1. New questions raised by school choice How to do tie breaking? Tradeoffs between Pareto optimality, stability, strategy proofness—what are the ‘costs’ of each? Evaluating welfare from different points in time Restricted domains of preferences? 1

2. Matching with indifferences When we were mostly using matching models to think about labor markets, strict preferences didn’t seem like too costly an assumption – Strict preferences might be generic But that isn’t the case with school choice – We already saw that one of the first NYC design decisions we faced in 2003 was how to randomize to break ties. 2

3. 3 New Theoretical Issues Erdil, Aytek and Haluk Ergin, What's the Matter with Tie-breaking? Improving Efficiency in School Choice, American Economic Review, 98(3), June 2008, 669- 689What's the Matter with Tie-breaking? Improving Efficiency in School Choice Abdulkadiroglu, Atila, Parag A. Pathak, and Alvin E. Roth, " Strategy-proofness versus Efficiency in Matching with Indifferences: Redesigning the NYC High School Match,'' revised, November, 2008, American Economic Review, forthcoming.Strategy-proofness versus Efficiency in Matching with Indifferences: Redesigning the NYC High School Match Featherstone, Clayton and Muriel Niederle, “ Manipulation in School Choice Mechanisms ”, October 2008. Manipulation in School Choice Mechanisms

4. Other new issues we won’t get to today… Pathak, Parag and Tayfun Sönmez “Leveling the Playing Field: Sincere and Strategic Players in the Boston Mechanism”, American Economic Review, 98(4), 1636-52, 2008“Leveling the Playing Field: Sincere and Strategic Players in the Boston Mechanism”, Ergin, Haluk and Tayfun Sonmez, Games of School Choice under the Boston Mechanism ”, Journal of Public Economics, 90: 215- 237, January 2006.Games of School Choice under the Boston Mechanism ” Kesten, Onur On Two Kinds of Manipulation for School Choice Problems March, 2008.On Two Kinds of Manipulation for School Choice Problems Kesten, Onur, " An Alternative Mechanism Design Approach to School Choice in the United States," March, 2008.An Alternative Mechanism Design Approach to School Choice in the United States Abdulkadiroglu, Atila, Yeon-Koo Che, and Yosuke Yasuda, " Expanding ‘Choice’ in School Choice,, " November, 2008. Expanding ‘Choice’ in School Choice Kesten and Unver—random matchings--in progress... Sotomayor, Marilda “The Pareto-Stability Concept … for Discrete Matching Markets with Indifferences,” working paper, August 2008 4

5. Matching with indifferences I: a finite set of students (individuals) with (strict) preferences P i over school places. S: a finite set of schools with responsive weak preferences/priorities R s over students (i.e. can include indifferences: P s ( ≻ s ) is the asymmetric part of R s ). As before: q = (q s ) sєS : a vector of quotas (q s ≥ 1, integer). A matching is a correspondence μ: I U S → S U I satisfying: (i) For all i є I : μ(i) є S U {i} (ii) For all s є S : |μ(s)| ≤ q s, and i ∈  (s) implies μ(i) = s. We’ll mostly concentrate on student welfare and student strategy, and regard R S as fixed. 5

6. Matchings and student welfare A matching μ is individually rational if it matches every x ∈ I ∪ S with agent(s) that is(are) acceptable for x. A matching μ is blocked by (i, s) if sP i μ(i), and either [|μ(s)| < q s and i ≻ s s] or [i ≻ s i′ for some i′ ∈ μ(s)]. μ is stable if μ is individually rational and not blocked by any student-school pair (i, s). A matching μ dominates matching if μ(i)R i (i) for all i ∈ I, and μ(i)P i (i) for some i ∈ I. (Weak Pareto domination for students.) A stable matching μ is a student-optimal stable matching if it is not dominated by any other stable matching. “A” not “the”: When school preferences aren’t strict, there won’t generally be a unique optimal stable match for each side, rather there will be a non-empty set of stable matches that are weakly Pareto optimal for agents on that side. 6

7. Example: multiple optimal stable matchings 7

8. Weak Pareto optimality generalizes… Proposition 1. If μ is a student-optimal stable matching, there is no individually rational matching (stable or not) such that (i)P i μ(i) for all i ∈ I. (terminology: a student optimal stable matching is weakly Pareto optimal because it can’t be strictly Pareto dominated, but the outcome of student proposing deferred acceptance algorithm might not be strongly Pareto optimal, i.e. might not be student optimal, because it can be weakly Pareto dominated) 8

9. Tie breaking A tie-breaker is a bijection r:I→N, that breaks ties at school s by associating R s with a strict preference relation P s : iP s j ⇔ [(i ≻ s j) or (i ∼ s j and r(i) < r(j))]. 9

