Designing for Diversity: Matching with Slot-Specific Priorities

Transcription

1 Designing for Diversity: Matching with Slot-Specific Priorities Scott Duke Kominers Becker Friedman Institute University of Chicago Tayfun Sönmez Department of Economics Boston College July 2, 2012 Astract To encourage diversity, ranches may vary contracts priorities across slots. The agents who match to ranches, however, have preferences only over match partners and contractual terms. Ad hoc approaches to resolving agents indifferences across slots in the Chicago and Boston school choice programs have introduced iases, which can e corrected with more careful market design. Slot-specific priorities can fail the sustitutaility condition typically crucial for outcome staility. Nevertheless, an emedding into a one-to-one agent slot matching market shows that stale outcomes exist and can e found y a cumulative offer mechanism that is strategy-proof and respects unamiguous improvements in priority. JEL classification: C78, D63, D78. Keywords: Market Design, Matching with Contracts, Staility, Strategy-Proofness, School Choice, Affirmative Action. The authors are deeply grateful to Parag Pathak for his comments and for providing access to data from the Chicago pulic school choice program. They appreciate the helpful suggestions of Eric Budish, John William Hatfield, Ellen Kominers, Sonia Jaffe, Jonathan Levin, Alexandru Nichifor, Philip J. Reny, Assaf Romm, Alvin E. Roth, Alexander Westkamp, seminar audiences at Chicago, Haifa, and St. Andrews, and memers of the Measuring and Interpreting Inequality Working Group. Kominers gratefully acknowledges the support of an AMS Simons travel grant and the Human Capital and Economic Opportunity Working Group sponsored y the Institute for New Economic Thinking.

2 1 Introduction The deferred acceptance algorithm of Gale and Shapley (1962) has een applied in a variety of real-world matching prolems, stailizing laor markets and school choice programs, and contriuting significantly to the emergence and success of the field of market design. Until the last decade, deferred acceptance was primarily applied in two-sided, many-to-one matching markets such as the United States doctor residency match (Roth (1984); Roth and Peranson (1999)) in which oth sides of the market may ehave strategically. This trend has changed since the introduction of the student placement (Balinski and Sönmez (1999)) and school choice (Adulkadiroğlu and Sönmez (2003)) models, which, following school matching traditions in the United States and elsewhere, focus on settings in which seats at institutions are simply a set of ojects to e allocated to agents and institution has an exogenous priority ranking over students which determines claims over seats. As Duins and Freedman (1981) and Roth (1982) showed, truth-telling is a dominant strategy for all agents when agent-proposing deferred acceptance is run. Meanwhile, there exists no stale mechanism that is dominant-strategy incentive compatile for institutions (Roth (1982)). Market designers have thus advocated the use of agent-proposing deferred acceptance, even though use of this mechanism may conflict with the interests of institutions. 1 The case for agent-proposing deferred acceptance is in a sense stronger for priority matching applications than it is for two-sided laor markets, as in such settings agent-proposing deferred acceptance is dominant-strategy incentive compatile for all strategic players, i.e. it is strategy-proof. 2 Staility and strategy-proofness are attractive in priority matching: the former has a natural normative interpretation as elimination of justified envy, and the latter 1 In the framework of Gale and Shapley (1962), the proposing side of the market is significantly advantaged: the deferred acceptance algorithm produces an outcome which the proposing side of the market weakly prefers to all other stale outcomes; this outcome is also pessimal among stale outcomes for the non-proposing side of the market. 2 As efore, conflict of interest etween agents and institutions arises, ut there is often less concern for institution welfare in priority matching contexts. Institution welfare is not irrelevant, however, as it can affect roader policy goals: Hatfield et al. (2012) show, for example, how welfare effects of school choice mechanism selection can impact schools incentives for self-improvement. 1

3 levels the playing field y eliminating gains to strategic sophistication and enales the collection of true preference data for planning purposes (Adulkadiroğlu et al. (2006); Pathak and Sönmez (2008)). Because of these advantages, a large numer of school districts have recently adopted centralized school choice programs ased on the agent-proposing deferred acceptance. 3 Many of these school districts (e.g., Boston, New York City, and Chicago) are concerned with issues of student diversity and have therefore emedded affirmative action systems into their school choice programs. At present, however, these diversity efforts are handled in a somewhat ad hoc manner. In this paper, we oserve that diversity, financial aid, or other concerns might cause priorities to vary across slots at a given institution. We argue that to effectively handle these slot-specific priority structures, market designers should go eyond the traditional deferred acceptance algorithm and use a more detailed design approach ased on the Kelso and Crawford (1982)/Hatfield and Milgrom (2005) theory of many-to-one matching with contracts. To make this case, we first illustrate how existing ad hoc deferred acceptance implementations impact student welfare in the Boston and Chicago school choice programs. Then, we introduce a model of matching with slot-specific priorities, which emeds classical priority matching frameworks (e.g., Balinski and Sönmez (1999); Adulkadiroğlu and Sönmez (2003)), models of affirmative action (e.g., Kojima (2012); Hafalir et al. (forthcoming)), and the cadet ranch matching framework (Sönmez and Switzer (2011); Sönmez (2011)). We advocate for a specific implementation of the cumulative offer mechanism of Hatfield and Milgrom (2005) and Hatfield and Kojima (2010), which generalizes agent-proposing deferred acceptance. Previous priority matching models have relied on the existence of agentoptimal stale outcomes to guarantee that this mechanism is strategy-proof. In markets 3 These school choice reforms include assignment of high school students in New York City in 2003 Adulkadiroğlu et al. (2005, 2009), assignment of K 12 students to pulic schools in Boston in 2005 Adulkadiroğlu et al. (2005a), assignment of high school students to selective enrollment schools in Chicago in 2009 Pathak and Sönmez (forthcoming), and assignment of K 12 students to pulic schools in Denver in Perhaps most significantly, a version of the agent-proposing deferred acceptance has een recently een adopted y all (more than 150) local authorities in England (Pathak and Sönmez (forthcoming)). 2

4 with slot-specific priorities, however, agent-optimal stale outcomes may not exist. Nevertheless, as we show, the cumulative offer mechanism is still strategy-proof in such markets; this oservation is perhaps the most important theoretical contriution of our work. We show moreover that the cumulative offer mechanism has two other features essential for applications: the cumulative offer mechanism yields stale outcomes and respects unamiguous improvements of agent priority. 4 Our work demonstrates that the existence of a plausile mechanism for real-world manyto-one matching with contracts does not rely on the existence of an agent-optimal stale outcome. However, it is worth emphasizing that the cumulative offer mechanism selects the agent-optimal stale outcome whenever such an outcome exists. This is important ecause the existence of an agent-optimal stale outcome can e considered quite relevant for applications ecause it removes all conflict of interest among agents in the context of stale assignment. The existence of an agent-optimal stale outcome in our general model may depend on several factors, including the numer of different contractual arrangements each agent can have with each institution and the precedence order according to which institutions prioritize individual slots aove others. 5 Finally, we note that our paper also has a methodological contriution: In general, slotspecific priorities fail the sustitutaility condition that has so far een key in analysis of most two-sided matching with contracts models (Kelso and Crawford (1982); Hatfield and Milgrom (2005); see also Adachi (2000); Fleiner (2003); Echenique and Oviedo (2004)). 6 Moreover, slot-specific priorities may fail the unilateral sustitutaility condition of Hatfield and Kojima (2010) that has een central to the analysis of cadet ranch matching (Sönmez and Switzer (2011); Sönmez (2011)). Nevertheless, the priority structure in our model gives rise to 4 These conclusions allow us to re-derive several main results of the innovative Hafalir et al. (forthcoming) approach to welfare-enhancing affirmative-action. 5 As we show in an application of our model (Proposition 5), the United States Military Academy cadet ranch matching mechanism in a sense uses the unique precedence order under which the cumulative offer mechanism is agent-optimal. 6 Thus, in particular, our model falls outside of the domain which Echenique (2012) has shown can e handled with only the Kelso and Crawford (1982) matching with salaries framework (see also Kominers (2012)). 3

5 a naturally associated one-to-one model of agent slot matching (with contracts). As the agent slot matching market is one-to-one, it trivially satisfies the sustitutaility condition. It follows that the set of outcomes stale in the agent slot market (called slot-stale outcomes to avoid confusion) has an agent-optimal element. We show that each slot-stale outcome corresponds to a stale outcome 7 ; moreover, we show that the cumulative offer mechanism in the true matching market gives the outcome which corresponds to the agent-optimal slot-stale outcome in the agent slot matching market. We utilize these relationships heavily in proving our main results. The remainder of this paper is organized as follows. In Section 2, we discuss the Chicago and Boston school choice programs, illustrating the importance of careful design in markets with slot-specific priorities. We present our model of matching with slot-specific priorities in Section 3. Then, in Section 4, we introduce the agent slot matching market and derive key properties of the cumulative offer mechanism. In Section 5, we revisit the affirmativeaction applications and riefly discuss applications to cadet ranch matching. Most proofs are contained in the Appendix. 2 Motivating Applications First advocated y Friedman (1955, 1962), school choice programs aim to enale parents to choose which schools their children attend. There is significant tension etween the proponents of school choice and the proponents of alternative, neighorhood assignment systems ased on students home addresses. The mechanics of producing the assignment of students to school seats received little attention in school choice deates until Adulkadiroğlu and Sönmez (2003) showed important shortcomings of several mechanisms adopted y United States school districts. Of particular concern was the vulneraility of school choice to preference manipulation: while parents were 7 The converse result is not true, in general there may e stale outcomes which are not associated to slot-stale outcomes. 4

6 allowed to express their preferences on paper, they were implicitly forced to play sophisticated admission games. Once this flaw ecame clear, several school districts adopted the (studentproposing) deferred acceptance mechanism, which was invented y Gale and Shapley (1962) and proposed as a school choice mechanism y Adulkadiroğlu and Sönmez (2003). 8 Deferred acceptance-ased mechanisms have een successful in part ecause they are fully flexile regarding the choice of student priorities at schools. In particular, priority rankings may vary across schools; hence, students can e given the option of school choice while retaining some priority for their neighorhood schools. Thus, deferred acceptance mechanisms provide a natural opportunity for policymakers to alance the concerns of oth school choice and neighorhood assignment advocates. However, deferred acceptance-style mechanisms are designed under the assumption that student priorities are identical across a given school s seats. While this assumption is natural for some school choice applications, it fails in several important cases: admissions to selective high schools in Chicago; K 12 school admissions in Boston; and pulic high school admission in New York. We next descrie how the Chicago and Boston matching prolems differ from the original school choice model of Adulkadiroğlu and Sönmez (2003), how policymakers in Chicago and Boston sought to transform their prolems into direct applications of the Adulkadiroğlu and Sönmez (2003) framework, and how these transformations introduced significant, yet hidden iases in the underlying priority structures. These oservations motivate the more general matching model with slot-specific priorities that we introduce in Section 3. 8 This mechanism is often called the student-optimal stale matching mechanism. 5

