Constrained Pseudo-market Equilibrium Federico Echenique Antonio Miralles Jun Zhang Caltech Universit` a degli Studi di Messina. Nanjing Audit U. UAB-BGSE Essex, May 14 2020
Antonio and Jun: Echenique-Miralles-Zhang Pseudomkts with constraints
Allocation problems ◮ Agents (w/ their preferences) ◮ Objects (goods, resources, “bads”) ◮ Who is allocated what? Echenique-Miralles-Zhang Pseudomkts with constraints
Allocation problems For example: ◮ Jobs to workers ◮ Courses to students ◮ Chores to family members. ◮ Organs to patients ◮ Schools to children ◮ Offices to professors. Echenique-Miralles-Zhang Pseudomkts with constraints
Normative desiderata ◮ Efficiency ◮ Fairness ◮ Property rights Echenique-Miralles-Zhang Pseudomkts with constraints
Efficiency Pareto optimality. An assignment is efficient if there is no alternative assignment that makes everyone better off and at least one agent strictly better off. Echenique-Miralles-Zhang Pseudomkts with constraints
Fairness Alice envies Bob at an assignment if she would like to have what Bob got. An assignment is fair if no agent envies another agent. Echenique-Miralles-Zhang Pseudomkts with constraints
Fairness When objects are indivisible, fairness requires randomization. If Alice and Bob want the same object = ⇒ flip a coin. Echenique-Miralles-Zhang Pseudomkts with constraints
Pseudomarkets ◮ Provide agents with a fixed budget in “Monopoly money.” ◮ Allow purchase of (fractions of) objects at given prices. Echenique-Miralles-Zhang Pseudomkts with constraints
Hylland-Zeckhauser (1979) Assign workers to jobs. An economy is a tuple Γ = ( I , L , ( u i ) i ∈ I ), where ◮ I is a finite set of agents ; ◮ L is the number of objects . ◮ Suppose L = | I | . ◮ u i : ∆ − = { x ∈ R L + : � l x l ≤ 1 } → R is i ’s utility function . An assignment is x = ( x i ) i ∈ I with x i ∈ ∆ − and � i x i ≤ 1 = (1 , . . . , 1). Echenique-Miralles-Zhang Pseudomkts with constraints
Hylland and Zeckhauser (1979) An HZ-equilibrium is a pair ( x , p ), with x ∈ ∆ N − and p = ( p l ) l ∈ [ L ] ≥ 0 s.t. 1. � N i =1 x i = (1 , . . . , 1) = 1 2. x i solves Max { u i ( z i ) : z i ∈ ∆ − and p · z i ≤ 1 } Condition (1): supply = demand. Condition (2): x i is i ’s demand at prices p and income = 1. Observe: ◮ Income is independent of prices ◮ Not a “closed” model (Monopoly money). Echenique-Miralles-Zhang Pseudomkts with constraints
Fairness and efficiency Suppose that each u i is linear (expected utility). Theorem (Hylland and Zeckhauser (1979)) There is an efficient HZ equilibrium. All HZ equilibrium assignments are fair. Rmk: w/endowments eqm. may not exist. Echenique-Miralles-Zhang Pseudomkts with constraints
This paper: ◮ Focus on allocation via pseudomarkets. ◮ With general constraints . ◮ With endowments . Echenique-Miralles-Zhang Pseudomkts with constraints
Example: Rural hospitals ◮ Agents: doctors ◮ Objects: positions in hospitals ◮ Constraints: each doctor gets at most one position. ◮ Constraints: UB on available positions. ◮ Constraints: LB on number of doctors/region. Problem: Some hospitals are undesirable. Challenge is to meet the LB on certain regions. Solution: “price” UB so that most desirable hospitals are too expensive. Demand “overflows” to meet the LB on undesirable hospitals. Echenique-Miralles-Zhang Pseudomkts with constraints
Example: Course bidding in B-schools ◮ Agents: MBA students. ◮ Objects: Courses. ◮ Constraints: UB on course enrollment. ◮ Constraints: LB on mandatory courses. Problem: Want efficiency; reflect student pref Solution: “price” UB so that most desirable courses are expensive. Demand “overflows” to meet the LB on less desirable. Properties: efficiency and fairness. Echenique-Miralles-Zhang Pseudomkts with constraints
Example: Roomates in college ◮ Agents: students ◮ Objects: students ◮ Constraints: At most one roommate (= “unit demand”). ◮ Constraints: symmetry ( i ’s purchase of j = j ’s purchase of i ). Problem: Non-existence of stable matchings. Equilibrium (a form of stability) + efficiency. Echenique-Miralles-Zhang Pseudomkts with constraints
Example: Endowments ◮ Agents: faculty. ◮ Objects: office. ◮ Constraints: Exactly one office for each faculty. ◮ Status quo: offices are currently assigned. New challenge: existing tenants must buy into the re-assignment = ⇒ individual rationality constraints. Echenique-Miralles-Zhang Pseudomkts with constraints
