Fairness and efficiency for probabilistic allocations with - PowerPoint PPT Presentation

Fairness and efficiency for probabilistic allocations with endowments Federico Echenique Antonio Miralles Jun Zhang Caltech Universit` a degli Studi di Messina. Nanjing Audit U. UAB-BGSE National University Singapore, Dec 4 2019

Antonio and Jun: Echenique-Miralles-Zhang Fairness & Efficiency

Discrete allocation 1 1 2 2 3 3 4 4 5 5 Echenique-Miralles-Zhang Fairness & Efficiency

For example ◮ Jobs to workers ◮ Courses to students ◮ Organs to patients ◮ Schools to children ◮ Offices to professors. Echenique-Miralles-Zhang Fairness & Efficiency

Desiderata ◮ Efficiency ◮ Fairness ◮ Property rights. Echenique-Miralles-Zhang Fairness & Efficiency

Efficiency Pareto optimality. An assignment is efficient if there is no alternative (feasible) assignment that makes everyone better off and at least one agent strictly better off. Echenique-Miralles-Zhang Fairness & Efficiency

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 Fairness & Efficiency

Fairness Fairness requires randomization. If Alice and Bob want the same office = ⇒ flip a coin. Echenique-Miralles-Zhang Fairness & Efficiency

Fairness vs. efficiency When there is a conflict between efficiency and fairness, policy makers (and society?) often prioritize fairness. Hence fairness is a priority in market design. So we’ll work with random assignments. Echenique-Miralles-Zhang Fairness & Efficiency

Pseudomarkets Can we be fair and efficient? Yes: use pseudomarkets Echenique-Miralles-Zhang Fairness & Efficiency

Pseudomarkets: Hylland and Zeckhauser 1979 Assign workers to jobs. ◮ L jobs. ◮ A lottery : x i = ( x i 1 , x i 2 , . . . , x i L ) ◮ x i l = probability that i is assigned job l . Echenique-Miralles-Zhang Fairness & Efficiency

Pseudomarkets: Hylland and Zeckhauser 1979 Assign workers to jobs. ◮ L jobs. ◮ A lottery : x i = ( x i 1 , x i 2 , . . . , x i L ) ◮ x i l = probability that i is assigned job l . ◮ utility function u i ( x i ) ◮ for ex. u i ( x i ) can be an exp. utility. Echenique-Miralles-Zhang Fairness & Efficiency

Pseudomarkets A lottery x i satisfies � x i l ≤ 1 l A lottery is an element of L ∆ − = { x ∈ R L � + : x j ≤ 1 } j =1 u i : ∆ − → R (cont. & mon.) Echenique-Miralles-Zhang Fairness & Efficiency

Model ◮ Agents: I = { 1 , . . . , N } . ◮ Objects: S = { s 1 , . . . , s L } . ◮ u i : ∆ − → R (cont. & mon.) Echenique-Miralles-Zhang Fairness & Efficiency

Allocations i =1 , with x i ∈ ∆ L An allocation is x = ( x i ) N − , s.t � x i s = 1 i ∈ I Echenique-Miralles-Zhang Fairness & Efficiency

Fairness i envies j at x if u i ( x j ) > u i ( x i ) An allocation x is fair if no agent envies another agent at x . Echenique-Miralles-Zhang Fairness & Efficiency

Fairness i envies j at x if u i ( x j ) > u i ( x i ) An allocation x is fair if no agent envies another agent at x . x i = (1 / L , . . . , 1 / L ) = ⇒ no envy Echenique-Miralles-Zhang Fairness & Efficiency

Efficiency An allocation x is Pareto optimal (PO) if there is no allocation y s.t u i ( y i ) ≥ u i ( x i ) for all i and u j ( y j ) > u j ( x j ) for some j . Echenique-Miralles-Zhang Fairness & Efficiency

Hylland and Zeckhauser (1979) Echenique-Miralles-Zhang Fairness & Efficiency

Hylland and Zeckhauser (1979) An HZ-equilibrium is a pair ( x , p ), with x ∈ ∆ N − and p = ( p s ) s ∈ S ≥ 0 s.t. i =1 x i = (1 , . . . , 1) 1. � N 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 Fairness & Efficiency

Fairness and efficiency Suppose that each u i is linear (expected utility). Theorem (Hylland and Zeckhauser (1979)) There is a HZ equilibrium allocation. It is envy-free and PO. Echenique-Miralles-Zhang Fairness & Efficiency

This paper: Fair assignment with endowments. Echenique-Miralles-Zhang Fairness & Efficiency

Why endowments? ◮ Endowments are relevant for any problem where we don’t start from scratch. ◮ Existing allocation matters. Want agents to buy into market design, hence respect property rights. Echenique-Miralles-Zhang Fairness & Efficiency

Why endowments? ◮ Endowments are relevant for any problem where we don’t start from scratch. ◮ Existing allocation matters. Want agents to buy into market design, hence respect property rights. ◮ School choice: ◮ Property rights are captured by priorities. ◮ As property rights, priorities are equivocal; not transparent. ◮ Endowments are explicit property rights. ◮ For ex., guarantee a: 1. chance at a good school; 2. neighborhood school; 3. slot for a sibling. Echenique-Miralles-Zhang Fairness & Efficiency