7 2.1 Affirmative Action at Chicago s Selective High Schools Since 2009, Chicago Pulic Schools (CPS) has adopted an affirmative action plan ased on socio-economic status (SES). Although CPS initially adopted a version of the Boston mechanism for selective enrollment high school admissions, they immediately aandoned it in favor of a deferred acceptance-ased approach. 9 Under the new assignment plan, the SES of each student is determined ased on home address; students are then divided into four roughly evenly sized tiers: 10 In 2009, 135,716 students living in 210 Tier 1 (lowest-ses) tracts had a median family income of $30,791, 136,073 students living in 203 Tier 2 tracts had a median family income of $41,038, 136,378 students living in 226 Tier 3 tracts had a median family income of $54,232, and 136,275 students living in 235 Tier 4 (highest-ses) tracts had a median family income of $76,829. Students in Chicago who apply to selective enrollment high schools take an admissions test as part of their application, and this test is used to determine a composite score. 11 Students from high-ses tiers typically have higher composite scores, and CPS has set aside seats as reserved for low-ses students in order to prevent the elite schools from ecoming accessile only to children from wealthy neighorhoods. In order to implement this ojective, CPS adopted the following priority structure at each of the nine selective enrollment high schools: priority for 40% of the seats is determined y students composite scores, while 9 Pathak and Sönmez (forthcoming) present a detailed account of this midstream reform. 10 SES scores are uniform across census tracts, and are ased on median family income, average adult educational attainment, percentage of single-parent households, percentage of owner-occupied homes, and percentage of non-english speakers. 11 If two students have the same score, then the younger student is coded y CPS as having a higher composite score. 6

8 15% of the seats are set aside for students in each of the four SES tiers t, with composite score determining relative priority among students in Tier t. 12 Because these priorities over seats are not uniform within schools, the Adulkadiroğlu and Sönmez (2003) school choice model does not fully capture all aspects of the Chicago admissions prolem. In order to implement deferred acceptance despite this difficulty, the CPS matching algorithm treats each selective enrollment high school as five hypothetical schools: The set S of seats at each school is partitioned into susets S = S o S 4 S 3 S 2 S 1. The seats in S o are open seats, for which students priorities are determined entirely y composite scores. Seats in S t are reserved for students of Tier t they give Tier t students priority over other students, and use composite scores to rank students within Tier t. Each set of seats is viewed as a separate school within the CPS algorithm. Because seat priorities are uniform within each set S e, the set of hypothetical schools satisfies the Adulkadiroğlu and Sönmez (2003) requirement of uniform within-school priorities. However, CPS students i sumit preferences P i over schools, and are indifferent among seats at a given school B. That is, if we denote a contract representing that i holds a seat in S e y i; se, then any Tier t student i is indifferent etween i; s o, i; s 4, i; s 3, i; s 2, and i; s 1, and prefers all these these contracts to any contract of the form i; s e if (and only if) i prefers school to school (i.e. P i ). As the Adulkadiroğlu and Sönmez (2003) model requires that students have strict preferences over schools, the CPS matching algorithm must 12 The priority structure we descrie here was used in the CPS match. The CPS priority structure has een revised slightly for school year Then, 5% of the seats are reserved for a second round to e hand picked y principals, the fraction of open competition seats is reduced to 28.5%, and the fraction of reserved seats is increased to % for each SES tier. 7

9 convert students true preferences P i into strict, extended preferences P i over the full set of hypothetical schools. In practice, CPS does this y assuming that i; s o P i i; s 4 P i i; s 3 P i i; s 2 P i i; s 1 for each student i and school. 13 That is, CPS assumes that students most prefer open seats, and then rank reserved seats. CPS thus picks for each student the (unique) preference ranking consistent with the student s sumitted preferences in which the open seats at each school are ranked immediately aove the reserved seats. With this transformation of preferences, the Chicago prolem finally fits within the standard school choice framework of Adulkadiroğlu and Sönmez (2003). Being an affirmative action plan, the priority structure in Chicago is designed so that students of lower SES tiers receive favorale treatment. What may e less clear is that the transformation used in the CPS implementation of deferred acceptance provides additional advantages to lower SES tier students. To understand this ias, we consider a simple example in which there is only one school. The CPS matching algorithm first assigns the open seats and susequently assigns the reserved seats. Therefore (under the CPS plan), 40% of the seats are assigned to the students from any SES tier with highest composite scores, and the remaining 60% of seats are shared evenly among the four SES tiers. Hence, students in each tier have access to 15% of the seats plus some fraction of the open seats depending on their composite scores. In order to see how this treatment favors for lower tiers, we consider an alternative mechanism in which reserved seats are allocated efore open competition seats. Under this counterfactual mechanism: First, the reserved seats are allocated to the students in each SES tier with the highest composite scores. Then, the students remaining unassigned are ranked 13 Note that as seats in S t are reserved for Tier t students, the only relevant part of this construction is the fact that i; s o P i i; s t for any Tier t student i. 8

10 Tier 4 Tiers 4 and 3 Tiers 4, 3, and 2 Tiers 4, 3, 2, and 1 Figure 1: Top-scoring students in the truncated distriution come from Tier 4; the next-highest score lock consists only of students from Tiers 4 and 3, and so forth. according to their composite scores and admitted in descending order of score until all the open seats are filled. High-SES students composite scores dominate low-ses students scores throughout the relevant part of the score distriution the numer of Tier 4 students with score σ high enough to gain admission is larger than the numer of Tier 3 students with score σ, and so forth. In practice, this means that after the reserved seats are filled, high-ses students fill most (if not all) of the open seats. Indeed, once the highest-scoring students in each tier are removed, the score distriution of students vying for the last 40% of seats takes the lock form illustrated in Figure 1. Thus, the open seats first fill only with Tier 4 students, then fill with oth Tiers 4 and 3, and then fill with students from Tiers 4, 3, and 2. Only after that (if seats remain) do Tier 1 students gain access. Of course, the size of the locks and hence the size of the effect of switching to the counterfactual mechanism is an empirical question. Using actual data from Chicago school choice admissions program, we now show that our intuition is accurate and that the magnitude of this effect is sustantial. We compare the outcome of the Chicago match under two alternative scenarios: 1. First we consider the current system, in which all open seats are allocated efore reserved seats. 2. Second we consider the counterfactual in which all open seats are allocated after reserved seats. 9

11 Tier 4 Tier 3 Tier 2 Tier 1 Prefer Actual Indifferent Prefer Counterfactual Tale 1: Comparison of individual student outcomes. Of the 4270 seats allocated at nine selective high schools, 771 of them change hands etween the two scenarios what CPS most likely considers a minor coding decision in fact impacts 18% of the seats at Chicago s selective high schools. Tale 1 shows that Tier 1 students unamiguously prefer the current mechanism; this confirms that the current implementation particularly enefits low-tier students. The effect is not uniform for all memers of higher SES tiers. Nevertheless, it is consistent with our simple example in terms of aggregate distriution: Of the 151 Tier 2 students who are affected, 108 prefer the current treatment and 43 prefer the counterfactual. Of the 158 Tier 3 students who are affected, 50 prefer the current treatment and 108 prefer the counterfactual. And of the 287 Tier 4 students who are affected, 62 prefer the current treatment and 225 prefer the counterfactual. Tale 2 shows how the composition of admitted classes varies etween the current and counterfactual mechanisms. We oserve that Tier 1 students enefit from the current implementation at the expense of students in all three higher tiers. Perhaps surprisingly, most of the enefit appears to e at the expense of Tier 3 students. 14 Finally, Tale 3 compares the numer of students receiving seats under the counterfactual mechanism to the sizes of schools reserved seat locks. We see that switching to the counterfactual mechanism can in effect convert reserves into quotas: Consistent with the intuition of our informal, one-school example, Tier 1 students receive no open seats under the 14 One reason for this might e the availaility of high quality outside options for some memers of Tier 4. 10

12 Current Mechanism Effect of Switching (Open Slots First) (to fill Open Slots Last) Tier 4 Tier 3 Tier 2 Tier 1 Tier 4 Tier 3 Tier 2 Tier TOT Tale 2: Effect of mechanism change on student ody composition at each of the nine selective high schools. Counterfactual Mechanism (Open Slots Last) Tier 4 Tier 3 Tier 2 Tier 1 Reserve/Tier Tale 3: Seat allocations under the counterfactual mechanism, in comparison to the 15% reserve. counterfactual mechanism at seven of the nine selective high schools. And again consistent with our example, we see that in four of the nine schools, even Tier 2 students receive no open seats under the counterfactual. 2.2 K 12 Admissions in Boston Pulic Schools In the Boston school choice program, the priority of a student for a given school depends on 1. whether the student has a siling at that school (siling priority), 2. whether the student lives within the (ojectively determined) walk-zone of that school (walk-zone priority), and 3. a random numer used to reak ties. 11

13 Students with walk-zone priority alone have higher claims for 50% of the seats at their neighorhood schools, ut have no priority advantage at other seats. This choice of priority structure in Boston is not aritrary it represents a delicate alance etween the interests of the school choice and neighorhood assignment advocates. As in the case of Chicago, Boston school choice priorities are not uniform across schools seats. And similarly to the Chicago mechanism, the BPS school choice algorithm treats each school as two hypothetical schools (of half the true capacity), one with walk-zone priority and one without. To convert student preferences over true schools into preferences over hypothetical schools, BPS chooses the (unique) ranking consistent with the ranking of the original schools such that seats with walk-zone priority are ranked aove seats without walk-zone priority. Because of this implementation decision, the BPS school choice program in fact systematically favors proponents of school choice over proponents of neighorhood assignment. 15 This oservation ecomes more interesting in light of the recent January 17, 2012 statement y Boston Mayor Thomas M. Menino: I m committing tonight that one year from now Boston will have adopted a radically different student assignment plan, one that puts a priority on children attending schools closer to their homes. 16 The current BPS mechanism design appears to e in conflict with Menino s statement. This conflict can e corrected y reversing the assumed order of the two types of seats, or y adopting a more alanced implementation ased on the model we develop in this paper. 2.3 Discussion The current Chicago school choice mechanism s treatment of low-ses students might well e consistent with the policy ojectives of CPS. However, it is not clear that the current BPS 15 Intuition for this fact is much the same as for the case of Chicago: Under the current system, walk-zone students with high enough priority to receive non-walk-zone seats are systematically awarded walk-zone seats, effectively wasting their high priority draws. 16 See transcript at 1 oston-police-officers-cityspeech-unemployment-rate. 12