Example: School choice ◮ Agents: children. ◮ Objects: slots in schools. ◮ Constraints: unit demand and school capacities. ◮ Endowment: neighborhood school (or sibling priority; etc.) New challenge: Respect option to attend neighborhood school = ⇒ individual rationality constraints. Echenique-Miralles-Zhang Pseudomkts with constraints
The approach we don’t take: ◮ Max SWF (e.g utilitarian) subject to constraints. ◮ Outcome can be decentralized (think 2nd Welfare Thm). ◮ Dual variables → prices. The decentralization will involve endogenous taxes/transfers. No hope of getting IR or fairness. Echenique-Miralles-Zhang Pseudomkts with constraints
Related Literature ◮ Mkts. & fairness: Varian (1974), Hylland-Zeckhauser (1979), Budish (2011). ◮ Allocations with constraints: Ehlers, Hafalir, Yenmez and Yildrim (2014), Kamada and Kojima (2015, 2017). ◮ Markets and constraints: Kojima, Sun and Yu (2019), Gul, Pesendorfer and Zhang (2019). ◮ Endowments: Mas-Colell (1992), He (2017) , and McLennan (2018). (Many) more references in the paper. . . Echenique-Miralles-Zhang Pseudomkts with constraints
Definitions ◮ A pair ( a , b ) ∈ R n × R defines a linear inequality a · x ≤ b . ◮ A linear inequality ( a , b ) has non-negative coefficients if a ≥ 0. ◮ A linear inequality ( a , b ) defines a (closed) half-space : { x ∈ R n : a · x ≤ b } . Echenique-Miralles-Zhang Pseudomkts with constraints
Definitions ◮ A polyhedron in R n is a set that is the intersection of a finite number of closed half-spaces. ◮ A polytope in R n is a bounded polyhedron. ◮ Two special polytopes are the simplex in R n : L ∆ n = { x ∈ R L � + : x l = 1 } , l =1 and the subsimplex L ∆ n − = { x ∈ R L � + : x l ≤ 1 } . l =1 ◮ When n is understood, we use the notation ∆ and ∆ − . Echenique-Miralles-Zhang Pseudomkts with constraints
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a 1 a 2 x 2 x 1 x 3 a 5 C x 5 a 3 x 4 a 4 Echenique-Miralles-Zhang Pseudomkts with constraints
Preliminary defns A function u : ∆ − → R is ◮ concave if ∀ x , z ∈ ∆ − , and ∀ λ ∈ (0 , 1), λ u ( z ) + (1 − λ ) u ( x ) ≤ u ( λ z + (1 − λ ) x ); ◮ quasi-concave if, ∀ x ∈ ∆ − , { z ∈ ∆ − : u ( z ) ≥ u ( x ) } is a convex set. ◮ semi-strictly quasi-concave if ∀ x , z ∈ ∆ − , u ( z ) < u ( x ) and λ ∈ (0 , 1) = ⇒ u ( z ) < u ( λ z + (1 − λ ) x ) ◮ expected utility if it is linear. Echenique-Miralles-Zhang Pseudomkts with constraints
The economy An economy is a tuple Γ = ( I , O , ( Z i , u i ) i ∈ I , ( q l ) l ∈ O ), where ◮ I is a finite set of agents ; ◮ O is a finite set of objects , with L = | O | ; ◮ Z i ⊆ R L + is i ’s consumption space ; ◮ u i : Z i → R is i ’s utility function ; ◮ q l ∈ R ++ is the amount of l ∈ O . Echenique-Miralles-Zhang Pseudomkts with constraints
Assignments An assignment in Γ is a vector x = ( x i , l ) i ∈ I , l ∈ O with x i ∈ Z i . A denotes the set of all assignments in Γ. x ∈ A is deterministic if ( ∀ i , j )( x i , l ∈ Z + ). Echenique-Miralles-Zhang Pseudomkts with constraints
Constraints Constraints are often imposed on deterministic assignments. For example: ◮ unit-demand constraints require � l ∈ O x i , l ≤ 1 ∀ i ∈ I ◮ supply constraints require � i ∈ I x i , l ≤ q l ∀ l ∈ O . Echenique-Miralles-Zhang Pseudomkts with constraints
Constraints Floor constraints may be used to capture distributional objectives. For example: ◮ A minimum number of doctors to be assigned to hospitals in rural areas, ◮ Lower bound on the number minority students that are assigned to a particular school. ◮ All students take at least two math courses. Echenique-Miralles-Zhang Pseudomkts with constraints
Constraints A deterministic assignment is feasible if it satisfies all exogenous constraints. An (random) assignment is feasible if it belongs to the convex hull of feasible deterministic assignments. The convex hull is a polytope since the number of feasible deterministic assignments is usually bounded, and therefore finite. Echenique-Miralles-Zhang Pseudomkts with constraints
Constraints We do not start from an explicit model of constraints. We introduce constraints implicitly through a primitive nonempty set C ⊆ A . The elements of C are the feasible assignments . Echenique-Miralles-Zhang Pseudomkts with constraints
Constrained allocation problems A constrained allocation problem is a pair (Γ , C ) in which ◮ Γ is an economy and ◮ C ⊆ A , a polytope, is the set of feasible assignments in Γ. Echenique-Miralles-Zhang Pseudomkts with constraints
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