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 2 3 4 5 1 2 3 4 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: xi = (xi

1, xi 2, . . . , xi L)

◮ xi

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: xi = (xi

1, xi 2, . . . , xi L)

◮ xi

l = probability that i is assigned job l.

◮ utility function ui(xi) ◮ for ex. ui(xi) can be an exp. utility.

Echenique-Miralles-Zhang Fairness & Efficiency

Pseudomarkets

A lottery xi satisfies

• l

xi

l ≤ 1

A lottery is an element of ∆− = {x ∈ RL

+ : L

• j=1

xj ≤ 1} ui : ∆− → R (cont. & mon.)

Echenique-Miralles-Zhang Fairness & Efficiency

Model

◮ Agents: I = {1, . . . , N}. ◮ Objects: S = {s1, . . . , sL}. ◮ ui : ∆− → R (cont. & mon.)

Echenique-Miralles-Zhang Fairness & Efficiency

Allocations

An allocation is x = (xi)N

i=1, with xi ∈ ∆L −, s.t

• i∈I

xi

s = 1

Echenique-Miralles-Zhang Fairness & Efficiency

Fairness

i envies j at x if ui(xj) > ui(xi) 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 ui(xj) > ui(xi) An allocation x is fair if no agent envies another agent at x. xi = (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 ui(yi) ≥ ui(xi) for all i and uj(yj) > uj(xj) 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 = (ps)s∈S ≥ 0 s.t.

• 1. N

i=1 xi = (1, . . . , 1)

• 2. xi solves

Max {ui(zi) : zi ∈ ∆− and p · zi ≤ 1} Condition (1): supply = demand. Condition (2): xi 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 ui 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 ωi ∈ ∆. ◮ ωi is an initial lottery. ◮ Suppose that

i ωi = (1, . . . , 1).

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 = {s1, . . . , sL}. Suppose N = L. ◮ For each i ∈ I,

◮ ui : ∆− → R ◮ ωi ∈ ∆.

◮

i ωi = (1, . . . , 1).

Echenique-Miralles-Zhang Fairness & Efficiency

Walrasian equilibrium

A Walrasian equilibrium is a pair (x, p) with x ∈ ∆N

−, p ≥ 0 s.t

• 1. N

i=1 xi = N i=1 ωi; and

• 2. xi solves

Max {ui(zi) : zi ∈ ∆− and p · zi ≤ p · ωi}

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

ωi p

Budget set

ωi p (1, 1) simplex

Budget set

ωi no Walras’ Law non-responsive demand

Echenique-Miralles-Zhang Fairness & Efficiency

HZ Example

3 agents; exp. utility u1 u2 u3 sA 10 10 1 sB 1 1 10 Endowments: ωi = (1/3, 2/3).

Echenique-Miralles-Zhang Fairness & Efficiency

HZ Example

3 agents; exp. utility u1 u2 u3 sA 10 10 1 sB 1 1 10 Endowments: ωi = (1/3, 2/3). Obvious allocation: x1 = x2 = (1/2, 1/2) x3 = (0, 1)

Echenique-Miralles-Zhang Fairness & Efficiency

HZ Example

simplex

Echenique-Miralles-Zhang Fairness & Efficiency

HZ Example

1/2 1/2 2/3 1/3 ωi Obvious allocation

Echenique-Miralles-Zhang Fairness & Efficiency

HZ Example

1/2 1/2 2/3 1/3 ωi

Echenique-Miralles-Zhang Fairness & Efficiency

HZ Example

1/2 1/2 2/3 1/3 ωi

Echenique-Miralles-Zhang Fairness & Efficiency

HZ Example

1/2 1/2 2/3 1/3 ωi

Echenique-Miralles-Zhang Fairness & Efficiency

HZ Example

1/2 1/2 2/3 1/3 ωi

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 ui(yi) > ui(xi) for all i ◮ ε-weak Pareto optimal (ε-PO), for ε > 0, if ∃ an allocation y s.t ui(yi) > ui(xi) + ε for all i.