14 school choice mechanism is consistent with Mayor Menino s stated goal of increasing the emphasis on neighorhood assignment. In any event, it is clear that ad hoc implementations of deferred acceptance have introduced implicit iases in oth the Chicago and Boston school choice programs. The general model we provide in the remainder of this paper encapsulates the Chicago and Boston school choice settings, and provides machinery which can virtually eliminate the iases we have illustrated. 3 Basic Model 3.1 Agents, Branches, Contracts, and Slots A matching prolem with slot-specific priorities is comprised of a set of agents I, a set of ranches B, and a (finite) set of contracts X. 17 Each contract x X is etween an agent i(x) I and ranch (x) B 18 ; we extend the notations i( ) and ( ) to sets of contracts y setting i(y ) y Y {i(y)} and (Y ) y Y {(y)}. Each agent i I has a (linear) preference order P i (with weak order R i ) over contracts in X i {x X : i(x) = i}. For ease of notation, we assume that each i also ranks a null contract i which represents remaining unmatched (and hence is always availale), so that we may assume that i ranks all the contracts in X i. We say that the contracts x X for which i P i x are unacceptale to i. Each ranch B has a set S of slots which can e assigned contracts in X {x X : (x) = }. Each slot s S has a (linear) priority order Π s (with weak order Γ s ) over contracts in X ; for convenience, we use the convention that X s X for s S. As with agents, we assume that each slot s ranks a null contract s which represents remaining 17 Here, a ranch could represent a ranch of the military (as in cadet ranch matching), or a school (as in the Chicago and Boston examples). 18 A contract may have additional terms in addition to an agent and a ranch. For concreteness, X may e considered a suset of I B T for some set T of potential contract terms. 13

15 unassigned. We set S B S. 3.2 Choice and the Order of Precedence We assume that for each B, the slots in S are ordered according to a (linear) order of precedence. We denote S {s 1,..., sq } with q S and the understanding that s l s l+1 unless otherwise noted. The interpretation of, defined formally in the choice function construction elow, is that if s l s l s l efore filling sl. then whenever possile ranch fills slot To simplify our exposition and notation in the sequel, we treat linear orders over contracts as interchangeale with orders over singleton contract sets. We denote y max P iy the P i -maximal element of Y i, using the convention that max P iy = i if i P i y for all y Y i ; similarly, we denote y max ΠsY the Π s -maximal element of Y s, using the convention that max ΠsY = s if s Πs y for all y Y s. 19 Agents have unit demand, that is, they choose at most one contract from a set of contract offers. We assume also that agents always choose the maximal availale contract, so that the choice C i (Y ) of an agent i I from contract set Y X is defined y C i (Y ) max P iy. Meanwhile, ranches B may e assigned as many as q contracts from an offer set Y X one for each slot in S ut may hold no more than one contract with a given agent. Slots at ranch are filled in the order of precedence : First, slot s 1 is assigned the contract x1 which is Π s1 -maximal among contracts in Y. Then, slot s 2 is assigned the contract x2 which is Π s2 -maximal among contracts in the set Y \ Y i(x 1 ) of contracts in Y with agents other than i(x 1 ). 19 Here, we use the notations P i and Π s ecause we will sometimes need to maximize over orders other than P i and Π s. 14

16 This process continues, with each slot s l eing assigned the contract xl which is Π sl - maximal among contracts in the set Y \ Y i({x 1,...,x }). l 1 Formally, the choice (set) C (Y ) of a ranch B from Y is defined y the following algorithm: 1. Let H 0, and let V 1 Y. 2. For each l = 1,..., q : (a) Let x l max Π s l V l e the Πsl -maximal contract in V l. () Set H l = Hl 1 {x l } and set V l+1 = V l \ Y i(x ). l 3. Set C (Y ) = H q. We say that a contract x Y is assigned to slot s l S in the computation of C (Y ) if {x} = H l \ Hl 1 in the running of the algorithm defining C (Y ). If no contract x Y is assigned to slot s l S in the computation of C (Y ), then we say that s l null contract s l. is assigned the 3.3 Staility An outcome is a set of contracts Y X. We follow the Gale and Shapley (1962) tradition in focusing on match outcomes which are stale in the sense that neither agents nor ranches wish to unilaterally walk away from their assignments, and agents and ranches cannot enefit y recontracting outside of the match. Formally, we say that an outcome Y is stale if it is 1. individually rational C i (Y ) = Y i for all i I and C (Y ) = Y for all B and 2. unlocked there does not exist a ranch B and locking set Z C (Y ) such that Z = C (Y Z) and Z i = C i (Y Z) for all i i(z). 15

17 3.4 Conditions on the Structure of Branch Choice Sustitutaility Conditions Definition. A choice function C is sustitutale if for all z, z X and Y X, z / C (Y {z}) = z / C (Y {z, z }). Hatfield and Milgrom (2005) introduced this sustitutaility condition, which generalizes the earlier gross sustitutes condition of Kelso and Crawford (1982). Hatfield and Milgrom (2005) also showed that sustitutaility is sufficient to guarantee the existence of stale outcomes. 20 Choice function sustitutaility is necessary for the guaranteed existence of stale outcomes in a variety of settings, including many-to-many matching with contracts (Hatfield and Kominers (2010)) and the Ostrovsky (2008) supply chain matching framework (Hatfield and Kominers (2012)). However, sustitutaility is not necessary for the guaranteed existence of stale outcomes in settings where agents have unit demand (Hatfield and Kojima (2008, 2010)). Indeed, as Hatfield and Kojima (2010) showed, the following condition weaker than sustitutaility suffices not only for the existence of stale outcomes, ut also to guarantee that there is no conflict of interest among agents. 21 Definition. A choice function C is unilaterally sustitutale if z / C (Y {z}) = z / C (Y {z, z }) 20 Their analysis of Hatfield and Milgrom (2005) implicitly assumes irrelevance of rejected contracts, the requirement that z / C (Y {z}) = C (Y ) = C (Y {z}) for all B, Y X, and z X \ Y (Aygün and Sönmez (2012)). This condition is naturally satisfied in most economic environments including ours. (The fact that all ranch choice functions in our setting satisfy the irrelevance of rejected contracts condition is immediate from the algorithm defining ranch choice see Lemma D.1.) 21 As in the work of Hatfield and Milgrom (2005), an irrelevance of rejected contracts condition (which is naturally satisfied in our setting see Footnote 20) is implicitly assumed throughout the work of Hatfield and Kojima (2010) (Aygün and Sönmez (2012a)). 16

18 for all z, z X and Y X for which i(z) / i(y ) (i.e. no contract in Y is associated to agent i(z)). Unilateral sustitutaility is a powerful condition, and has een applied in the study of cadet ranch matching mechanisms (Sönmez and Switzer (2011); Sönmez (2011)). Although cadet ranch matching arises as a special case of our framework, the choice functions C which arise in markets with slot-specific priorities are not unilaterally sustitutale, in general. Our next example illustrates this fact; this also shows (a fortiori) that the ranch choice functions in our framework may e non-sustitutale. Example 1. Let X = {i 0, i 1, j 1 }, with B = {}, I = {i, j}, i(i 0 ) = i = i(i 1 ) and i(j 1 ) = j. 22 If has two slots, s 1 s 2, with priorities given y Π s1 : i0 s 1, Π s2 : i1 j 1 s 2, then C fails the unilateral sustitutes condition: j 1 / C ({i 1, j 1 }), ut j 1 C ({i 0, i 1, j 1 }). Nevertheless, the choice functions C do ehave sustitutaly whenever each agent offers at most one contract to. Definition. A choice function C is sustitutale on no-negotiation contract sets if z / C (Y {z}) = z / C (Y {z, z }) for any z, z X and Y X such that Y {z, z } = i(y {z, z }). (1) 22 Clearly (in order for ( ) to e well-defined), we must have (i 0 ) = (i 1 ) = (j 1 ) =, as B = 1. 17

19 Proposition 1. Every ranch choice function C is sustitutale on no-negotiation contract sets. Also, the choice functions C satisfy the weakest form of the sustitutaility condition introduced y Hatfield and Kojima (2010). Definition. A choice function C is ilaterally sustitutale if z / C (Y {z}) = z / C (Y {z, z }) for all z, z X and Y X with i(z), i(z ) / i(y ). Proposition 2. Every choice function C is ilaterally sustitutale. In addition to illustrating some of the structure underlying the choice functions induced y slot-specific priorities, Proposition 1 is also useful in our proofs. Meanwhile, Proposition 2 is not used directly in the sequel as with Example 1, we present Proposition 2 only to illustrate the relationship etween our work and the conditions introduced in the prior literature The Law of Aggregate Demand A numer of structural results in two-sided matching theory rely on the following monotonicity condition introduced y Hatfield and Milgrom (2005). 24 Definition. A choice function C satisfies the Law of Aggregate Demand if Y Y = C (Y ) C (Y ). 23 Comining Proposition 2 with Theorem 1 of Hatfield and Kojima (2010) can e used to prove the existence of stale outcomes in our setting, although (as oserved in Footnote 21) this logic implicitly requires an irrelevance of rejected contracts condition which ranch choice functions satisfy (Aygün and Sönmez (2012a)). However, as Hatfield and Kojima (2010) pointed out, the ilateral sustitutes condition is not sufficient for the other key results necessary for matching market design, such as the existence of strategyproof matching mechanisms. To otain these additional results in our framework, we draw upon structures present in our specific model (see Section 4.1); these structures give rise to a self-contained existence proof, which does not make use of the ilateral sustitutaility condition. 24 Alkan (2002) and Alkan and Gale (2003) introduced a related cardinal monotonocity condition. 18

