to the best room. Therefore the most valuable pairs need not form under φL , which may harm students’ ex ante payoffs relative to φR . This is true in the following simple example. Example 2. Let there be 6 students, and let roommate indices be drawn i.i.d. from U (0, 1). Let room values be such that v(r1 ) = 1 and v(r2 ) = v(r3 ) = 0 for all students. Therefore v(r1 ) + f i (w(si , s j )) > v(rl ) + f i (w(si , sk )) for any i 6= j, k and l = 2, 3. Under φR , the stable match includes the pair with the highest roommate index of the fifteen possible pairs, and the pair with the highest roommate index of the six remaining pairs. Thus the expected total payoff is 2(15/16 + (15/16)(6/7) + (1/2)(15/16)(6/7) + 1). Under φL , the pair including the student with the highest priority and her favorite roommate is in the stable match as well as the pair with the highest roommate value among the remaining six pairs. Thus the expected total payoff is 2(3/4 + 6/7 + (1/2)(6/7) + 1). Because the students’ payoffs depend symmetrically on their roommate indices, each student expects to receive 1/6 of the total payoff in either case. Therefore each student prefers φR ex ante. The example illustrates the key insight of this section. Although after preferences and priorities are determined we generally get ambiguous preferences over the two mechanisms, from an ex ante perspective all students may prefer φR to φL . Our main result of this section, Theorem 5, generalizes this example to incorporate a broad class of distributions for the roommate indices. The construction of the example above guarantees that a stable match always exists in φL , but we know from Theorem 3 that this need not be the case. Our generalization of the example explicitly allows for the possibility that for some realizations of roommate indices no stable match exists. We do not take a stance 12

on whom a student should expect to be paired with in such a situation, and only consider realizations of roommate indices for which a stable match exists under both mechanisms. Theorem 5 also restricts attention to distribution functions that are logconcave, or G such that log( G ) is a concave function. This regularity condition is satisfied by distribution functions that are commonly used in economic models.9 For our purposes, we make use of the implication of log-concavity that x − E[Y |Y ≤ x ] is an increasing function of x. Theorem 5. If roommate indices are i.i.d. according to the log-concave distribution G, f s ( x ) = x, and vs (r ) = vs0 (r ) for all s, s0 ∈ S and r ∈ R, each student’s expected payoff from the unique stable match under φR is greater than that of any stable match under an alternative φ ∈ Φ L . To show that some distributional assumption is required for the previous result to hold, we next provide a counter example in which φL may be preferred ex ante to φR . Example 3. Let there be 6 students and suppose G puts probability 1/2 on the interval Ia = (0, ε) and probability 1/2 on the interval Ib = (1 − ε, 1) for some small ε. Let room values be v(r1 ) = 1 − 2ε and v(r2 ) = 0. Let µ∗R (resp. µ∗L ) be the match which is stable with respect to φR (φL ).10 Notice that either mechanism will always choose a match containing a roommate pair which has a value in Ib if one exists, and that if µ∗L and µ∗R contain the same number of roommate indices drawn from Ib the difference in total payoffs is approximately zero. Therefore there is only a significant difference in total payoffs between µ∗L and µ∗R if one of them has two roommate indices from Ib while the other has only one. Finally, notice that whenever the roommate pair in µ∗R which contains the student with the highest priority under φL has a value in Ia , then any match with two roommate pairs that each have a value in Ib will block µ∗R under φL . It follows that in this setting φL , relative to φR , shifts some probability away from selecting a match with one roommate value in Ia and one in Ib , and towards a match with two roommate indices in Ib , which increases the expected total payoff. Similar to Example 2, the 9 Bergstrom

and Bagnoli (2005 2005) discusses the key implications of this assumption and its use in economic models. 10 With 4 students a match which is stable with respect to φ always exists. L

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symmetry of the students’ preferences implies that each expects to receive 1/4 of the expected total payoff, and hence every student prefers φL ex ante in this example.

