How to control controlled school choice Federico Echenique M. Bumin - PowerPoint PPT Presentation

How to control controlled school choice Federico Echenique M. Bumin Yenmez Caltech Carnegie Mellon WZB Matching Workshop – Aug 29, 2014

School choice

Example Two schools/colleges: c 1 , c 2 Two students: s 1 , s 2 . c 1 c 2 s 1 s 2 s 1 , s 2 s 1 c 1 c 2 s 2 c 2 c 1 s 1 and s 2 are of different “type” and c 1 must be balanced.

Example Two schools/colleges: c 1 , c 2 Two students: s 1 , s 2 . c 1 c 2 s 1 s 2 s 1 , s 2 s 1 c 1 c 2 s 2 c 2 c 1 s 1 and s 2 are of different “type” and c 1 must be balanced.

Example Two schools/colleges: c 1 , c 2 Two students: s 1 , s 2 . c 1 c 2 s 1 s 2 s 1 , s 2 s 1 c 1 c 2 s 2 c 2 c 1 s 1 and s 2 are of different “type” and c 1 must be balanced.

Example Two schools/colleges: c 1 , c 2 Two students: s 1 , s 2 . c 1 c 2 s 1 s 2 s 1 , s 2 s 1 c 1 c 2 s 2 c 2 c 1 s 1 and s 2 are of different “type” and c 1 must be balanced.

Main results Tension between: ◮ “package” preferences, ◮ “item” preferences, Pure item preferences → GS. Pure package preferences → complements.

Main results ◮ GS + Axioms on how to resolve tension ⇐ ⇒ specific “utility function” (or procedure) for schools. ◮ Implications for matching: some procedures are better for students than others (Pareto ranking of school choice procedures).

Literature ◮ School choice: Abdulkadiro˘ glu and S¨ onmez (2003), Abdulkadiro˘ glu, Pathak, S¨ onmez and Roth (2005) (Boston), Abdulkadiro˘ glu, Pathak and Roth (2005) (NYC) ◮ Controlled school choice: Abdulkadiro˘ glu and S¨ onmez (2003), Abdulkadiro˘ glu (2005), Kamada and Kojima (2010), Kojima (2010), Hafalir, Yenmez, Yildirim (2011), Ehlers, Hafalir, Yenmez, Yildirim (2011), Budish, Che, Kojima, and Milgrom (2011), Erdil and Kurino (2012), Kominers and S¨ onmez (2012), Ayg¨ un and Bo (2013), Westkamp (2013).

One School

The model: primitives ◮ A finite set S of students . ◮ A choice rule C on S . ◮ A strict priority ≻ on S . ◮ Students partitioned into types.

The model: primitives 1. A finite set S of students . 2. A choice rule : C : 2 S \ {∅} → 2 S s.t. C ( S ) ⊆ S . 3. A number q > 0 s.t. | C ( S ) | ≤ q .

The model: primitives 1. A finite set S of students . 2. A choice rule : C : 2 S \ {∅} → 2 S s.t. C ( S ) ⊆ S . 3. A number q > 0 s.t. | C ( S ) | ≤ q . Note: 1. Choice C ( S ) is a “package” 2. Allow C ( S ) = ∅ . 3. q = school capacity.

The model: primitives ◮ A finite set S of students . ◮ A strict priority ≻ on S .

Axioms: Gross substitutes Axiom (Gross Substitutes (GS)) s ∈ S ⊆ S ′ and s ∈ C ( S ′ ) ⇒ s ∈ C ( S ) .

Axioms: Gross substitutes Equivalently: Axiom (Gross Substitutes (GS)) S ⊆ S ′ and s ∈ S \ C ( S ) ⇒ s ∈ S ′ \ C ( S ′ ) . Here: substitutes = absence of complements. When schools satisfy GS, there is a stable matching & the DA algorithm finds one.