20 Unfortunately, like the sustitutaility and unilateral sustitutaility conditions, the ranch choice functions in our framework may fail to satisfy the law of aggregate demand. Example 2. Let X = {i 0, i 1, j 0 }, with B = {}, I = {i, j}, i(i 0 ) = i = i(i 1 ) and i(j 0 ) = j. 25 If has two slots, s 1 s 2, with priorities given y Π s1 : i0 j 0 s 1, Π s2 : i1 s 2, then C does not satisfy the law aggregate demand: C ({i 1, j 0 }) = {i 1, j 0 } = 2 > 1 = {i 0 } = C ({i 0, i 1, j 0 }). 4 Basic Theory 4.1 Associated Agent Slot Matching Market We now associate our original market to a (one-to-one) agent slot matching market, in which, slots, rather than ranches, compete for contracts. The contract set X is extended to the set X defined y X { x; s : x X and s S (x) }. Slot priorities Π s over contracts in X exactly correspond to the priorities Π s over contracts in X: x; s Π s x ; s xπ s x ; s Πs x; s [ s Π s x or s s]. 25 Clearly (in order for ( ) to e well-defined), we must have (i 0 ) = (i 1 ) = (j 0 ) =, as B = 1. 19

21 Meawhile, the preferences P i of i I over contracts in X respect the order P i, while using orders of precedence to reak ties among slots: x; s P i x ; s xp i x or [x = x and s (x) s ]; i P i x; s [ i P i x or i(x) i]. These extended priorities Π s and preferences P i induce choice functions over X: C s (Ỹ ) max ΠsỸ ; C i (Ỹ ) max P iỹ. To avoid terminology confusion, we call a set Ỹ X a slot-outcome. It is clear that slot-outcomes Ỹ X correspond to outcomes Y X according to the natural projection ϖ : X X defined y ϖ(ỹ ) {x : x; s Ỹ for some s S (x)}. Our contract set restriction notation extends naturally to slot-outcomes Ỹ : Ỹ i = { y; s Ỹ : i(y) = i}; Ỹ s = { y; s Ỹ : s = s}. Definition. A slot-outcome Ỹ X is slot-stale if it is 1. individually rational for agents and slots C i (Ỹ ) = Ỹi for all i I and C s (Ỹ ) = Ỹ s for all s S and 2. not locked at any slot there does not exist a slot-lock z; s X such that z; s = C i(z) (Ỹ { z; s }) and z; s = C s (Ỹ { z; s }). Lemma 1. If Ỹ X is slot-stale, then ϖ(ỹ ) is stale. 20

22 Theorem 3 of Hatfield and Milgrom (2005) implies that one-to-one matching with contracts markets have stale outcomes. Comining this oservation with Lemma 1 gives a second proof that the set of stale outcomes is always nonempty in our framework. In the next section, we refine this oservation y focusing on the stale outcome associated to the slot-outcome of agent-optimal slot-stale mechanism for the agent slot market. 4.2 The Cumulative Offer Process We now introduce the cumulative offer process for matching with contracts (see Hatfield and Kojima (2010); Hatfield and Milgrom (2005); Kelso and Crawford (1982)), which generalizes the agent-proposing deferred acceptance algorithm of Gale and Shapley (1962). We provide an intuitive description of this algorithm here; a more technical statement is given in Appendix A. Definition. In the cumulative offer process, agents propose contracts to ranches in a sequence of steps l = 1, 2,...: Step 1. Some agent i 1 I proposes his most-preferred contract, x 1 X i 1. Branch (x 1 ) holds x 1 if x 1 C (x1) ({x 1 }), and rejects x 1 otherwise. Set A 2 (x 1 ) = {x 1 }, and set A 2 = for each (x 1 ); these are the sets of contracts availale to ranches at the eginning of Step 2. Step 2. Some agent i 2 I for whom no contract is currently held y any ranch proposes his most-preferred contract which has not yet een rejected, x 2 X i 2. Branch (x 2 ) holds the contracts in C (x2) (A 2 (x 2 ) {x2 }) and rejects all other contracts in A 2 (x 2 ) {x2 }; ranches (x 2 ) continue to hold all contracts they held at the end of Step 1. Set A 3 (x 2 ) = A2 (x 2 ) {x2 }, and set A 3 = A2 for each (x 2 ). Step l. Some agent i l I for whom no contract is currently held y any ranch proposes his most-preferred contract which has not yet een rejected, x l 21

23 X i l. Branch (x l ) holds the contracts in C (xl) (A l (x l ) {xl }) and rejects all other contracts in A l (x l ) {xl }; ranches (x l ) continue to hold all contracts they held at the end of Step l 1. Set A l+1 (x l ) = Al (x l ) {xl }, and set A l+1 = A l for each (x l ). If at any time no agent is ale to propose a new contract, 26 then the algorithm terminates. The outcome of the cumulative offer process is the set of contracts held y ranches at the end of the last step efore termination. In the cumulative offer process, agents propose contracts sequentially. Branches accumulate offers, choosing at each step (according to C ) a set of contracts to hold from the set of all previous offers. The process terminates with no agent wishes to propose contracts that is, when all agents for whom no contracts are held have proposed all contracts they find acceptale. Note that we do not explicitly specify the order in which agents make proposals. This is ecause in our setting, the cumulative offer process outcome is in fact independent of the order of proposal. 27 An analogous order-independence result is known for settings where priorities induce unilaterally sustitutale ranch choice functions (Hatfield and Kojima (2010)). However, as our discussion in Section notes, slot-specific priorities may not induce unilaterally sustitutale choice functions. Meanwhile, no order-independence result is known for the general class of ilaterally sustitutale preferences. 28 Our first main result shows that the cumulative offer process outcome has a natural interpretation: it corresponds to the outcome of the agent-optimal slot-stale slot-outcome in the agent slot matching market. Theorem 1. The slot-outcome of the agent-optimal slot-stale mehanism in the agent slot matching market corresponds (under projection ϖ) to the outcome of the cumulative offer 26 That is, if all agents for whom no contracts are on hold have proposed all contracts they find acceptale. 27 We make this statement concrete in Theorem B.1 of Appendix B. 28 Although Hatfield and Kojima (2010) state their cumulative offer process algorithm without attention to the proposal order, they only prove that the choice of proposal order has no impact on outcomes in the case of unilaterally sustitutale preferences. 22

24 process. The proof of Theorem 1 proceeds in three steps. First, we show that the contracts held y each slot improve (with respect to slot priority order) over the course of the cumulative offer process. 29 This oservation implies that no contract held y a slot s S at some step of the cumulative offer process has higher priority than the contract s holds at the end of the process; it follows that the cumulative offer process outcome Y is the ϖ-projection of a slot-stale slot-outcome Ỹ. Then, we demonstrate that agents (weakly) prefer Ỹ to the agent-optimal slot-stale slot outcome Z (which exists y use Theorem 3 of Hatfield and Milgrom (2005)); this implies that Ỹ = Z, proving Theorem 1 as ϖ(ỹ ) = Y. Theorem 1 implies that the cumulative offer process always terminates. 30 Moreover, it shows (y Lemma 1) that the cumulative offer process outcome is stale and somewhat distinguished among stale outcomes. Theorem 2. The cumulative offer process produces an outcome which is stale. Moreover, for any slot-stale Z X, each agent (weakly) prefers the outcome of the cumulative offer process to ϖ( Z). Note that Theorem 2 shows only that agents weakly prefer the cumulative offer process outcome to any other stale outcome associated to a slot-stale slot-outcome. As not all stale outcomes are associated to slot-stale slot-outcomes, this need not imply that each agent prefers the cumulative offer process outcome to all other stale outcomes; we demonstrate this explicitly in the next section. 4.3 Agent-Optimal Stale Outcomes We say that an outcome Y X Pareto dominates Y X if Y i R i Y i for all i I, and Y i P i Y i for at least one i I. An outcome Y X which Pareto dominates all other stale 29 Here, y the contract held y a slot s S in step l, we mean the contract assigned to s in the computation of C (A l+1 ). 30 This fact can also e oserved directly, as the set X is finite, and the availale contract sets A l grow monotonically in l. 23

25 outcomes is called an agent-optimal stale outcome. 31 For general slot-specific priorities, agent-optimal stale outcomes need not exist, as the following example shows. Example 3. Let X = {i 0, i 1, j 0, j 1, k 0, k 1 }, with B = {}, I = {i, j, k} and i(h 0 ) = h = i(h 1 ) for each h I. 32 We suppose that h 0 P h h 1 P h h for each h I, and that has two slots, s 1 s 2, with slot priorities given y Π s1 : i1 j 1 k 1 i 0 j 0 k 0 s 1, Π s2 : i0 i 1 j 0 j 1 k 0 k 1 s 2. In this setting, the outcomes Y {j 1, i 0 } and Y {i 1, j 0 } are oth stale. However, Y i P i Y i while Y j P j Y j, so there is no agent-optimal stale outcome. Here, Y is associated to a slot-stale slot-outcome, ut Y is not. As we expect from Theorem 2, the cumulative offer process produces the former of these two outcomes, Y. Although matching markets with slot-specific priorities may not have agent-optimal stale outcomes, the cumulative offer process finds agent-optimal stale outcomes when they exist. Theorem 3. If an agent-optimal stale outcome exists, then it is the outcome of the cumulative offer process. In our proof of Theorem 3, we show that no stale outcome can Pareto dominate the cumulative offer process outcome. That is: for any stale outcome Y which is not equal to the outcome Z of the cumulative offer process, there is some agent i I such that Z i P i Y i. This quickly implies Theorem 3, as agent-optimal stale outcomes Pareto dominate all other stale outcomes. 31 That is, an agent-optimal stale outcome is a stale outcome such that such that Y i R i Y i for any agent i I and stale outcome Y X. 32 Clearly (in order for ( ) to e well-defined), we must have (h 0 ) = = (h 1 ) for each h I, as B = 1. 24