10. 10 Step 0: arbitrarily break all ties in preferences Step 1: Each student “proposes” to her first choice. Each school tentatively assigns its seats to its proposers one at a time in their priority order. Any remaining proposers are rejected. … Step k: Each student who was rejected in the previous step proposes to her next choice if one remains. Each school considers the students it has been holding together with its new proposers and tentatively assigns its seats to these students one at a time in priority order. Any remaining proposers are rejected. The algorithm terminates when no student proposal is rejected, and each student is assigned her final tentative assignment. Basic Deferred Acceptance (Gale and Shapley 1962)

11. Deferred acceptance algorithm with tie breaking: DA τ A single tie breaking rule uses the same tie- breaker r s = r at each school, while a multiple tie breaking rule may use a different tie breaker r s at each school s. For a particular set of tie breakers τ=(r s ) s ∈ S, let the mechanism DA τ be the student-proposing deferred acceptance algorithm acting on the preferences (P I,P S ), where P s is obtained from R s by breaking ties using r s, for each school s. 11

12. Single and Multiple tie breaking The dominant strategy incentive compatibility of the student-proposing deferred acceptance mechanism for every student implies that DA τ is strategy-proof for any τ. But the outcome of DA τ may not be a student optimal stable matching. – We already saw this is true even for single tie breaking. 12

13. Single versus multiple tie breaking (NYC Grade 8 applicants in 2006-07) 13

14. Proposition: For any (P I,R S ), any matching that can be produced by deferred acceptance with multiple tie breaking, but not by deferred acceptance with single tie breaking is not a student-optimal stable matching. 14

15. Dominating stable matchings Lemma: Suppose μ is a stable matching, and ν is some matching (stable or not) that dominates μ. Then the same set of students are matched in both ν and μ 15

16. Proof If there exists a student who is assigned under μ and unassigned under ν, then ν(i)=iP i μ(i), which implies that μ is not individually rational, a contradiction. So every i assigned under μ is also assigned under ν. Therefore |ν(S)|≥|μ(S)|. If |ν(S)|>|μ(S)| then there exists some s ∈ S and i ∈ I such that |ν(s)|>|μ(s)| and ν(i)=s≠μ(i). This implies there is a vacancy at s under μ and i is acceptable for s. Furthermore, sP i μ(i) since ν dominates μ. These together imply that μ is not stable, a contradiction. So |ν(S)|=|μ(S)|. Then the same set of students are matched in both ν and μ since |ν(S)|=|μ(S)| and every student assigned under μ is also assigned under ν. 16

17. Stable Improvement Cycles (Erdil and Ergin, 08) Fix a stable matching μ w.r.t. given preferences P and priorities R. Student i desires s if sP i μ(i). Let B s = the set of highest R s -priority students among those who desire school s. Definition: A stable improvement cycle C consists of distinct students i 1,..., i n = i 0 (n ≥ 2) such that (i) μ(i k ) є S (each student in the cycle is assigned to a school), (ii) i k desires μ(i k+1 ), and (iii) i k є Bμ(i k+1 ), for any ) k = 0,..., n − 1. Given a stable improvement cycle define a new matching μ’ by:  ’(j) = μ(j) if j is not one of {i 1,..., i n }  ’(j) = μ(i k+1 ) if j = i k Proposition: μ’ is stable and it (weakly) Pareto dominates μ. 17

18. Improving on DA τ Theorem (Erdil and Ergin, 2008): Fix P and R, and let μ be a stable matching. If μ is Pareto dominated by another stable matching, then μ admits a stable improvement cycle. Algorithm for finding a student optimal matching: start with a stable matching. Find and implement a stable improvement cycle, as long as one exists. 18

19. Outline of proof Fix P and R. Suppose μ is a stable matching Pareto dominated by another stable matching. Simplifying assumption: Each school has one seat. 1. I’ := {i є I | (i)P i μ(i)} = {i є I | (i) ≠ μ(i)}. 2. All students in I’ are matched to a school at. 3. S’ := (  ’)=μ(I’). Hence, I [S] can be partitioned into two subsets I’ and I\I’ [S’ and S \ S’] such that Those in I \ I’ [S \ S’] have the same match under μ and. The matches of those in I’ [S’] have been “shuffled” among themselves to obtain from μ. 19

20. 4. For all s є S’: I’ s := (i є I’|i desires s at μ, and no j є I’ desires s at μ and j P s i) is nonempty;. 5. Construct a directed graph on S’: For each s є S’, arbitrarily choose and fix i s є I’ s. i s є B s : i.e., i s desires s at μ, and there is no j є I who desires s at μ and j P s i. (from stability of ) For all s, t є S’, let t →s if t = μ(i s ). 6. The directed graph has a cycle of n ≥ 2 distinct schools: s 1 → s 2 → · · → s n → s 1 7. The students i s1, i s2,..., i sn constitute a stable improvement cycle at μ 20

