The Double-Edged Sword of Information: Revealed versus Hidden Lotteries in School Choice

1 Introduction

• Because students often cannot rank every school and choose a subset to apply to, their strategies may depend on lottery information.In NYC, students receive lottery-aware guidance, and the DOE began revealing lottery numbers in 2022 for transparency and FOIL compliance.Source: §1 · p. 2
• With hidden lotteries, all equilibria produce the same match rate and social welfare; the same holds for revealed and partial lotteries under additional assumptions.Thus, shifting students between equilibria without changing the information policy cannot improve outcomes.Source: §1 · pp. 2–3
• Revealing the lottery can either raise or lower match rate under heterogeneous preferences, unlike the always-positive result under homogeneous preferences.The paper also finds that revealed versus hidden lotteries can differ in either direction for social welfare, with larger effects possible under heterogeneous preferences.Source: §1 · p. 3
• All six orderings of revealed, hidden, and partial-information lotteries are possible for both metrics under heterogeneous preferences.Even with homogeneous preferences, partial information can yield a lower match rate than both hidden and revealed lotteries.Source: §1 · p. 3
• Lottery information has a double-edged effect: students can choose better strategies, while their competitors’ improved strategies may create matching inefficiencies.The paper attributes the trade-off to strategic responses by both individual students and the wider market.Source: §1 · p. 3
• Partial information may serve as a policy option balancing transparency and matching efficiency, but bounded reasoning can make its effects unintended.The paper encourages scrutiny before disclosure-policy changes and notes that similar non-monotonic effects may apply to exam-based admissions and job scheduling.Source: §1 · pp. 3–4

School choice

• Deferred acceptance is strategy-proof only when students submit complete rankings; application limits can induce strategic behavior, while cognitive biases may also shape school choices.Prior work studies hedging with reach, match, and safety schools and identifies several biases.Source: §2 · p. 4
• Formal comparisons of lottery disclosure remain limited; Huang and Zhang [22] find revealed lotteries improve match rate under identical preferences, while other work studies information structures or NYC lottery data.The related work also distinguishes single lotteries from independent lotteries at each school.Source: §2 · p. 4

Related information timing

• Pre- versus post-exam preference-submission studies treat exam scores as noisy measures of ability, unlike this paper’s lottery number as the true priority ordering.Source: §2 · p. 4

Game theory

• The model is a simultaneous Bayesian game: students do not observe others’ actions, and its restricted signals differ from Bayesian persuasion’s broader information structures.Revealed-lottery strategies also resemble cognitive hierarchy, but worse-priority students face greater reasoning burdens.Source: §2 · p. 5

3 Problem formulation

• The model allows T student types with normalized school valuations, private knowledge of one’s own type, and a commonly known type distribution and valuation profile.Student types are drawn independently from the population distribution.Source: §3 · p. 5
• With one school choice per student and one seat per school, deferred acceptance reduces to serial dictatorship in lottery-number order.Each student receives their most-preferred available school, or remains unmatched if their selected school is unavailable.Source: §3 · pp. 5–6
• Students may know their exact lottery number, know only its ordered block, or have no lottery information; these policies correspond to revealed, partial, and hidden lotteries.Partial information becomes revealed information with singleton blocks and hidden information with one block containing every number.Source: §3 · p. 6

3.2 Matching outcomes under information policies

• The paper evaluates outcomes by match rate, the fraction of students matched, and social welfare, the sum of valuations for matched students.It defines expected metrics over student types and any randomization in mixed strategies.Source: §3 · p. 6, §3.2 · p. 8

3.1 Equilibrium strategies

• Under partial information, students of the same type in the same lottery block use a shared possibly mixed strategy, choosing applications to maximize expected utility in a symmetric Bayesian Nash equilibrium.Strategies can be computed block by block, but the paper notes that this iterative calculation is unrealistic for students to perform in practice.Source: §3.1 · pp. 6–7

3.3 Illustrative example

• The example has three students, three schools, and two types: common type-1 and rare type-2 students both prefer school 1, while both value school 3 negligibly.Type-1 students value school 1 strongly; type-2 students also value school 2.Source: §3.3 · p. 8
• Rare type-2 students choose school 1 when revealed at lottery number 1, choose school 2 under hidden information, and randomize between schools 1 and 2 under partial information.With partial information, they face a better prospect than under a uniform prior but a worse one than knowing they have lottery number 1.Source: §3.3 · p. 9
• Revealing fills every seat, while hidden and partial information leave school 3 empty; yielding can raise welfare by letting higher-valuing type-1 students receive school 1.The example therefore shows that match rate and social welfare need not move together across information policies.Source: §3.3 · p. 9