26 4.4 The Cumulative Offer Mechanism A mechanism consists of a strategy space S i for each agent i I, along with an outcome function ϕ Π : i I Si X that selects an outcome for each choice of agent strategies. We confine our attention to direct mechanisms, i.e. mechanisms for which the strategy spaces correspond to the preference domains: S i P i, where P i denotes the set of all possile preference relations for agent i I. Such mechanisms are entirely determined y their outcome functions, hence in the sequel we identify mechanisms with their outcome functions and use the term mechanism ϕ Π to refer to the mechanism with outcome function ϕ Π and S i P i (for all i I). All mechanisms we discuss implicitly depend on the priority profile under consideration; we often suppress the priority profile from the mechanism notation, writing ϕ instead of ϕ Π, if doing so will not introduce confusion. In this section, we undertake analysis of the cumulative offer mechanism (associated to slot priorities Π), which selects the outcome otained y running the cumulative offer process (with respect to sumitted preferences and priorities Π). We denote this mechanism y Φ Π : i I Pi X Staility and Strategy-Proofness of Φ Π A mechanism ϕ is stale if it always selects an outcome stale with respect to the input preferences. We say that a mechanism ϕ is strategy-proof if there is no agent i I, P I i I Pi, and P i P i such that ϕ ( ) i P, P i P i ϕ ( ) P i, P i. Similarly, we say that a mechanism ϕ is group strategy-proof if there is no set of agents I I, P I i I Pi, and P I P I such that ϕ ( ) I P, P I P i ϕ ( ) P I, P I for all i I. It follows immediately from Theorem 2 that the cumulative offer mechanism is stale. Meanwhile, Theorem 1 of Hatfield and Kojima (2009) implies that the agent-optimal slotstale mechanism is (group) strategy-proof in the agent slot matching market. Thus, we see that the cumulative offer mechanism is (group) strategy-proof, as any P I P I such 25

27 that ϕ ( P I, P I ) P i ϕ ( P I, P I ) for all i I would give rise to a profitale manipulation ( P I P I ) of the agent-optimal slot-stale mechanism. These oservations are summarized in the following theorem. Theorem 4. The cumulative offer mechanism Φ Π is 1. stale, and 2. (group) strategy-proof Φ Π Respects Unamiguous Improvements We say that priority profile Π is an unamiguous improvement over priority profile Π for i I if 1. for all x X i and y X I\{i}, if xπ s y, then x Π s y; and 2. for all y, z X I\{i}, yπ s z if and only if y Π s z. That is, Π is an unamiguous improvement over priority profile Π for i I if Π is otained from Π y increasing the priorities of some of i s contracts (at some slots) while leaving the relative priority orders of other agents contracts unchanged. We say that a mechanism ϕ respects unamiguous improvements for i if for any preference profile P I, (ϕ Π(P I )) i R i (ϕ Π (P I )) i whenever Π is an unamiguous improvement over Π for i. And we say that ϕ respects unamiguous improvements if it respects unamiguous improvements for each agent i I. Theorem 5. The cumulative offer mechanism Φ Π respects unamiguous improvements. Our proof of Theorem 5 makes use of the fact that the cumulative offer process outcome is independent of the contract proposal order. In particular, we focus on a proposal order 26

28 in which i proposes contracts only when no other agent is ale to propose. This choice of proposal order guarantees that i is always the last agent to propose a contract in the running of the cumulative offer process (for any priority profile). As Π is an unamiguous improvement over Π for i, we can show that the last contract i proposes in the cumulative offer process with priority profile Π must also e proposed in the cumulative offer process with priority profile Π. This yields the desired result ecause it implies that i is at least as well off under the outcome of cumulative offer process with priority profile Π as under the cumulative offer process with priority profile Π. As an alternative to this approach to the proof of Theorem 5, we could instead show that the agent-optimal slot-stale mechanism for the agent slot matching market satisfies a condition analogous to respecting unamiguous improvements. Theorem 5 would then follow from Theorem 1. One note: A natural strengthening of our notion of an unamiguous improvement for i I would include the condition that i s preferred contracts (weakly) increase in priority formally, for all B, s S, and x, x (X i X ), if x Π s x and x Π s x, then xp i x. (2) As Theorem 5 shows that Φ Π respects unamiguous improvements, we see a fortiori that Φ Π respects unamiguous improvements that satisfy the additional condition (2). 5 Further Applications 5.1 Design of Affirmative Action Mechanisms We say that a matching prolem has agent types if the contract set X is a suset of I B T for some type set T, and for each i I, X i = {i} B {t} for some t T, 27

29 so that each i is associated to exactly one type t. 33 For such a prolem, we identify agents with their types, writing t(i) for the unique type t T such that X i = {i} B {t}. For consistency with the prior literature on school choice, we ause notation slighly y writing i to denote, for each ranch B, the unique contract (i,, t(i)) (X i X ). Imposing this additional structure on our general model simplifies the form of slot-specific priorities, rendering ranches choice functions sustitutale. Proposition 3. In a matching prolem with slot-specific priorities and agent types, the ranch choice functions C are sustitutale and satisfy the law of aggregate demand. In settings with agent types, sustitutaility coincides with sustitutaility on no-negotialility contract sets this conclusion otains whenever X i X 1 for all i I and B. Thus, Proposition 3 follows directly from Proposition 1. Comining Proposition 3 with Theorems 15, 3, and 4 of Hatfield and Milgrom (2005) shows that there exists an agent-optimal stale outcome in any matching prolem with slotspecific priorities and agent types. The following result then follows upon comining this oservation with our Theorems 3, 4, and 5. Corollary 1. In a matching prolem with slot-specific priorities and agent types, the cumulative offer mechanism Φ Π is an agent-optimal stale mechanism, which is (group) strategyproof and respects unamiguous improvements. As our discussion in Section 2.1 suggests, decreases in the precedence of slots that rank agents of type t highly can improve type-t agents cumulative offer outcomes. Unfortunately, while we elieve this comparative static should hold in reasonaly-sized marketplaces, it may fail in small markets Note that we may assusme without loss of generality that X i = {i} B {t}, as any case in which X i {i} B {t} can then e captured y assuming some contracts x {i} B {t} to e unacceptale to slots at their associated ranches (x). 34 Example C.1 of Appendix C illustrates this fact. 28

30 5.1.1 Soft Minority Quotas in School Choice Many affirmative action programs impose quotas on majority agents. However, as Kojima (2012) showed, quota policies can have perverse effects: some quota-ased affirmative action policies hurt all minority students under any stale matching mechanism. Despite these discouraging oservations, Hafalir et al. (forthcoming) recently introduced a novel approach to affirmative action, affirmative action with minority reserves, which compares favoraly to the more standard majority-quota policies. In the Hafalir et al. (forthcoming) approach, certain slots at each school are reserved for minorities ut convert into regular slots if not claimed y minority students. Formally, the model of Hafalir et al. (forthcoming) emeds into the framework of matching with slotspecific priorities as follows: The agents i I are students and the ranches B are schools. Each student i I as a strict linear preference order P i over schools, and is of either minority (m) or majority (M) type (i.e. t(i) {M, m} = T). Each school B has a strict linear order π over students and a numer of slots q corresponding to its capacity. Under affirmative action with minority reserves, each school B has an associated minority reserve r m q such that prefers any minority applicant to any majority applicant if the numer of minority students admitted to is elow r m (Hafalir et al. (forthcoming)). This policy can e implemented y choosing slot-specific priorities Π such that 1. for all l r m, i Π sl i Πs l s l (a) t(i) = m and t(i ) = M, or () t(i) = t(i ) and iπ i ; 2. for all l > r m, i Π sl i Πs l s l iπ i. Under affirmative action with majority quotas, meanwhile, each school B has an associated majority quota q M q such that cannot admit more than q M majority applicants. This policy can e implemented y choosing slot-specific priorities Π such that 29

31 1. for all l < q q M, iπsl i Π sl s l t(i) = t(i ) = m and iπ i ; 2. for all l q q M, iπsl i Π sl s l iπ i. With these oservations, we may derive two of the main results of Hafalir et al. (forthcoming) as consequences of our general results for slot-specific priority structures. Proposition 4 (Hafalir et al. (forthcoming)). 1. In the presence of affirmative action with minority reserves, the cumulative offer mechanism produces the student-optimal stale outcome and is (group) strategy-proof. 2. Given a vector q M of majority quotas, set r m = q q M for each B, and let Y e an outcome which is stale under the priorities Π induced y the quotas q M and tiereakers π. Either: (a) Y is stale under the priorities Π induced y reserves r m and tiereakers π, or () there exists an outcome Z which is stale under priorities Π and Pareto dominates Y. Proof. Part 1 follows directly from Proposition 3. Part 2 also follows quickly: By Corollary 1, we know that (Φ Π (P I )) i R i Y i for each i I. Meanwhile, Π is an unamiguous (weak) improvement for each i I, hence (Φ Π(P I )) i R i (Φ Π (P I )) i for each i I (y Theorem 5). Thus, taking Z Φ Π(P I ) gives Z i = (Φ Π(P I )) i R i (Φ Π (P I )) i R i Y i, (3) for each i I. Now, if Y = Φ Π (P I ) = Φ Π(P I ) = Z, then Y is stale under the priorities Π. Otherwise, there is at least one i I for whom the identity (3) is strict. In that case, Z is stale under priorities Π and Pareto dominates Y. 30

32 < l 40 q q 100 < l 55 q q 100 < l 70 q q < l 85 q q 100 < l 100q 100. Figure 2: Structure of slot priorities in the Chicago selective high school match. The top 40% of slots are open slots; the ottom 60% feature socioeconomic reserves for each of the four student types ({4, 3, 2, 1}) Socioeconomic Affirmative Action in Chicago School Choice We now demonstrate that our framework emeds the real-world structure of the Chicago selective high school choice affirmative action program discussed in Section 2.1. Here, the agents i I and ranches B again correspond to students and schools. Each student i I has a strict linear preference order P i over schools, and there are four agent types representing the different SES tiers: T = {4, 3, 2, 1}. The top 40% of the q slots at shool B are open slots, assigned ased on a strict linear merit order π over students, which is determined y composite test scores and thus uniform across schools. The remaining 60% of the slots at each school B feature socioeconomic reserves: the first 15% of these slots are reserved for students i I of type t(i) = 4; the next 15% are reserved for students i I of type t(i) = 3; and so forth. These type priority structures are illustrated in Figure 2. Formally, the set S of slots at school is partitioned into susets S = S o S 4 S 3 S 2 S 1, with S o 40 consisting of q 100 slots, and each set S t 15 consisting of q 100 slots. 35 The priorities of slots s S o are such that iπ s i Π s s iπ i. 35 We assume for simplicity that q is a multiple of 20, so that q, q Z. 31