This paper: ◮ Assignment with endowments ◮ Make agents unequal ◮ Confict between no-envy and property rights. Echenique-Miralles-Zhang Fairness & Efficiency

No envy: fairness for equals Echenique-Miralles-Zhang Fairness & Efficiency

Fairness for unequal agents? ◮ Agents have unequal endowments ◮ No envy may violate property rights. Echenique-Miralles-Zhang Fairness & Efficiency

This paper: ◮ We propose a notion of fairness for unequally endowed agents ◮ Prove it can be achieved with efficiency and individual rationality. ◮ Can be obtained as a market outcome. ◮ And respecting general constraint structures. Echenique-Miralles-Zhang Fairness & Efficiency

Related Literature ◮ Mkts. & fairness: Varian (1974), Hylland-Zeckhauser (1979), Budish (2011). ◮ Justified envy w/endowments: Yilmaz (2010) ◮ Allocations with constraints: Ehlers, Hafalir, Yenmez and Yildrim (2014), Kamada and Kojima (2015, 2017). More references in the paper. . . Echenique-Miralles-Zhang Fairness & Efficiency

Fairness among unequals ◮ Each i has an endowment ω i ∈ ∆. ◮ ω i is an initial lottery. i ω i = (1 , . . . , 1). ◮ Suppose that � For example, suppose schools are allocated via a lottery. Admission probabilities reflect: neighborhood school (walk-zone priority), sibling priority, or test scores. Echenique-Miralles-Zhang Fairness & Efficiency

Model ◮ Agents: I = { 1 , . . . , N } . ◮ Objects: S = { s 1 , . . . , s L } . Suppose N = L . ◮ For each i ∈ I , ◮ u i : ∆ − → R ◮ ω i ∈ ∆. i ω i = (1 , . . . , 1). ◮ � Echenique-Miralles-Zhang Fairness & Efficiency

Walrasian equilibrium A Walrasian equilibrium is a pair ( x , p ) with x ∈ ∆ N − , p ≥ 0 s.t i =1 x i = � N 1. � N i =1 ω i ; and 2. x i solves Max { u i ( z i ) : z i ∈ ∆ − and p · z i ≤ p · ω i } Echenique-Miralles-Zhang Fairness & Efficiency

Proposition (Hylland and Zeckhauser (1979)) There are economies in which all agents’ utility functions are expected utility, that posses no Walrasian equilibria. Echenique-Miralles-Zhang Fairness & Efficiency

Budget set p ω i

Budget set (1 , 1) p simplex ω i

Budget set no Walras’ Law non-responsive demand ω i Echenique-Miralles-Zhang Fairness & Efficiency

HZ Example 3 agents; exp. utility u 1 u 2 u 3 s A 10 10 1 s B 1 1 10 Endowments: ω i = (1 / 3 , 2 / 3). Echenique-Miralles-Zhang Fairness & Efficiency

HZ Example 3 agents; exp. utility u 1 u 2 u 3 s A 10 10 1 s B 1 1 10 Endowments: ω i = (1 / 3 , 2 / 3). Obvious allocation: x 1 = x 2 = (1 / 2 , 1 / 2) x 3 = (0 , 1) Echenique-Miralles-Zhang Fairness & Efficiency

HZ Example simplex Echenique-Miralles-Zhang Fairness & Efficiency

HZ Example Obvious allocation 1 / 2 ω i 1 / 3 1 / 2 2 / 3 Echenique-Miralles-Zhang Fairness & Efficiency

HZ Example 1 / 2 ω i 1 / 3 1 / 2 2 / 3 Echenique-Miralles-Zhang Fairness & Efficiency

HZ Example 1 / 2 ω i 1 / 3 1 / 2 2 / 3 Echenique-Miralles-Zhang Fairness & Efficiency

HZ Example 1 / 2 ω i 1 / 3 1 / 2 2 / 3 Echenique-Miralles-Zhang Fairness & Efficiency

HZ Example 1 / 2 ω i 1 / 3 1 / 2 2 / 3 Echenique-Miralles-Zhang Fairness & Efficiency

Moreover, . . . ◮ the first welfare theorem fails. ◮ There are Pareto ranked Walrasian equilibria. Echenique-Miralles-Zhang Fairness & Efficiency

Our results Echenique-Miralles-Zhang Fairness & Efficiency

Preliminary defns Let x be an allocation. ◮ x is weak Pareto optimal (wPO) if � ∃ an allocation y s.t u i ( y i ) > u i ( x i ) for all i ◮ ε -weak Pareto optimal ( ε -PO), for ε > 0, if � ∃ an allocation y s.t u i ( y i ) > u i ( x i ) + ε for all i . Echenique-Miralles-Zhang Fairness & Efficiency

Property rights Let x be an allocation. ◮ x is acceptable to i if u i ( x i ) ≥ u i ( ω i ). ◮ x is individually rational (IR) if it is acceptable to all agents. Echenique-Miralles-Zhang Fairness & Efficiency