4.1 Invariance of multiple equilibria

• Symmetric Bayesian Nash equilibria always exist, and under the stated singleton-block condition all equilibria yield the same match rate and welfare, despite equilibrium multiplicity.For any information policy, the number of symmetric equilibria is one or infinite; because outcomes are invariant across equilibria, nudging students between them cannot improve either metric.Source: §4.1 · pp. 10–12

4.2.1 Ratio of match rate

• Under heterogeneous rankings, hidden information can beat revealed information by at most a log factor, whereas revealed information can beat hidden information by a linear factor n.Mildly preferring students may avoid congestion by applying to less desirable schools under hidden information; when everyone strongly prefers the same school, revealing lottery numbers can spread applications and produce a perfect match rate rather than MR(H) = 1/n.Source: §4.2.1 · p. 14

4.2.2 Ratio of social welfare

• The hidden-lottery advantage of Θ(log n) versus the revealed-lottery advantage of a constant ≈2 under homogeneous rankings shifts to Θ(√n) versus O(log2 n) under heterogeneous rankings.Revealing lottery numbers can hurt welfare when students with high priority strategically take matches with low utility; under heterogeneous rankings, common-type students can also block rare-type students from schools that benefit them more.Source: §4.2.2 · p. 15

4.2.3 Difference of social welfare

• Under heterogeneous rankings, both lotteries can yield social welfare linear in n while their attainable welfare difference reaches Θ(n) in either direction.Under homogeneous rankings, revealed-lottery welfare is SW(R) = 1, so maximizing the welfare difference is equivalent to maximizing the ratio.Source: §4.2.3 · p. 16

4.3 Partial information lottery

• Partial information is non-monotonic: under heterogeneous rankings any of the six match-rate orderings is attainable, and under homogeneous rankings any of the six welfare orderings is attainable.Even with homogeneous rankings, partial information may fail to increase match rate over the hidden lottery, although the revealed lottery has MR(R) = 1.Source: §4.3 · pp. 16–17
• Partial lottery information can improve matching when revealing exact lottery numbers hurts, suggesting a way to improve efficiency without sacrificing too much transparency.The passage also notes the reverse case: partial information can hurt when exact information improves matching.Source: §4.3 · p. 17
• Partial information can instead hurt matching when revealing exact lottery numbers improves the outcome.These opposite cases demonstrate non-monotonic effects of the amount of lottery information available.Source: §4.3 · p. 17
• Students’ cognitive biases may lead them to process revealed lottery numbers coarsely, making them behave as if only partial information were available and potentially causing unintended harm.This risk arises when students do not optimally process lottery information.Source: §4.3 · p. 17

5 Conclusion and discussion

• Under stated assumptions, all equilibria yield the same match rate and social welfare, while revealing the lottery can worsen both relative to hiding it.The paper also reports non-monotonic effects from partial information.Source: §5 · p. 17
• Partial disclosure coarsens the true lottery number rather than strategically designing a signal, and is consistent with current NYC practice.Students are often advised to use the first two digits of their 32-character hexadecimal number for a rough percentile estimate.Source: §5 · p. 17
• The authors leave case-specific disclosure guidelines, fairness criteria, richer preference lists, school-specific priorities, and alternative behavioral models as open problems.They also identify historical-data simulations and personalized recommendation tools as valuable directions.Source: §5 · p. 18
• Because school choice is a simultaneous game with no observation of others’ submitted preferences, richer policies using earlier players’ actions remain an open direction for sequential allocation.The authors cite national-park permit allocation as a potential application.Source: §5 · p. 18

A.1 Partial information lottery

• In the partial-information analysis, flow is the probability that a student in a block applies to a school, while admission probabilities account for availability and better-numbered applicants.Availability after earlier blocks and within-block competition are treated separately.Source: §A.1 · pp. 21–22

A.2 Revealed lottery; A.3 Hidden lottery

• The appendix tracks application flows separately under revealed and hidden lotteries, including revealed-lottery flows for students with particular lottery numbers.The hidden-lottery passage identifies the flow as the probability a student applies to a school.Source: §A.2 · p. 22, §A.3 · p. 23

A.4 Connection between revealed and hidden lotteries

• Lemma 4 relates revealed and hidden lotteries by showing that, for a given lottery number, positive hidden-lottery applications within a school set constrain revealed-lottery applications to that set.The set is defined using schools where revealed-lottery admission probability exceeds the hidden-lottery probability by at least a factor c.Source: §A.4 · pp. 24–25

A.5 Construction of economies

• The section constructs two economies in which matching outcomes are better under the hidden lottery than under the revealed lottery.The examples are used to establish that disclosure need not improve outcomes.Source: §A.5 · p. 25