Example Two schools/colleges: c 1 , c 2 Two students: s 1 , s 2 . c 1 c 2 s 1 s 2 s 1 , s 2 s 1 c 1 c 2 s 2 c 2 c 1 s 1 / ∈ C c 1 ( { s 1 } ) while s 1 ∈ C c 1 ( { s 1 , s 2 } ).

The model: primitives 1. S is partitioned into students of different types. 2. Set T ≡ { t 1 , . . . , t d } of types , 3. τ : S → T

The model: primitives 1. S is partitioned into students of different types. 2. Set T ≡ { t 1 , . . . , t d } of types , 3. τ : S → T Define function ξ : 2 S → Z d + . Let ξ ( S ) = ( | S ∩ τ − 1 ( t ) | ) t ∈ T ; ξ ( S ) is the type distribution of students in S .

Recall: tension between ◮ “package” preferences, ◮ “item” preferences,

Recall: tension between ◮ “package” preferences, ◮ “item” preferences, When will you admit a high priority student over a low priority student?

Recall: tension between ◮ “package” preferences, ◮ “item” preferences, When will you admit a high priority student over a low priority student? First resolution of this tension: never when of different types. Put package (distributional) preferences first.

Axiom (Monotonicity) ξ ( S ) ≤ ξ ( S ′ ) implies that ξ ( C ( S )) ≤ ξ ( C ( S ′ )) .

Axiom (Within-type ≻ -compatibility) s ∈ C ( S ) , s ′ ∈ S \ C ( S ) and τ ( s ) = τ ( s ′ ) ⇒ s ≻ s ′ .

Ideal point ( S , C , ≻ ) is generated by an ideal point if: Given an ideal z ∗ ∈ Z d + , 1. Chose closest feasible distribution of types to z ∗ . 2. For each type, chose “best” (highest priority) available students.

Ideal point ( S , C , ≻ ) is generated by an ideal point if: ∃ z ∗ ∈ Z d + such that � z ∗ � ≤ q s.t, 1. ξ ( C ( S )) min. Euclidean distance to z ∗ in B ( ξ ( S )) where B ( x ) = { z ∈ Z d + : z ≤ x and | z | ≤ q } ; 2. students of type t in C ( S ) have higher priority than students of type t in S \ C ( S ).

Ideal point Theorem ( S , C , ≻ ) satisfies ◮ GS ◮ Monotonicity ◮ and within-type ≻ -compatibility iff it is generated by an ideal point.

Ideal point rule may be wasteful. Axiom (Acceptance) A student is rejected only when all seats are filled. | C ( S ) | = min {| S | , q } .

Recall: tension between ◮ “package” preferences, ◮ “item” preferences, When will you admit a high priority student over a low priority student?

Recall: tension between ◮ “package” preferences, ◮ “item” preferences, When will you admit a high priority student over a low priority student? Second resolution of tension: some times; depending on the number of students of each type.

t ∈ T is saturated at S if there is S ′ such that | S t | = | S ′ t | with S ′ t \ C ( S ′ ) t � = ∅ .

t ∈ T is saturated at S if there is S ′ such that | S t | = | S ′ t | with S ′ t \ C ( S ′ ) t � = ∅ . Axiom (Saturated ≻ -compatibility) s ∈ C ( S ) , s ′ ∈ S \ C ( S ) and τ ( s ) is saturated at S imply s ≻ s ′ .

Reserves ( S , C , ≻ ) is generated by reserves if: Lower bound on each student type that school tries to fill: “painted seats.” Students compete openly for the unfilled seats.

Reserves ( S , C , ≻ ) is generated by reserves if: ∃ vector ( r t ) t ∈ T ∈ Z d + with � r � ≤ q such that for any S ⊆ S , 1. | C ( S ) t | ≥ r t ∧ | S t | ; 2. if s ∈ C ( S ), s ′ ∈ S \ C ( S ) and s ′ ≻ s , then it must be the case that τ ( s ) � = τ ( s ′ ) and | C ( S ) τ ( s ) | ≤ r τ ( s ) ; and 3. if ∅ � = S \ C ( S ), then | C ( S ) | = q .