21. How much room is there to improve on deferred acceptance? Are there costs to Pareto improvements in welfare? 21

22. Strategy-proof mechanisms A direct mechanism φ is a function that maps every (PI,R S ) to a matching. For x ∈ I ∪ S, let φ x (P I ;R S ) denote the set of agents that are matched to x by φ. A mechanism φ is dominant strategy incentive compatible (DSIC) for i ∈ I if for every (P I,R S ) and every P′ i, φ i (P I ;R S )R i φ i (P′ i, P −i ;R S ). A mechanism will be called strategy-proof if it is DSIC for all students. 22

23. Pareto improvement and strategy proofness Fix R S. We say that a mechanism φ dominates ψ if for all P I : φ i (P I ;R S )R i ψ i (P I ;R S ) for all i ∈ I, and for some P I : φ i (P I ;R S )P i ψ i (P I ;R S ) for some i ∈ I. Theorem (Abdulkadiroglu, Pathak, Roth): For any tie breaking rule τ, there is no mechanism that is strategy-proof and dominates DA τ. 23

24. Proof Suppose that there exists a strategy-proof mechanism ϕ and tie-breaking rule r such that ϕ dominates DA τ. There exists a profile P I such that ϕ i (P I ;R S )R i DA τ (P I ;R S ) for all i ∈ I, and ϕ i (P I ;R S )P i DA τ (P I ;R S ) for some I ∈ I. Let s i =DA i τ (P I ;R S ) and s’ i =ϕi(P I ;R S ) be i's assignment under DA τ (P I ;R S ) and ϕ(P I ;R S ), respectively, where s’ i P i s i. 24

25. …continued Consider profile P I ′=(P i ′,P -i ), where P i ′ ranks s’ i as the only acceptable school. Since DA τ is strategy-proof, s i =DA i τ (P I ;R S )R i DA i τ (P I ′;R S ), and since DA i τ (P I ′;R S ) is either s’ i or i, we conclude that DA i τ (P I ′;R S )=i. Then the Lemma implies ϕ i (P I ′;R S )=i. Now let (P I ′;R S ) be the actual preferences. In this case, i could state P i and be matched to ϕ i (P I ;R S )=s’ i, which under P i ′ she prefers to ϕ(P I ′;R S )=i. So ϕ is not strategy-proof. 25

26. Let’s look at some data We can’t tell what preferences would have been submitted with a different (non strategy- proof) mechanism, but we can ask, given the preferences that were submitted, how big an apparent welfare loss there might be due to not producing a student optimal stable matching. 26

27. Inefficiency in the NYC match (cost of strategy-proofness) 27

28. Cost of stability in NYC 28

29. Comparison with Boston 29

30. Open questions (Equilibrium) misrepresentation in stable improvement cycles? (Can potential gains be realized?) – It appears there will be an incentive to raise popular schools in your preferences, since they become tradeable endowments… Restricted domains of preference? – Manipulation will be easier on some domains than others, and potential welfare gains greater on some domains than others. 30

31. Ex post versus ex ante evaluation? E.g. Boston mechanism in uncorrelated environment, where you don’t have to pay the cost for lack of strategy proofness…Featherstone and Niederle 2008 Recall that DA is strategy-proof (DSIC) while the Boston mechanism is not. (The following slides are adapted from F&N’s) 31

32. Example; correlated preferences (likely the general case…) 32

33. Boston mechanism in the correlated environment—complex eq. strategies 33

34. Uncorrelated preferences: (a conceptually illuminating simple environment) 34 Consider a student after he knows his own type, and before he knows the types of the others. Then (because the environment is uncorrelated) his type gives him no information about the popularity of each school. So, under the Boston mechanism, truthtelling is an equilibrium. (Note that for some utilities this wouldn’t be true e.g. of the school-proposing DA, even in this environment.) 2 schools, one for Art, one for Science, each with one seat 3 students, each iid a Scientist with p=1/2 and Artist with p=1/2. Artists prefer the art school, scientists the science school. The (single) tie breaking lottery is equiprobable over all orderings of the three students.

35. Boston can stochastically dominate DA in an uncorrelated environment Example: 3 students, 2 schools each with one seat 35

36. Things to note The uncorrelated environment let’s us look at Boston and DA in a way that we aren’t likely to see them in naturally occurring school choice. In this environment, there’s no incentive not to state preferences truthfully in the Boston mechanism, even though it isn’t a dominant strategy. (So on this restricted domain, there’s no corresponding benefit to compensate for the cost of strategyproofness.) Boston stochastically dominates DA, even though it doesn’t dominate it ex-post (ex post the two mechanisms just redistribute who is unassigned) 36

37. Recap: New questions raised by school choice How to do tie breaking? Tradeoffs between Pareto optimality, stability, strategy proofness—what are the ‘costs’ of each? Evaluating welfare from different points in time Restricted domains of preferences? 37