V

Conclusion

In a model of student housing allocation mechanisms in which students have preferences over roommates and rooms, we can only guarantee that a stable allocation exists if priority for housing is determined after a match forms, as happens in φR . We show that although there is no mechanism which is unanimously preferred by the students given their preferences, if roommate values are independently and identically distributed according to a log-concave distribution and students have the same preferences over rooms, then φR is preferred by all students ex ante over any mechanism which determines housing priority through a varying function of the roommate match.

References Abdulkadiroglu, A. and Sönmez, T. (1999). House allocation with existing tenants. Journal of Economic Theory, 88(2):233 – 260. Arkin, E. M., Bae, S. W., Efrat, A., Okamoto, K., Mitchell, J. S., and Polishchuk, V. (2009). Geometric stable roommates. Information Processing Letters, 109(4):219 – 224. Bartholdi, J. and Trick, M. A. (1986). Stable matching with preferences derived from a psychological model. Operations Research Letters, 5(4):165 – 169. Becker, G. S. (1973). A theory of marriage: Part i. Journal of Political Economy, 81:813–846. Bergstrom, T. and Bagnoli, M. (2005). Log-concave probability and its applications. Economic theory, 26:445–469.

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Boisjoly, J., Duncan, G., Kremer, M., Levy, D., and Eccles, J. (2006). Empathy or antipathy? the impact of diversity. American Economic Review, 96(5):1890–1905. Carrell, S., Fullerton, R., and West, J. (2009). Does your cohort matter? measuring peer effects in college achievement. Journal of Labor Economics, 27(3):439–464. Chung, K.-S. (2000). On the existence of stable roommate matchings. Games and Economic Behavior, 33(2):206 – 230. Dogan, O., Laffond, G., and Laine, J. (2011). The core of shapley-scarf markets with couples. Journal of Mathematical Economics, 47(1):60 – 67. Foster, G. (2006). It’s not your ppeer, and it’s not your friends: Some progress toward understaunder the educational peer effect mechanism. Journal of Public Economics, 90(8–9):1455–1475. Gale, D. and Shapley, L. S. (1962). College admissions and the stability of marriage. The American Mathematical Monthly, 69(1):pp. 9–15. Griffith, A. and Rask, K. (2014). Peer effects in higher education: A look at heterogeneous impacts. Economics of Education Review, 39(1):65–77. Hylland, A. and Zeckhauser, R. (1979). The efficient allocation of individuals to positions. Journal of Political Economy, 87(2):293 – 314. Kremer, M. and Levy, D. (2008). Peer effects and alcohol use among college students. Journal of Economic Perspectives, 22(3):189–206. Morrill, T. (2010). The roommates problem revisited. Journal of Economic Theory, 145(5):1739–1756. Pycia, M. (2012). Stability and preference alignment in matching and coalition formation. Econometrica, 80(1):pp. 323–362. Sacerdote, B. (2001). Peer effects with random assignment: Results for dartmouth roommates. Quarterly Journal of Economics, 116(2):681–704. Shapley, L. and Scarf, H. (1974). On cores and indivisibility. Journal of Mathematical Economics, 1(1):23 – 37.

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Zimmerman, D. (2003). Peer effects in academic outcomes: Evidence from a natural experiment. Review of Economics and Statistics, 85(1):9–23. Zimmerman, D., Rosenblum, D., and Hillman, P. (2004). Institutional ethos, peers and individual outcomes. Discussion Paper No. 68, Williams Project on the Economics of Higher Education.

Appendix Proof of Theorem 1. Construct match µn as follows: let µ0 = ∅ and µi = µi−1 ∪ {(s, s0 )i }, where (s, s0 )i ∈ arg max{s j ,sk }⊂S\µi−1 w(s j , sk ), with any ties broken randomly if (s, s0 ) is not unique. We claim that µn is stable with respect to φR . Under φR the expected room value of each student is independent of the match, thus any blocking pair (s, t) must be such that f s (w(s, t)) > f s (w(s, s0 )) and f t (w(s, t)) > f t (w(t, t0 )), where s0 = µn (s) and t0 = µn (t). This implies that w(s, t) > max{w(s, s0 ), w(t, t0 )}, which is clearly a contradiction. Proof of Theorem 2. Suppose that µ is stable with respect to φR , that (s1 , s2 ) ∈ µ is assigned to r1 , and that (t1 , t2 ) ∈ µ is assigned to r2 . Without loss of generality, assume vi (r1 ) ≥ vi (r2 ) for all i ∈ {s1 , s2 , t1 , t2 }. For a contradiction, assume the allocation is not ex post stable, or that f s1 (w(s1 , t1 )) + vs1 (r1 ) ≥ f s1 (w(s1 , s2 )) + vs1 (r1 ) =⇒ w(s1 , t1 ) ≥ w(s1 , s2 ) (1) f s2 (w(s2 , t2 )) + vs2 (r2 ) ≥ f s2 (w(s1 , s2 )) + vs2 (r1 )