Echenique-Miralles-Zhang Fairness & Efficiency

Property rights

Let x be an allocation. ◮ x is acceptable to i if ui(xi) ≥ ui(ωi). ◮ x is individually rational (IR) if it is acceptable to all agents.

Echenique-Miralles-Zhang Fairness & Efficiency

Justified envy

i envies j at x if ui(xj) > ui(xi). Such envy will be tolerated (i.e not be justified) only if j’s endowment is “good enough.”

Echenique-Miralles-Zhang Fairness & Efficiency

Justified envy

i envies j at x if ui(xj) > ui(xi). Such envy will be tolerated (i.e not be justified) only if j regards xi as unacceptable.

Echenique-Miralles-Zhang Fairness & Efficiency

Justified envy

i envies j at x if ui(xj) > ui(xi). Such envy will be tolerated (i.e not be justified) only if uj(ωj) > uj(xi)

Echenique-Miralles-Zhang Fairness & Efficiency

Justified envy: defn

i has justified envy towards j at allocation x if ui(xj) > ui(xi) and uj(xi) ≥ uj(ωj).

Echenique-Miralles-Zhang Fairness & Efficiency

Fairness

Let x be an allocation. x has no justified envy (NJE) if no agent has justified envy towards any other agent at x.

Echenique-Miralles-Zhang Fairness & Efficiency

Fairness

Observe: NJE and IR imply equal treatment of equals.

Echenique-Miralles-Zhang Fairness & Efficiency

Fairness

Let x be an allocation. x has no justified envy (NJE) if no agent has justified envy towards any other agent at x.

Echenique-Miralles-Zhang Fairness & Efficiency

Justified envy

◮ i has strong justified envy (SJE) towards j at x if ui(xj) > ui(xi) and uj(xi) > uj(ωj). ◮ For ε > 0, i has ε-justified envy (ε-JE) towards j at x if ui(xj) > ui(xi) and uj(xi) > uj(ωj) − ε.

Echenique-Miralles-Zhang Fairness & Efficiency

Justified envy

no ε-justified envy = ⇒ no justified envy = ⇒ no strong just. envy

Echenique-Miralles-Zhang Fairness & Efficiency

Fairness, property rights and efficiency

Theorem

Suppose utility functions are concave.

• 1. ∃ an allocation that is ε-IR, ε-PO and has no ε-justified envy;
• 2. ∃ an allocation that is IR, wPO and has no strong justified

envy.

• 3. Moreover, if utility functions are expected utility ∃ an

allocation that is IR, PO and has no strong justified envy.

Echenique-Miralles-Zhang Fairness & Efficiency

Fairness, property rights and efficiency

Theorem

Suppose utility functions are quasi-concave, and that ♠ Then there exists continuous functions mi : ∆ → R+ and (x, p) = ((xi)I

i=1, p) ∈ (∆I −) × ∆, such that

1.

i xi = i ωi (x is an allocation; or, “supply equals

demand”).

• 2. x is Pareto optimal, individually rational and has no justified

envy.

• 3. xi ∈ argmax{ui(zi) : zi ∈ ∆− and p · zi ≤ mi(p)}

Echenique-Miralles-Zhang Fairness & Efficiency

Constraints

◮ Given as primitive a set AC of allocations. ◮ The feasible allocations. ◮ Assume AC is convex and compact. For example: ◮ Distributional constraints. ◮ Geographical constraints. ◮ etc.

Echenique-Miralles-Zhang Fairness & Efficiency

Constraints

i has an justified envy towards j at an allocation x ∈ AC if ui(xj) > ui(xi), uj(xi) ≥ uj(ωj) and xi↔j ∈ AC.

Echenique-Miralles-Zhang Fairness & Efficiency

Constraints

i, j ∈ I are of equal type if for all x ∈ AC, xi↔j ∈ AC.

Echenique-Miralles-Zhang Fairness & Efficiency

Constraints

Theorem

Suppose agents’ utility functions are concave and that ω ∈ AC.

• 1. For any ε > 0, there exists an allocation that is ε-IR, ε-PO

and has no equal-type ε-justified envy;

• 2. There exists an allocation that is IR, wPO, and has no strong

equal-type justified envy.