Uniform-prefix economy

• In the uniform-prefix economy, each type applying to its corresponding school is an equilibrium under the hidden lottery.The economy has n types and a uniform prior.Source: §A.5 · p. 26

Common-rare economy

• In the common-rare economy, the hidden-lottery equilibrium sends common and rare types to their respective schools, while revealed-lottery diversion of common students to rare schools reduces social welfare.Those diverted seats could otherwise go to rare types, who value the rare schools more.Source: §A.5 · pp. 26–27

Revealed Lottery

• For the revealed lottery, the example invokes the homogeneous-preference result to obtain its equilibrium and outcome calculations.The supplied passage does not state the resulting match rate or social welfare values.Source: B Derivations for Example 1 · p. 28

Hidden Lottery

• Under the hidden lottery, students of types 1 and 2 apply to their corresponding schools in the verified equilibrium.The passage also introduces the resulting match-rate and social-welfare calculations without giving readable values.Source: B Derivations for Example 1 · p. 28

Partial Information Lottery

• Under partial information, all students optimally apply to school 2 in the described example.The appendix separately derives the resulting match rate and social welfare.Source: B Derivations for Example 1 · p. 29

C Example with multiple equilibria

• In the three-student, two-type example, the hidden lottery admits multiple equilibria, and the full equilibrium set is characterized by linear combinations of two equilibria.The example specifies school valuations and equal type priors.Source: C Example with multiple equilibria · pp. 29–30

D.1 Proof of Theorem 1

• Under a partial-information policy, any two equilibria generate the same match rate and social welfare because their flows coincide and determine both outcomes.The proof establishes identical flows across equilibria, then shows that match rate and welfare depend on those flows.Source: §D.1 · pp. 30, 32

D.2 Proof of Corollary 1

• If two equilibria differ under a partial-information policy, the appendix constructs infinitely many equilibria by mixing their strategies in the earliest differing block.The mixed strategy remains a best response in that block.Source: §D.2 · pp. 32–33

D.3 Proof of Corollary 2

• In the hidden-lottery setting examined for Corollary 2, a unique equilibrium strategy profile corresponds to the unique equilibrium flow.The proof links the strategy profile to the flow on school 1.Source: §D.3 · p. 33

D.4 Proof of Lemma 2

• With homogeneous preferences and lexicographic tie-breaking, students apply in lottery-number order to distinct schools, yielding a match rate of 1.The proof states the equilibrium strategy and its resulting match rate.Source: §D.4 · pp. 33–34

D.5 Proof of Theorem 2

• The appendix states that the revealed-to-hidden match-rate ratio equals n under homogeneous preferences and that the corresponding heterogeneous-preference proof is identical.It also notes that this ratio is at most n because match rate lies between 1/n and 1.Source: §D.5.2 · pp. 34–35

D.5.3 Proof of bound (6c)

• H/R > 1.038 in a constructed economy with heterogeneous preference rankings.The construction analyzes how students with different lottery numbers allocate applications across schools.Source: §D.5.3 · pp. 35, 37, 38
• H/R = O(log n) under heterogeneous preference rankings.The proof compares matches from schools whose revealed-lottery admission probabilities are relatively high with those from the remaining schools.Source: §D.5.3 · p. 35

D.6.1 Proof of bound (8a)

• Under homogeneous rankings, hidden-lottery social welfare is Θ(log n), while revealed-lottery social welfare is 1.The proof establishes matching upper and lower bounds for hidden-lottery welfare.Source: §D.6.1 · p. 40

D.6.2 Proof of bound (8b)

• Under homogeneous rankings, revealed-lottery social welfare is 1, while hidden-lottery welfare can approach n/(2n−1) in the worst case.The proof gives a lower bound for every economy and constructs an equilibrium approaching that bound.Source: §D.6.2 · pp. 40–41

D.6.3 Proof of bound (8c)

• In the common-rare economy, H/R = Ω(√n).The proof also establishes SW(R) = Ω(1) in every economy, then uses the revealed-equilibrium incentive condition to relate welfare contributions under the two lotteries.Source: §D.6.3 · pp. 42, 44, 46

D.6.4 Proof of bound (8d)

• Under heterogeneous rankings, R/H = O(log2 n), proved by bounding school admission-probability ratios across revealed and hidden lotteries.The proof outlines two steps, using those ratios to relate social welfare under the two policies and then bounding the ratios.Source: §D.6.4 · pp. 49, 51
• For each interval Ik, lottery numbers r whose blue squares lie there form Rk, and associated school sets Jr cannot persist across many r, implying |Rk| cannot be very large.Source: §D.6.4 · p. 52