(2)

f t1 (w(s1 , t1 )) + vt1 (r1 ) ≥ f t1 (w(t1 , t2 )) + vt1 (r2 )

(3)

f t2 (w(s2 , t2 )) + vt2 (r2 ) ≥ f t2 (w(t1 , t2 )) + vt2 (r2 ) =⇒ w(s2 , t2 ) ≥ w(t1 , t2 ), (4) where at least one inequality is strict. First, suppose vi (r1 ) = vi (r2 ) for all i ∈ {s1 , s2 , t1 , t2 }. Then all of the room values cancel and the strictness of one of the inequalities implies that a pair blocks µ under φR , a contradiction. Therefore, we must have vi (r1 ) > vi (r2 ) for all i ∈ {s1 , s2 , t1 , t2 }. Then (22) implies that w(s2 , t2 ) > w(s1 , s2 ), which in combination with (44) implies that (s2 , t2 ) is a blocking pair for µ under φR , a contradiction.

16

Proof of Theorem 3. Keeping φ ∈ Φ L fixed, let Vs (φ, µ) denote the value of the room assigned to s under φ when the match is µ. Recall that any φ ∈ Φ L selects some priority order over pairs with probability one. Combined with the assumption of common ordinal preferences this implies, that given the mechanism φ and match µ completely determine the room assigned to each pair. In the rest of the proof, to simplify notation, we may denote Vsi as Vi . Lemma 1. Suppose φ ∈ Φ L . For i 6= j, i 6= k and j 6= k, there exist four matches µ1 , µ2 , µ3 , and µ4 such that: (i) µ1 ↔ µ2 ↔ µ3 ↔ µ4 ↔ µ1 , (ii) |µ1 \ µ3 | = 3, (iii) (si , s j ) ∈ µ1 ∩ µ4 , (iv) (si , sk ) ∈ µ2 ∩ µ3 , and (v) Vi (φ, µ2 ) − Vi (φ, µ1 ) 6= Vi (φ, µ3 ) − Vi (φ, µ4 ). Proof. To generate a contradiction, assume that the statement is not true. It is straightforward to find four matches satisfying conditions (i)–(iv) in the lemma. Therefore, we assume that for any such set (v) is violated, i.e., Vi (φ, µ2 ) − Vi (φ, µ1 ) = Vi (φ, µ3 ) − Vi (φ, µ4 ). Let i = 1, j = 2 and k = 3. Take a match ν1 and three pairs {(s1 , s2 ), (s3 , s4 ), (s5 , s6 )} ⊆ ν1 . Holding all other pairs fixed, restrict attention to the subset M0 ⊆ M of fifteen matches which differ only in how these six students are paired. Consider the following arrangement of members of this set:

{(s1 , s2 ), (s3 , s4 ), (s5 , s6 )} ⊆ ν1

{(s1 , s2 ), (s3 , s6 ), (s4 , s5 )} ⊆ ν10

{(s1 , s4 ), (s2 , s3 ), (s5 , s6 )} ⊆ ν2

{(s1 , s4 ), (s3 , s6 ), (s2 , s5 )} ⊆ ν20

{(s1 , s3 ), (s2 , s4 ), (s5 , s6 )} ⊆ ν3

{(s1 , s3 ), (s2 , s5 ), (s4 , s6 )} ⊆ ν30

{(s1 , s2 ), (s3 , s4 ), (s5 , s6 )} ⊆ ν1

{(s1 , s2 ), (s3 , s5 ), (s4 , s6 )} ⊆ ν100

Note that they are arranged so that each match is adjacent to the one in the same row as well as to any match immediately above or below it. By the assumption that the lemma is false, we have the following equalities:

V1 φ, ν10 − V1 (φ, ν1 ) = V1 φ, ν20 − V1 (φ, ν2 )
= V1 φ, ν30 − V1 (φ, ν3 )
= V1 φ, ν100 − V1 (φ, ν1 )

(5) (6) (7)

This implies V1 (φ, ν10 ) = V1 (φ, ν100 ). By exchanging s4 and s6 in all of the above matches, we find that V1 (φ, ν1 ) = V1 (φ, ν100 ). Note that swapping two students 17

consistently across all matches does not change any adjacency relationships. We therefore have that for matches in M0 , s1 must receive a constant room value when paired with s2 independently of the arrangement of the other four students. By swapping s1 with si and s2 with s j for any i, j ∈ {1, . . . , 6} with i 6= j, a similar argument establishes that the pair (si , s j ) must receive the same room value for all matches in M0 . Because the original match ν1 was arbitrary, we conclude that the same is true of any such subset M0 . We show next that it is impossible to give a pair the same room value for all matches in some subset M0 , and hence that there must exist four matches satisfying the conditions laid out in the lemma. We assume in the model that there exist three rooms with room values that differ from all others. We refer to these rooms as r1 , r2 and r3 . Without loss of generality, we assign (s1 , s2 ), (s3 , s4 ) and (s5 , s6 ) to rooms r1 , r2 and r3 respectively. To derive a contradiction, assume that we can assign each pair, (si , s j ), to the same room in every match that includes (si , s j ). Consider assigning (s1 , s6 ) to r1 . We deduce that the following assignments must be made, where the column of the pair corresponds to the assigned room, r2 r3 r1 (1) (s1 , s2 ) (s3 , s4 ) (s5 , s6 ) by assumption (2) (s1 , s6 ) (s3 , s4 ) (s2 , s5 ) (s1 , s6 ) is in r1 and (s3 , s4 ) is in r2 (3) (s1 , s6 ) (s2 , s3 ) (s4 , s5 ) (s2 , s3 ) cannot be in r3 when (s5 , s6 ) is a pair (4) (s1 , s6 ) (s2 , s4 ) (s3 , s5 ) (s2 , s4 ) cannot be in r3 when (s5 , s6 ) is a pair (5) (s1 , s2 ) (s3 , s6 ) (s4 , s5 ) using (1) and (3). But this requires that we assign (s2 , s4 ) and (s3 , s6 ) to r2 in all matches, which is impossible because these two pairs appear in the same match with (s1 , s5 ). The assumption that (s1 , s6 ) is in r1 was not without loss, but a similar set of steps shows that assigning (s1 , s6 ) to r3 also creates a contradiction. Note that (s1 , s6 ) cannot occupy r2 . We conclude that it is impossible to assign a pair (si , s j ) with i 6= j to the same room for all µ ∈ M0 . Take four matches, µ1 , µ2 , µ3 , and µ4 , satisfying conditions (i)–(iv) of the lemma with i = 1, j = 2 and k = 3. Then s2 = µ1 (s1 ) and s3 = µ2 (s1 ). Let s4 = µ1 (s3 ) and s5 = µ3 (s2 ). Assume f s (w(s, s0 )) = w(s, s0 ) for all s, s0 ∈ S. Without loss of generality we may assume that V1 (φ, µ2 ) > V1 (φ, µ1 ) and V1 (φ, µ2 ) − V1 (φ, µ1 ) >

18

V1 (φ, µ3 ) − V1 (φ, µ4 ). Whenever V1 (φ, µ2 ) − V1 (φ, µ1 )

> w ( s1 , s2 ) − w ( s1 , s3 )

(8)