33 < l 15 q q 100 < l 30 q q 100 < l 45 q q < l 60 q q 100 < l 100q 100. Figure 3: Counterfactual slot priority structure for the Chicago selective high school match. Here, the top 60% of slots feature socioeconomic reserves, while the ottom 40% are open. Meanwhile, the priorities of slots s S t are such that iπ s i Π s s iπ i whenever t(i) = t(i ) = t, and s Π s i whenever t(i) t. 36 The order of precedence is such that s o s 4 s 3 s 2 s 1 (4) for all s o S o, s4 S 4, s3 S 3, s2 S 2, and s1 S 1.37 Tales 1 3 show the effect of switching from the current precedence orders to the alternate orders, illustrated in Figure 3, in which open slots are filled after reserve slots. Formally, these counterfactual precedence orders are such that s 4 s 3 s 2 s 1 s o for all s o S o, s4 S 4, s3 S 3, s2 S 2, and s1 S 1. Our model suggests a natural precedence order which gives rise to priorities in etween the current CPS priority structure under which all open slots are filled first and the counterfactual structure discussed in Section 2.1 under which all reserved slots are filled 36 Because all the slots in Chicago s selective high schools are overdemanded, all seats reserved for students of type t are claimed y students in type t, hence we may assume for expositional simplicity that slots s S t find students of types t t unacceptale. 37 Since all slots s e S e have idential priorities, (4) suffices to specify the precedence order up to equivalence. 32

34 first. Instead of filling all the open slots at once, CPS could alternate etween filling open slots and filling reserve slots, in proportion to the total numers of each slot type availale. For example, intermediate priorities could e designed so as to fill three open slots at each school, then one of each type of reserved slot at each school, then three more open slots, then four more reserved slots, and so forth. 38 This approach spreads the access to open seats evenly throughout the priority structure, virtually eliminating the iases of the current CPS system. 39 Simulation results presented in Tale 4 shows that student outcomes under the intermediate priorities are almost identical to those arising when all reserved seats are filled efore the open seats (the counterfactual discussed in Section 2.1). While this might at first seem surprising, is in fact quite natural, given the distriution of CPS students test scores. As we pointed out in Section 2.1, sigh-ses students composite scores dominate low-ses students scores throughout the relevant part of the score distriution. As a result, high-ses students fill the first availale open seats, then the highest-scoring low-ses students receive the first reserved seats. Then, once again, the top of the truncated scoring distriution consists of high-ses students; the students take the next open slots. Once again, then, the highest-scoring low-ses students who remain unassigned receive reserved seats, leaving even fewer low-ses students with high scores in the pool. This process produces an outcome very similar to that found when all reserved seats are filled efore the open seats. 5.2 Precedence Order Changes in Cadet Branch Matching The cadet ranch matching prolem studied y Sönmez and Switzer (2011) and Sönmez (2011) is a matching prolem with X = I B {t 0, t + }. Here, the agents i I correspond to cadets, who must e assigned to ranches of service B. Contracts with term t 0 represent standard service contracts; contracts with term t + represent the standard contract 38 When using this approach, every third lock of slots should have only two open slots, so as to maintain the overall 40%-15%-15%-15%-15% proportions in every lock of 20 slots. 39 As the numer of slots at each school is finite, completely eliminating the ias would require randomizing the precedence relation to some extent. 33

35 Intermediate Mechanism Effect of Switching (Open Slots First in Group) (to fill Open Slots Last) Tier 4 Tier 3 Tier 2 Tier 1 Tier 4 Tier 3 Tier 2 Tier TOT Tale 4: Comparison etween the uniased, intermediate mechanism for CPS selective high school enrollment and the counterfactual discussed in Section 2.1, in which open slots are filled last. supplemented with a three-year service extension. Cadets are ranked according to a strict linear order of merit ranking π. The slots of each ranch B are partitioned into two sets, S 0 S and S + S, of sizes (1 λ)q and λq, respectively. Slots s S 0 are regular slots, whose priority rankings follow the order of merit list exactly: for any B, s S 0, i i I, and t, t {t 0, t + }, (i,, t)π s (i,, t ) iπ i. Our results are independent of how regular slots relative priorities of contracts (i,, t) and (i,, t ) are specified; for concreteness, we follow the military s convention of assuming that (i,, t 0 )Π s (i,, t + ) for all slots s S 0. Slots s S+ are ranch-of-choice slots, which give priority to t + contracts: for any B, s S +, and i i I, (i,, t + )Π s (i,, t 0 ), and (i,, t)π s (i,, t) iπ i 34

36 for any t {t 0, t + }. In the settings of Sönmez and Switzer (2011) and Sönmez (2011), the ranch-of-choice slots have lowest precedence at each ranch. That is, the slots s, s S at ranch B follow a precedence order such that s S 0 and s S + = s s. (5) In our model, all precedence orders satisfying condition (5) are equivalent; hence, we identify the full class of such orders with a single precedence order. Sönmez and Switzer (2011) demonstrated that the ranch choice functions induced y precedence order are unilaterally sustitutale. This implies the existence of a cadetoptimal stale outcome, and allowed Sönmez and Switzer (2011) to propose the use of a cadet-optimal stale mechanism for cadet ranch matching. 40 Our next result shows that the structure found y Sönmez and Switzer (2011) is unique to the specific precedence order the United States military selected: up to equivalence, is the only precedence order which guarantees the existence of cadet-optimal stale outcomes in general. Proposition 5. For any cadet ranch matching prolem precedence order for which there exists B, s S 0, and s S + such that s s, there exist preference orders for all cadets such that no outcome stale with respect to the ranch choice functions C induced y the slot priorities Π s (s S ) and precedence order is cadet-optimal. References Adulkadiroğlu, Atila and Tayfun Sönmez, School Choice: A Mechanism Design Approach, American Economic Review, 2003, 93, In his discussion of market design for the ROTC cadet ranch match, Sönmez (2011) extended these results to the case in which more than two distinct contract terms are availale. 35

37 , Parag A. Pathak, Alvin E. Roth, and Tayfun Sönmez, The Boston pulic school match, American Economic Review, 2005, 95, ,,, and, Changing the Boston School Choice Mechanism: Strategy-proofness as Equal Access, Mimeo, Massachusetts Institute of Technology.,, and, The New York City high school match, American Economic Review, 2005, 95, ,, and, Strategyproofness versus Efficiency in Matching with Indifferences: Redesigning the NYC High School Match, American Economic Review, 2009, 99 (5), Adachi, H., On a characterization of stale matchings, Economics Letters, 2000, 68, Alkan, Ahmet, A class of multipartner matching markets with a strong lattice structure, Economic Theory, 2002, 19, and David Gale, Stale Schedule Matching under Revealed Preference, Journal of Economic Theory, 2003, 112, Aygün, Orhan and Tayfun Sönmez, The Importance of Irrelevance of Rejected Contracts in Matching under Weakened Sustitutes Conditions, Mimeo, Boston College. and, Matching with Contracts: The Critical Role of Irrelevance of Rejected Contracts, Mimeo, Boston College. Balinski, Michel and Tayfun Sönmez, A tale of two mechanisms: Student placement, Journal of Economic Theory, 1999, 84, Duins, L. E. and D. A. Freedman, Machiavelli and the Gale-Shapley Algorithm, American Mathematical Monthly, 1981, 88,

38 Echenique, Federico, Contracts vs. Salaries in Matching, American Economic Review, 2012, 102, and Jorge Oviedo, Core many-to-one matchings y fixed-point methods, Journal of Economic Theory, 2004, 115, Fleiner, T., A fixed point approach to stale matchings and some applications, Mathematics of Operations Research, 2003, 28, Friedman, Milton, The Role of Government in Education, in Roert A. Solo, ed., Economics and the Pulic Interest, Rutgers University Press 1955, pp , Capitalism and Freedom, University of Chicago Press, Gale, David and Lloyd S. Shapley, College admissions and the staility of marriage, American Mathematical Monthly, 1962, 69, Hafalir, Isa Emin, M. Bumin Yenmez, and Muhammed Ali Yildirim, Effective Affirmative Action in School Choice, Theoretical Economics, forthcoming. Hatfield, John William and Fuhito Kojima, Matching with Contracts: Comment, American Economic Review, 2008, 98, and, Group Incentive Compatiility for Matching with Contracts, Games and Economic Behavior, 2009, 67, and, Sustitutes and Staility for Matching with Contracts, Journal of Economic Theory, 2010, 145, and Paul Milgrom, Matching with Contracts, American Economic Review, 2005, 95, and Scott Duke Kominers, Contract Design and Staility in Matching Markets, Mimeo, Harvard Business School. 37

39 and, Matching in Networks with Bilateral Contracts, American Economic Journal: Microeconomics, 2012, 4, , Fuhito Kojima, and Yusuke Narita, Promoting School Competition Through School Choice: A Market Design Approach, Mimeo, Stanford Graduate School of Business. Kelso, Alexander S. and Vincent P. Crawford, Jo Matching, Coalition Formation, and Gross Sustitutes, Econometrica, 1982, 50, Kojima, Fuhito, School Choice: Impossiilities for Affirmative Action, Games and Economic Behavior, 2012, 75, Kominers, Scott Duke, On the Correspondence of Contracts to Salaries in (Many-to- Many) Matching, Games and Economic Behavior, 2012, 75, Ostrovsky, Michael, Staility in supply chain networks, American Economic Review, 2008, 98, Pathak, Parag A. and Tayfun Sönmez, Leveling the Playing Field: Sincere and Sophisticated Players in the Boston Mechanism, The American Economic Review, 2008, 98, and, School Admissions Reform in Chicago and England: Comparing Mechanisms y Their Vulneraility to Manipulation, American Economic Review, forthcoming. Roth, Alvin E., The economics of matching: Staility and incentives, Mathematics of Operations Research, 1982, 7, , The Evolution of the Laor Market for Medical Interns and Residents: A Case Study in Game Theory, Journal of Political Economy, 1984, 92, and Elliott Peranson, The effects of the change in the NRMP matching algorithm, American Economic Review, 1999, 89,