D.7 Proof of Theorem 4

• Under homogeneous rankings, since SW(R)=1, maximizing the social-welfare difference SW*H−R (respectively SW*R−H) is equivalent to maximizing the ratio SW*H/R (respectively SW*R/H).Source: §D.7 · p. 55

D.7.1 Proof of bound (10c)

• Hidden-lottery social welfare exceeds revealed-lottery social welfare by Θ(n) in the constructed common-rare economy.The proof attributes the revealed-lottery loss to common-type students blocking rare-type students at rare schools.Source: §D.7.1 · p. 56
• Under the hidden lottery, each type applies to its preferred school in equilibrium.The revealed-lottery equilibrium is characterized by iteratively choosing a school for each lottery number.Source: §D.7.1 · p. 56
• The revealed-lottery algorithm tracks prior common-type applications and selects a school maximizing expected utility for each lottery number.The counts determine school availability and expected utility; the resulting assignments form an equilibrium.Source: §D.7.1 · pp. 56–57

D.7.2 Proof of bound (10d)

• Revealed-lottery social welfare exceeds hidden-lottery social welfare by Θ(n) under heterogeneous rankings.The proof establishes a linear lower bound alongside the stated linear upper bound.Source: §D.7.2 · p. 59
• Under the hidden lottery, type-1 students apply to school 1 in equilibrium, and the proof extends the welfare calculation across types.The revealed-lottery strategy instead compares expected utility at two schools for each lottery number.Source: §D.7.2 · p. 60

11 end

• The revealed-strategy count difference oscillates around threshold τ_m and ends within [τ_m−1, τ_m+1].The proof uses this bound to characterize the final application counts.Source: §11 · p. 63

Ordering 1: H < P < R

• When H < P < R, the match rates are MR(H) ≈0.493, MR(P) ≈0.666, and MR(R) = 1.Source: §D.8 · p. 63

Ordering 2: P < H < R

• When P < H < R, the match numbers are MN(P) = 1.8092, MN(H) = 1.8448, and MN(R) = 4.Source: §D.8 · p. 63

Partial information lottery

• For the partial-information equilibrium, types 1 and 3 apply to school 1 in block P1, while types 2 and 3 apply to school 2 in block P2.Type 2 applies to school 2 in P1; the reported P2,2 flow is 0.4.Source: §D.8 · p. 64

• The constructions cover all six possible orderings; the economies for Orderings 1 and 2 are the same as those in Appendix D.8.Source: §D.9 · p. 64

Ordering 3: H < R < P

• For Ordering 3, H < R < P, the match numbers are MN(H) = 1.72, MN(R) = 2.224, and MN(P) = 2.48.Source: §D.9 · p. 64

Ordering 4: R < H < P

• For Ordering 4, R < H < P, the reported match numbers are MN(R) ≈1.653, MN(H) ≈1.694, and MN(P) ≈1.959.Source: §D.9 · p. 65

Ordering 5: R < P < H

• For Ordering 5, R < P < H, the match numbers are MN(R) = 1.874, MN(P) = 1.904, and MN(H) = 1.93.Source: §D.9 · p. 65

Ordering 6: P < R < H

• For Ordering 6, P < R < H, the match numbers are MN(P) ≈1.822, MN(R) = 1.8875, and MN(H) ≈1.938.Source: §D.9 · p. 66

Ordering 1: R < H < P

• For Ordering 1, R < H < P, social welfare is SW(R) = 1, SW(H) ≈1.006, and SW(P) ≈1.044.Source: §D.10 · p. 67

Ordering 2: H < R < P

• For Ordering 2, H < R < P, social welfare is SW(H) = 0.954, SW(R) = 1, and SW(P) ≈1.007.Source: §D.10 · p. 68

Ordering 3: H < P < R

• For Ordering 3, H < P < R, social welfare is SW(H) = 0.75, SW(P) = 0.95, and SW(R) = 1.Source: §D.10 · p. 68

Ordering 4: P < H < R

• For Ordering 4, P < H < R, social welfare is SW(P) ≈0.878, SW(H) ≈0.890, and SW(R) = 1.Source: §D.10 · p. 69

Ordering 5: R < P < H

• For Ordering 5, R < P < H, social welfare is SW(R) = 1, SW(P) ≈1.012, and SW(H) ≈1.128.Source: §D.10 · p. 69

Ordering 6: P < R < H

• For Ordering 6, P < R < H, social welfare is SW(P) = 0.99, SW(R) = 1, and SW(H) ≈1.128.Source: §D.10 · p. 69