> V1 (φ, µ3 ) − V1 (φ, µ4 ) we have U1 (φ, µ2 ) > U1 (φ, µ1 ) but U1 (φ, µ3 ) < U1 (φ, µ4 ). Find some M such that M > max |Vi (φ, µ) − Vi (φ, µ0 )|, i,µ↔µ0

and let w(s1 , s2 ) = 3M + δ and w(s1 , s3 ) = 3M to satisfy (88). Let w(s3 , s4 ) = w(s2 , s5 ) = 2M. For all pairs (si , s j ) remaining in µ1 ∩ µ3 let w(si , s j ) = M. For any other pair let w(si , s j ) = 0. We claim that these preferences are such that there does not exist a stable match under φ. First note that any match µ in which there is some s ∈ S such that µ(s) ∈ / {µ1 (s), µ3 (s)} is not stable. Next, since µ1 and µ3 differ by exactly three pairs, they are the only two matches in which every s ∈ S is such that µ(s) ∈ {µ1 (s), µ3 (s)}. Finally, note that µ1 is blocked by µ2 , and µ3 is blocked by µ4 . Proof of Theorem 4. Suppose that φ is such that (s1 , s2 ) receives r1 , (t1 , t2 ) receives r2 , (s1 , t1 ) receives r3 , and (s2 , t2 ) receives r4 . Let µ be stable with respect to φ, and (s1 , s2 ), (t1 , t2 ) ∈ µ. Consistent with µ being stable with respect to φ, assume that w(s1 , t1 ) > w(t1 , t2 ) > w(s1 , s2 ) > w(s2 , t2 ) while f s1 (w(s1 , t1 )) + vs1 (r3 ) < f s1 (w(s1 , s2 )) + vs1 (r1 ) ≤ f s1 (w(s1 , t1 )) + vs1 (r1 ) f t1 (w(s1 , t1 )) + vt1 (r3 ) < f t1 (w(t1 , t2 )) + vt1 (r2 ) ≤ f t1 (w(s1 , t1 )) + vt1 (r1 ) f s2 (w(s2 , t2 )) + vs2 (r4 ) < f s2 (w(s1 , s2 )) + vs2 (r1 ) < f s2 (w(s2 , t2 )) + vs2 (r2 ) f t2 (w(s2 , t2 )) + vt2 (r4 ) < f t2 (w(t1 , t2 )) + vt2 (r2 ) < f t2 (w(s2 , t2 )) + vt2 (r2 ) with vi (r1 ) > vi (r2 ) > vi (r3 ) > vi (r4 ) for all i ∈ {s1 , s2 , t1 , t2 }. For example, these inequalities are satisfied if for i ∈ {s1 , s2 , t1 , t2 } and j ∈ {1, 2, 3, 4} vi (r j ) = 4 − j, f i ( x ) = x, w(s1 , s2 ) = w(s1 , t1 ) = 1, w(t1 , t2 ) = 2, and w(s2 , t2 ) = 3. They also provide a case where the match is stable with respect to φ yet not ex post stable. To complete the example, for all s, s0 ∈ S \ {s1 , s2 , t1 , t2 } and k ∈ / {1, 2, 3, 4} set 19

vs (rk ) = vs0 (rk ) = 0, f s ( x ) = f s0 ( x ) = x, w(s, s0 ) = 10 if (s, s0 ) ∈ µ and w(s, s0 ) = 0 if (s, s0 ) ∈ / µ. Proof of Theorem 5. Let the roommate index for the pair (si , s j ) be determined by the random variable Xij ∼ G. Let x be a realization of roommate indices corresponding to a stable match under φ and let y the realized roommate indices of an adjacent match that blocks x under φR . There need not be any match that blocks x under φR , but in this event there is no difference in roommate indices between the mechanisms. Because they are adjacent, all but two elements each of x and y are identical. Let the four elements that differ with positive probability be y1 , y2 , x1 , and x2 . Denote the value of the rooms received by the pairs corresponding to x1 and x2 by v x1 and v x2 . For y1 and y2 , we need the values of the rooms received under φ if these pairs form, and we denote those by vy1 and vy2 respectively. At this point we can only rule out that v x1 and v x2 (or vy1 and vy2 ) refer to the same room. The inequalities that characterize the event, E, that y blocks x under φR but x blocks under φ are x 1 ≤ y 1 ≤ x 1 + ( v x1 − v y1 ) x2 ≤ min{y1 , x1 + (v x1 − v x2 )}

(9)

y2 ≤ min{y1 , x1 + (v x1 − vy2 )}.