40 Sönmez, Tayfun, Bidding for Army Career Specialties: Improving the ROTC Branching Mechanism, Mimeo, Boston College. and Toias B. Switzer, Matching with (Branch-of-Choice) Contracts at United States Military Academy, Mimeo, Boston College. A Formal Description of the Cumulative Offer Process The cumulative offer process associated to proposal order is the following algorithm: 1. Let l = 0. For each B, let D 0, and let A1. 2. For each l = 1,..., X : (a) Let i e the l -maximal agent i I such that i / i( B D l 1 ) and max P i(x \ ( B A l )) i i that is, the agent highest in the proposal order who wants to propose a new contract if such an agent exists. (If no such agent exists, then proceed to Step 3, elow.) i. Let x max P i(x \( B A l )) i e i s most-preferred contract that has not yet een proposed. ii. Let (x). Set D l = C (A l, set D l = Dl 1 and set A l+1 = A l. {x}) and set Al+1 = A l {x}. For each 3. Return the outcome Y ( B D l 1 ) ( ) = C (A l ) B consisting of contracts held y ranches at the point when no agent wants to propose additional contracts. 39

41 Here, the sets D l 1 and A l denote the sets of contracts held y and availale to ranch at the eginning of cumulative offer process step l. We say that a contract z is rejected during the cumulative offer process if z A l (z) ut z / Dl 1 (z) for some l. B Proofs Omitted from the Main Text Proof of Proposition 1 We prove the following auxilarly lemma which directly implies Proposition 1. Lemma B.1. Suppose that Y Y X, Y = i(y ), and Y = i(y ). Then, if y Y and y Y are the contracts assigned to s S in the computations of C (Y ) and C (Y ), respectively, we have y Γ s y. Proof. The hypotheses on Y and Y imply that Y i(x) = {x} for each x Y and that Y i(x ) = {x } for each x Y. With this oservation, the following claim follows quickly. Claim. Let V l l (Z) denote the set V defined in step l 1 of the computation of C (Z). Then, V l (Y ) V l (Y ). Proof. We proceed y induction. We have V 1 (Y ) = V 1 (Y ) a priori, so we assume that V l l (Y ) V (Y ) for all l < l + 1 for some l > 0. We now show that this hypothesis ( implies that V l+1 (Y ) V l+1 (Y ): Let x max s l Π V l ( V l (Y ) ) ; hence, x = max Π s l (Y ) ). If x V l+1 (Y ), then clearly V l+1 (Y ) = (V l (Y )) \ Y i(x ) = (V l (Y )) \ {x } (V l (Y )) \ {x } = (V l (Y )) \ Y i(x ) = V l+1 (Y ) as desired. Otherwise, we have x / V l+1 (Y ), so that x V l (Y ) V l (Y ) \ {x }, we have ( ( max s l Π V l (Y ) )) x x. As V l+1 (Y ) = (V l (Y )) \ Y i(x) = (V l (Y )) \ {x} (V l (Y )) \ {x } = (V l (Y )) \ Y i(x ) = V l+1 (Y ). 40

42 The claim implies that ( ) ( ) max V l s l Π (Y ) Γ sl max V l s l Π (Y ) (6) for all l; this shows the result. To see that Proposition 1 follows from Lemma B.1, we suppose that (1) holds for some z, z X and Y X, and note that (1) also implies that Y {z} = i(y {z}). (7) Now, given (7), we know that if z / C (Y {z}), then for each s S, the contract y assigned to s in the computation of C (Y {z}) must e higher-priority than z under Π s, that is, yπ s z. But then, it follows from (7) and Lemma B.1 that each such s must e assigned a contract y for which y Γ s yπ s z in the computation of C (Y {z, z }). Thus, we must have z / C (Y {z, z }). Hence, we see that each C is sustitutale on no-negotiation contract sets. Proof of Proposition 2 See Appendix D. Proof of Lemma 1 It is immediate that if ϖ(ỹ ) is not individually rational, then Ỹ is not individually rational for agents and slots. Thus, we need only consider the locking conditions. For Ỹ X, suppose that Z X is a set of contracts which locks ϖ(ỹ ). We fix some (Z), and oserve that there must e a contract z Z \ ϖ(ỹ ) for which there is some step l of the computation of C (ϖ(ỹ ) Z) such that Dl \ Dl 1 = {z}. (That is, there 41

43 must exist a contract z Z \ ϖ(ỹ ) which is assigned to the highest-precedence slot, sl, among those slots which are assigned contracts in Z \ ϖ(ỹ ) during the computation of C (ϖ(ỹ ) Z).) We let x ϖ(ỹ ) e the (possily null) contract which is assigned to slot s l in the computation of C (ϖ(ỹ )). It is clear that zπ sl x, y construction. Thus, we have z; s l Πs l x; s l. Meanwhile, we know that zp i(z) (ϖ(ỹ )) i(z) ecause Z locks ϖ(ỹ ). It follows that z; s l is a slot-lock for Ỹ. Thus, if Ỹ is not locked at any slot, then ϖ(ỹ ) is unlocked; the result follows. Proof of Theorem 1 We prove the following result, which is slightly more general than Theorem 1. Theorem B.1. For any proposal order, the slot-outcome of the agent-optimal slot-stale mehanism in the agent slot matching market corresponds (under projection ϖ) to the outcome of the cumulative offer process associated to proposal order. Proof. We suppress the dependence on, as doing so will not introduce confusion. We egin with a simple lemma which shows that slots assigned contracts improve (with respect to slot priorities) over the course of the cumulative offer process. Lemma B.2. Fix l and l with l < l, and let x l and x l (with (x l ) = = (x l )) e the contracts assigned to s S in the computations of C (A l+1 ) = D l and C (A l +1 ) = D l, respectively. Then, x l Γ s x l. Proof. The result follows immediately from Lemma B.1, as A l+1 A l +1 X y construction. We denote the outcome of the cumulative offer process y Y, and let Ỹ { y; s : y Y and s is assigned z in the computation of C (z) (Y )}. By construction, we have ϖ(ỹ ) = Y. 42

44 Lemma B.3. The slot-outcome Ỹ is slot-stale. Proof. We suppose that z; s slot-locks Ỹ, so that zp i(z) (ϖ(ỹ )) i(z) = Y i(z), (8) zπ s (ϖ(ỹ )) s = Y s. (9) Now, y (8) and the fact that Y is the cumulative offer process outcome, we know that z must e proposed in some step l of the cumulative offer process. We let l l e the first step of the cumulative offer process for which no slot s S (z) with s (z) s is assigned z in the computation of C (z) (A l+1 (z) ) = Dl (z). (Such a step l must exist since z / Y.) We let xl e the contract assigned to s in the computation of C (z) (A l+1 (z) ). Since xl z, we know that x l Π s z. But then, we know y Lemma B.2 that for each l l, the contract x l assigned to s in the computation of C (z) (A l +1 (z) ) has (weakly) higher Πs -priority than x l, and hence has (strictly) higher Π s -priority than z: x l Γ s x l Π s z. In particular, then, we must have Y i(z) Π s z, contradicting (9). Thus, there cannot e a slot-contract z; s which slot-locks Ỹ. Now, we let Z e the agent-optimal slot-stale slot-outcome. (Such an outcome exists y Theorem 3 of Hatfield and Milgrom (2005).) Lemma B.4. For each agent i I, Ỹi R i Zi. Proof. It suffices to show that no contract z ϖ( Z) is ever rejected during the cumulative offer process. To see this, we suppose the contrary, and consider the first step l at which some contract z ϖ( Z) is rejected. We let s S (z) e the slot such that z; s Z, and let x z e the contract assigned to s in the computation of C (z) (A l+1 (z) ). Now, as z is the first contract in ϖ( Z) to e rejected, we know that xp i(x) (ϖ( Z)) i(x). Moreover, as z / C (z) (A l+1 (z) ) and x is assigned to s in the computation of C(z) (A l+1 (z) ), we know that xπ s z. But then, it follows that x; s slot-locks Z, contradicting the fact that Z is slot-stale. 43

45 Now, y Lemma B.3, we know that Ỹ is slot-stale; it then follows from Lemma B.4 that Ỹ must e the agent-optimal slot-stale slot-outcome. The theorem then follows directly, as ϖ(ỹ ) = Y. Proof of Theorem 2 The result follows immediately from Lemma 1 and Theorem 1, as the agent-proposing deferred acceptance algorithm in the agent slot matching market yields the agent-optimal slot-outcome (y Theorem 3 of Hatfield and Milgrom (2005)). Proof of Theorem 3 We show Theorem 3 y way of the following more general result. Theorem B.2. For any stale outcome Y which is not equal to the outcome Z of the cumulative offer process, there is some agent i I such that Z i P i Y i. Proof. We suppose to the contrary that there is some stale outcome Y Z such that Y i R i Z i for all i I. We prove the theorem y considering an alternative proposal order for the cumulative offer process, and showing that the outcome Z of the cumulative offer process associated to must e (weakly) preferred to Y y all agents. We now descrie the alternative proposal order: 41 Let e any order such that at each step of the cumulative offer process associated to, 1. all agents in i(y ) who wish to propose contracts weakly preferred to those in Y have the opportunity to propose contracts efore any agents in I \ i(y ) do, and 2. all agents in i(y ) who wish to propose contracts not weakly preferred to those in Y are not allowed to propose unless no agents in I \ i(y ) wish to propose contracts. Claim. We have Z ir i Y i for all i I. 41 For ease of exposition, we do not provide a fully formal specification of ; nonetheless, our description directly yields an algorithm to compute such a specification. 44

46 Proof. We oserve that if there is some step of the cumulative offer process associated to at which all contracts in Y are held, then all future proposals will e rejected, and so the process produces the outcome Z = Y. 42 To see this, we oserve that no agent in i(y ) will propose while Y is held. Meanwhile, if at that time some agent i I \ i(y ) proposes a contract x which is held following its proposal, then {x} locks Y, contradicting the fact that Y is stale. Now, we note that y our choice of, at any step in the cumulative offer process associated to when there exists at least one agent i i(y ) who wishes to propose a contract weakly preferred to Y i, some such agent is given the opportunity to propose. It follows that, if there is no step of the cumulative offer process associated to at which all contracts in Y are held, then the process must terminate efore all contracts in Y have een proposed. In this case, the process outcome Z must e weakly-preferred to Y y all agents; comining this fact with our previous oservations shows the claim. Now, y Theorem B.1, Z must e the outcome of the cumulative offer process associated to the original proposal order that is, Z = Z. But, y the aove claim, we then have Z i = Z ir i Y i R i Z i for all i I, contradicting the assumption that Z Y. Theorem 2 shows that the outcome Z of the cumulative offer process is stale. But then, if there were an agent-optimal stale outcome Y Z, we would have a stale outcome Y for which Y i R i Z i for all i I, contradicting Theorem B Formally, our argument shows that an analogous statement is true under any proposal order; we state the claim for proposal order ecause that statement is important in the sequel. 45