Echenique-Miralles-Zhang Fairness & Efficiency

Ideas

Echenique-Miralles-Zhang Fairness & Efficiency

Theorem

Suppose utility functions are concave.

• 1. ∃ an allocation that is ε-IR, ε-PO and has no ε-justified envy;
• 2. ∃ an allocation that is IR, wPO and has no strong justified

envy.

• 3. Moreover, if utility functions are expected utility ∃ an

allocation that is IR, PO and has no strong justified envy.

Echenique-Miralles-Zhang Fairness & Efficiency

Idea

Consider problem Max

• i

λiui(xi) s.t. x is an allocation. Obtain a NJE allocation from this problem by choosing right welfare weights, (λi) ∈ ∆N. (Actual proof uses an approximation to this problem, hence the ε).

Echenique-Miralles-Zhang Fairness & Efficiency

KKM Lemma

(1, 0, 0) (0, 0, 1) (0, 1, 0)

KKM Lemma

(1, 0, 0) (0, 0, 1) (0, 1, 0)

KKM Lemma

(1, 0, 0) (0, 0, 1) (0, 1, 0)

Echenique-Miralles-Zhang Fairness & Efficiency

Theorem

Suppose utility functions are quasi-concave, and that ♠ Then there exists continuous functions mi : ∆ → R+ and (x, p) = ((xi)I

i=1, p) ∈ (∆I −) × ∆, such that

1.

i xi = i ωi (x is an allocation; or, “supply equals

demand”).

• 2. x is Pareto optimal, individually rational and has no justified

envy.

• 3. xi ∈ argmax{ui(zi) : zi ∈ ∆− and p · zi ≤ mi(p)}

♠: ∃l s.t. for any i ∈ I and xi ∈ ∆L

−, decreasing consumption of

any object k = l in favor of l leads to an increase in ui; and ωi

l > 0.

Echenique-Miralles-Zhang Fairness & Efficiency

ei(v, p) = inf{p · x : ui(x) ≥ v}, for p ∈ ∆L and v ∈ R. Let vi = sup ui(∆L

−) be the utility of agent i when she is satiated.

Echenique-Miralles-Zhang Fairness & Efficiency

For any scalar m ≥ 0 and p ∈ ∆L, let µi(m, p) = median({ei(ui(ωi), p), m, ei(vi, p)}). Consider the function ϕ(m, p) =

• i

µi(m, p) −

• i

p · ωi. Observe that ◮ ei(ui(ωi), p) ≤ ei(vi, p). ◮ µi is continuous and m → µi(m, p) weakly monotone increasing. ◮ ϕ is continuous and m → ϕ(m, p) weakly monotone increasing. ◮ ϕ(m, p) ≤ 0 for m ≥ 0 small enough as ei(ui(ωi), p) ≤ p · ωi.

Echenique-Miralles-Zhang Fairness & Efficiency

◮ in the case that

i ei(vi, p) < i p · ωi, we let

mi(p) = ei(vi, p). ◮ in the case that

i ei(vi, p) ≥ i p · ωi, we have that

ϕ(m, p) ≤ 0 for m ≥ 0 small enough, and ϕ(m, p) ≥ 0 for m ≥ 0 large enough. So ∃ m∗ ≥ 0 with ϕ(m∗, p) = 0. Now let mi(p) = µi(m∗, p).

Echenique-Miralles-Zhang Fairness & Efficiency

No justified envy

Suppose that i envies j at x∗. This implies that i is not satiated, hence mi(p∗) < ei(vi, p∗). It also implies that mi(p∗) < mj(p∗) as mi(p∗) < p∗ · xj = mj(p∗). On can then show that, by defn. of mj, mj(p∗) = ej(uj(ωj), p∗).

Echenique-Miralles-Zhang Fairness & Efficiency

No justified envy

We obtain that p∗ · xi = mi(p∗) < mj(p∗) = ej(uj(ωj), p∗), and hence uj(xi) < uj(ωj) by definition of expenditure function. So i’s envy is not justified.

Echenique-Miralles-Zhang Fairness & Efficiency