(10)

These follow from observing that it must be that max{y1 , y2 } ≥ max{ x1 , x2 } and max{y1 + vy1 , y2 + vy2 } ≤ max{ x1 + v x1 , x2 + v x2 }, and assuming without loss of generality that x1 + v x1 ≥ x2 + v x2 and y1 ≥ y2 . Note that it must be that v x1 > vy1 for this event to have non-zero probability. We first sign the difference in the sum of the payoffs to all of the students when the match is y compared to when it is x. Since the students have the same values for each of the rooms and x and y differ for only four pairs, this difference is simply ∆ = 2( y1 − x1 ) + 2( y2 − x2 ). The difference y1 − x1 is positive in these events but the sign on y2 − x2 is ambiguous ex post. To sign ∆, we condition on E and ( x1 , y1 , v x1 , vy1 ), taking expectations 20

over all x2 , y2 , v x2 and vy2 in E.11 Letting V \ {vi } be the set of room values not including vi , we find that the value of E[∆| E, x1 , y1 , v x1 , vy1 ] is 2( y1 − x1 ) +

2 n − 1 v ∈V∑ \{v j

−

2 n − 1 v ∈V∑ \{v j

= 2( y1 − x1 ) +

E[Y2 |Y2 ≤ min{y1 , x1 + v x1 − v j }] y1 }

E[ X2 | X2 ≤ min{y1 , x1 + v x1 − v j }] x1 }

2 E[Y2 |Y2 ≤ min{y1 , x1 }] n−1

2 E[ X2 | X2 ≤ min{y1 , x1 + v x1 − vy1 }] n−1 2 = 2( y1 − x1 ) + (E[Y2 |Y2 ≤ x1 ] − E[ X2 | X2 ≤ y1 ]) > 0. n−1

−

The first equality follows from canceling terms across the sums, and the inequality follows from the assumption of log-concavity, which guarantees that y − E[Y |Y < y] is a non-decreasing function of y. This shows that when the stable match under φ is blocked under φR by an adjacent match the total payoff improves on average. If there is no such blocking pair, there is no change in the total payoff. Since students are identical ex ante, each student receives the same fraction of the total expected payoff from a mechanism and hence each student’s expected payoff improves with the formation of such a blocking pair. We next show that forming subsequent blocking pairs under φR improves the total expected payoff and hence each individual’s expected payoff. From Theorem 1, the stable match under φR can be constructed by successively finding the pair with the highest roommate value among all remaining feasible pairs. Let the φR -stable match’s roommate indices be given by z, which may refer to the same match as y. Comparing some z to y, sort the vectors in descending order so that z1 is the largest element of z. If z and y do not refer to the same match, with positive probability there is some i such that either i = 1 and z1 > y1 , or i > 1 and ∀ j < i, z j = y j and zi > yi . That is, they differ at some point in the process of selecting the best pair from the remaining feasible pairs. The pair corresponding 11 Conditional

on x1 being assigned the room corresponding to v x1 , the anonymity of the mechanism implies an equal probability that x2 will be assigned any of the remaining rooms, and similarly for y1 and y2 respectively.

21

to zi blocks y under φR and hence there is a match y0 incorporating this blocking pair and the pair that results from this block, labeled y0j . Conditional on such a blocking pair forming, the expected difference in the total payoff between y0 and y is zi − yi + y0j − y j . Conditioning on zi and yi , this difference is positive, since zi − yi + E[Yj0 |Yj0 ≤ zi ] − E[Yj |Yj ≤ yi ] ≥ 0, because E[ X | X ≤ ·] is an increasing function. If y0 6= z, then one can form a y00 adjacent to y0 in the same manner. This process can be continued until z is reached. At each step, there is a nonnegative change in the expected total payoff, and hence each student’s expected payoff improves ex ante.

22