47 Proof of Theorem 4 Part 1 is immediate from Theorem 2. Meanwhile, for Part 2, we suppose that there is some set of agents I I and P I P I such that ϕ ( P I, P I ) P i ϕ ( P I, P I ) (10) for all i I. Now, if Z is the outcome of the agent-optimal slot-stale mechanism run on preferences ( P I, P I ) and Ỹ is the outcome of the agent-optimal slot-stale mechanism run on preferences ( P I, P I ), then we have Z P i Ỹ for all i I, as we have ϖ( Z) = ϕ ( P I, P I ) P i ϕ ( P I, P I ) = ϖ( Ỹ ) y (10) and Theorem B.1. But this implies that the agent-optimal slot-stale mechanism is not group strategy-proof (in the agent slot market), contradicting Theorem 1 of Hatfield and Kojima (2009). 43 Proof of Theorem 5 To see this, we fix an agent i and let Π B e an unamiguous improvement over Π B for i. We let e the proposal order used in computing Φ, and let e the alternative (uniquely defined) proposal order such that for all l, j l k j l k (for all j, k i) j l i (for all j i), that is, the order otained from y moving i to the ottom of each linear order l. 43 Theorem 1 of Hatfield and Kojima (2009) implies that the agent slot matching mechanism which selects the (slot-)outcome of the agent-optimal slot-stale mechanism is group strategy-proof for agents. To see this, it suffices to note that oth the sustitutes condition and the law of aggregate demand hold automatically (for all preferences) in one-to-one matching markets such as the agent slot matching market. 46

48 By Theorem B.1, the outcome Φ Π(P I ) is equal to that of the cumulative offer process associated to under priorities Π. Likewise, the outcome Φ Π (P I ) is equal to that of the cumulative offer process associated to under priorities Π. These oservations essentially prove the result: In any cumulative offer process associated to, agent i always proposes after all other agents are unwilling to propose new contracts. Hence, under priority structure Π, there is some contract x which i proposes in the last step efore the cumulative offer process associated to terminates. As Π is an unamiguous improvement over Π for i, we see that i proposes x in the cumulative offer process associated to, under priorities Π. 44 It follows that (Φ Π(P I )) i = xr i (Φ Π (P I )) i, as desired. Proof of Proposition 3 In any prolem with agent types, X i X = 1 for all i I and B. It follows immediately that in such a prolem, Y = i(y ) for all B and Y X. (11) The sustitutaility of each ranch choice function C in the presence of agent types then follows from Proposition 1, as sustitutaility on no-negotiation contract sets is equivalent to sustitutaility under condition (11). Additionally, we see that in the presence of agent types, each choice function C satisfies the law of aggregate demand, as condition (11) and Lemma B.1 together show that for all l and Y Y, max Π s l V l (Y ) = s l whenever 44 As in the cumulative offer process associated to, any contract x with i(x ) = i and x P i x is proposed efore x is proposed. Moreover, y our choice of, the process state at the time of the proposal of such a contract x is exactly the same under priorities Π as it is under priorities Π. Then, such x must e rejected under priorities Π, as otherwise Π would not e an unamiguous improvement over Π for i. 47

49 max Π s l V l (Y ) = s l ; hence, C (Y ) = C (Y ) C (Y ) = C (Y ). Proof of Proposition 5 We let B e some ranch for which, and let l e the minimal value such that s l S+. As, there are s S 0 and s S + such that s s. In particular, then, there must e some slot s S 0 for which sl s. We let l e such that s l such slot s S 0. is the -minimal We lael the cadets in I as i 1, i 2,... y the ranking π, so that i m π i m m m. We assume that (i m,, t 0 ) i m m < l P im = (i m,, t 0 ) (i m,, t + ) i m l m l (i m,, t 0 ) i m l < m. Claim. The outcomes Y {(i m,, t) : m < l} {(i m,, t) : l m < l } {(i m,, t) : l m S }, Y {(i m,, t) : m < l} {(i m,, t) : l < m l } {(i l,, t)} {(i m,, t) : l < m S } are oth stale under the priorities Π and preferences P I. Proof. Clearly, oth Y and Y are individually rational. Thus, it suffices to show that each is unlocked. Now under Y, all agents i m for which m < l or m l hold their most-preferred contract. Thus, any set locking Y must e a suset of Z {(i m,, t 0 ) : l m < l }. However, the 48

50 contracts (i m,, t + ) are assigned to slots s m in the computation of C (Y ) (for m S ), and τ sm = τ + for all l m < l. It follows that Y = C (Y Z) for any Z Z ; hence Y is unlocked, as desired. An analogous argument shows that Y is unlocked. The result follows directly from the claim, since Y i l Pil Y il ut Y P il Y i l i l. C Example Omitted from the Main Text Example C.1. Let X = {i, i, i, i, i, i, j, j }, with B = {, }, I = {i, i, i, j}, i(h ) = h = i(h ) for each h I, and (h ) = for each h I and B. We suppose that there are two types of agents T = {i, j} and that t(i) = t(i ) = t(i ) = i, while t(j) = j. We suppose that agents have preferences P i : i i i P i : i i P i : i i P j : j j j. We suppose further that has two slots, s 1 s 2, with priorities given y Π s1 : i i i j s 1, Π s2 : i j i i s 2, and that has one slot, s 1, with priority order identical to Πs2, Π s1 : i j i i s 2. In this example, the cumulative offer process outcome is {i, j, i }. If the precedence of slots s 1 and s2 were reversed, however, the cumulative offer process outcome would e 49

51 {i, i, j }; hence i is made worse off following a decrease in the precedence of a slot that favors agents of type i = t(i ). D Proof of Proposition 2 We first prove a lemma which shows that ranch choice functions satisfy the irrelevance of rejected contracts property of Aygün and Sönmez (2012,a). Lemma D.1. If z / C (Y {z, z }), for some z, z X and Y X, then C (Y {z, z }) = C (Y {z}). Proof. We suppose that z / C (Y {z, z }), and show the following claim. Claim. Let V l l (Z) denote the set V defined in step l 1 of the computation of C (Z). Then, V l (Y {z, z }) = V l (Y {z}) {z }. Proof. We proceed y induction. We have V 1 (Y {z, z }) = V 1 (Y {z}) {z } a priori, so we assume that V l (Y {z, z }) = V l (Y {z}) {z } for all l < l + 1 for some l > 0. We now show that this hypothesis implies that V l+1 (Y {z, z }) = V l+1 (Y {z}) {z }: Let ( x max s l Π V l (Y {z, z }) ). As z / C (Y {z, z }), we must have x z ; hence, x = max Π s l ( V l (Y {z, z }) \ {z } ) ( = max s l Π V l (Y {z}) ) (12) y the inductive hypothesis. It then follows that V l+1 (Y {z, z }) = V l (Y {z, z }) \ {x } = ( V l (Y {z}) {z } ) \ {x } = ( V l (Y {z}) \ {x } ) {z } = V l+1 (Y {z}) {z }, 50

52 where the second equality follows from the inductive hypothesis, and the fourth equality follows from (12). This completes our induction. The claim implies the desired result, as it shows that C (Y {z, z }) = (Y {z, z }) \ V q (Y {z, z }) = (Y {z, z }) \ (V q (Y {z}) {z }) = (Y {z}) \ (V q (Y {z})) = C (Y {z}). Now, we suppose that i(z), i(z ) / i(y ), and that z / C (Y {z}). Supposing that z C (Y {z, z }), we see y Lemma D.1 that z C (Y {z, z }). This implies immediately that i(z) i(z ), as no ranch ever selects two contracts with the same agent. Claim. If z l and y l are the contracts assigned to s l in the computation of C (Y {z, z }) and C (Y {z}), respectively, then z l Γ sl y l. Proof. Let H l (Z) denote the set Hl defined in step l of the computation of C (Z). We proceed y doule induction: Clearly, either z 1 = max s 1 (Y {z, z Π }) = max s 1 Π (Y {z}) = y1 or z 1 = max Π s 1 (Y {z, z }) = z, so z 1 Γ sl y 1 and i(h 1 (Y {z, z })) i(h 1 (Y {z}) {z }). Thus, we suppose that z l Γ sl y l and i(h l (Y {z, z })) i(h l (Y {z}) {z }) for all l < l. We have z l = max Π s l y l = max Π s l ( V l (Y {z, z }) ) ( = max s l Π ( V l (Y {z}) ) = max Π s l (Y {z, z }) \ ( ( (Y {z}) \ ( Y i(h l 1 (Y {z,z })) Y i(h l 1 (Y {z})) )). )), 51

53 Since i(z ) / i(y ) and i(z) i(z ), we have ( ( (Y {z}) \ Y i(h l 1 (Y {z})) )) ( ( = (Y {z}) \ ( ( (Y {z}) \ ( (Y {z, z }) \ Y i(h l 1 (Y {z}) {z }) Y i(h l 1 (Y {z,z })) )) ( Y i(h l 1 (Y {z,z })) )) )), (13) where the first inclusion follows from the hypothesis that i(h l 1 {z}) {z }). (Y {z, z })) i(h l 1 (Y The inclusion (13) implies that z l Π sl y l. This oservation completes the first part of the induction. Moreover, it quickly yields the second part. To see this, we oserve that if i(z l ) / i(h l 1 (Y {z})), then either z l = z or z l = y l, since z l Π sl y l and i(z ) / i(y {z}). In either case, we have i(h l (Y {z, z })) i(h l (Y {z}) {z }). And finally, if i(z l ) i(h l 1 (Y {z})), then i(h l (Y {z, z })) = (i(h l 1 (Y {z, z })) {i(z l )}) (i(h l 1 (Y {z}) {z }) {i(z l )}) = (i(h l 1 (Y {z})) {i(z )} {i(z l )}) = (i(h l 1 (Y {z})) {i(z )}) (i(h l (Y {z})) {i(z )}) = (i(h l (Y {z}) {z })), so the induction is complete. 45 Now, if z / C (Y {z}), we know that for each s S, the contract y assigned to s in the computation of C (Y {z}) must e higher-priority than z under Π s, that is, yπ s z. The preceding claim then shows that each such s must e assigned a contract y in the computation of C (Y {z, z }) for which y Γ s yπ s z. Thus, we must have z / C (Y {z, z }), contradicting our supposition to the contrary. 45 Here, the first inclusion follows from the inductive hypothesis. 52