Reserves Theorem ( S , C , ≻ ) satisfies ◮ GS, ◮ acceptance, ◮ saturated ≻ -compatibility, iff it is generated by reserves.

◮ Pathak - S¨ onmez ◮ Kominers - S¨ onmez

◮ First assign open seats based on priorities. ◮ Second, assign reserved seats based on priorities. The opposite order to Reserves.

Chicago Ex: ◮ S = { s 1 , s 2 , s 3 , s 4 } ◮ s 1 , s 2 of type 1 ◮ s 3 , s 4 of type 2 ◮ one school with three seats: one reserved for each type and one open. ◮ priorities are s 1 ≻ s 3 ≻ s 4 ≻ s 2 .

Chicago Ex: ◮ S = { s 1 , s 2 , s 3 , s 4 } ◮ s 1 , s 2 of type 1 ◮ s 3 , s 4 of type 2 ◮ one school with three seats: one reserved for each type and one open. ◮ priorities are s 1 ≻ s 3 ≻ s 4 ≻ s 2 . Reserves assign: s 1 , s 3 and s 4 Chicago: s 1 , s 2 and s 3 a violation of saturated ≻ -compatibility

Quotas Achieve diversity by upper bound: | C ( S ) t | ≤ r t

Quotas Achieve diversity by upper bound: | C ( S ) t | ≤ r t New axiom: Axiom Choice rule C satisfies rejection maximality (RM) if s ∈ S \ C ( S ) and | C ( S ) | < q imply for every S ′ such that | S ′ τ ( s ) | ≤ | S τ ( s ) | we have | C ( S ) τ ( s ) | ≥ | C ( S ′ ) τ ( s ) | .

Theorem ( S , C , ≻ ) satisfies ◮ GS, ◮ RM, ◮ demanded ≻ -compatibility, iff it is generated by quotas.

Proofs: Idea is to map C into f : Z d + → Z d + . Translate axioms into properties of f .

Proof sketch: Theorem ( S , C , ≻ ) satisfies ◮ GS ◮ Monotonicity ◮ and within-type ≻ -compatibility iff it is generated by an ideal point. Under Mon, { ξ ( C ( S )) : ξ ( S ) = x } is a singleton. So, map C into a function f : Z d + → Z d + by f ( x ) = { ξ ( C ( S )) : ξ ( S ) = x } .

Proof sketch: So, map C into a function f : Z d + → Z d + by f ( x ) = { ξ ( C ( S )) : ξ ( S ) = x } . Then C satisfies GS iff y ≤ x ⇒ f ( x ) ∧ y ≤ f ( y ) . (proof: . . . )

Proof sketch: So, map C into a function f : Z d + → Z d + by f ( x ) = { ξ ( C ( S )) : ξ ( S ) = x } . Then C satisfies GS iff y ≤ x ⇒ f ( x ) ∧ y ≤ f ( y ) . (proof: . . . ) Then C satisfies GS and Mon iff y ≤ x ⇒ f ( x ) ∧ y = f ( y ) .

Proof sketch: C satisfies GS and Mon iff y ≤ x ⇒ f ( x ) ∧ y = f ( y ) . Let z ∗ = ξ ( C ( S )). For any x , x ≤ ξ ( S ) implies f ( x ) = x ∧ f ( ξ ( S )) = x ∧ z ∗ . A “projection,” hence min. Euclidean distance.

z ∗

z ∗ y

z ∗ y

f ( y ) = z ∗ ∧ y z ∗ y f ( y )

Proof sketch: Theorem ( S , C , ≻ ) satisfies ◮ GS, ◮ acceptance, ◮ saturated ≻ -compatibility, iff it is generated by reserves. Map C into a function f : Z d + → Z d + by � f ( x ) = { ξ ( C ( S )) : ξ ( S ) = x } .