endogenous matching: adverse selection & moral...

MotivationNature of the Problem

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Where do we go from here?

Endogenous Matching:

Adverse Selection & Moral Hazard

On-Demand

Mihaela van der Schaar, Yuanzhang Xiao, William Zame

IPAMJuly 2015

Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR


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Economic Motivation

Technology (computers, internet, smartphones . . . ) has maderevolution in provision/delivery of

goods to consumers: Amazon, etc.

services to consumers: Uber, TaskRabbit, etc.



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Also making revolution in provision/delivery of services tofirms

surveys: Amazon Mechanical Turk

software: TopCoder

consulting: Capgemini, Eden McCallum

outsourcing

On-demand economy



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Issues

“On-demand companies . . . have difficulties training, managingand motivating workers.” (Economist Jan 3, 2015)



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Two-sided Matching with Transferable Utility

w ∈ W ↔ t ∈ T

buyers ↔ sellers

workers ↔ firms/jobs

men ↔ women

Output/Value Y (w , t)

Objective: match W with T to maximize total output∑match

Y (w , t)



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Linear Program: Primal

Let

ω = counting measure on W

τ = counting measure on T

Choose (positive) measure θ on W × T to maximize∫Y (w , t)dθ

subject to

θW ≤ ω

θT ≤ τ



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Linear Program: Dual

Choose functions ϕ : W → [0,∞), ψ : T → [0,∞) tominimize ∫

ϕ(w)dω +

∫ψ(t)dτ

subject toϕ(w) + ψ(t) ≥ Y (w , t)



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Interpretation: Housing Market

θ matching of buyers and sellers

ψ(t) is the price paid for house of seller t

ϕ(w) is the utility surplus obtained by buyer w



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Interpretation: Job Market

θ matching of workers and jobs

ϕ(w) is the wage paid to worker w

ψ(t) is the utility surplus remaining to owner of job t



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Competitive Formulation

Competitive environment

Many buyers, many sellers

Many workers, many jobs/firms

Competition drives matching and prices/wages

W ,T compact metric spaces; ω, τ non-atomic measuresY : W × T → [0,∞) is continuous

Many near-perfect substitutes for each buyer (worker)

Many near-perfect substitutes for each seller (job/firm)



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Example

W = T = [0, 1]

ω = τ = Lebesgue measure

Y (w , t) = wt supermodular



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Example: Optimal Matching

Supermodularity ⇒ Matching is assortative– better buyers are matched to better sellers

w1 ↔ t1,w2 ↔ t2,w1 < w2 ⇒ t1 ≤ t2

Optimality ⇒w1t1 + w2t2 ≥ w1t2 + w2t1

w1t1 − w1t2 ≥ w2t1 − w2t2

w1(t1 − t2) ≥ w2(t1 − t2)

t1 ≤ t2

⇒ unique optimal matchingθ = normalized Lebesgue measure on diagonal



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Example: Prices – Solving for ψ(t)

Suppose buyer w buys house from seller tpays price ψ(t)obtains net utility f (t) = wt − ψ(t)

Optimal matching = diagonal→ w buys house from seller w→ f (t) maximized when t = wFirst Order Condition

f ′(t)|t=w = 0

ψ′(t) = t

ψ(t) = (1/2)t2 + const

initial condition ψ(0) = 0 → const = 0



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More generally: W = [w0, 1],T = [t0, 1]

w0 < t0: more buyers than housesbuyers in [w0, t0) do not buy house, get 0 utilitydetermines initial conditionprices determinatew0 > t0: more houses than buyershouses in [t0,w0) do not sell, priced at 0determines initial conditionprices determinatew0 = t0price of house t0 indeterminateprice of house t0 determines initial conditioncompetition + price of house t0 determines all prices



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What is left out in this model

(and other matching models)?

Housing market: characteristics may not be observable→ adverse selection

Job market/marriage market: output must be producedProduction requires effortEffort is unobservable & costly→ moral hazard

Job market: Interaction is repeated/ongoing



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Setting I (future work)

Platform (Uber, TaskRabbit, etc.)

one platform

many clients arrive each period, each with a task and apayment schedule

clients matched to workers

workers choose effort, perform task, get paid

platorm seeks to maximize commission

ongoing



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Setting II (this paper)

Firm outsourcing

one firm

firm has many tasks to be performed each period

outsourced to many workers

firm sets payment schedule matches workers to tasks

workers choose effort, perform task

firm seeks to maximize profit = total output - payments

ongoing



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Who knows/observes what?

Firm knows/observes

Task types

History of output

Firm does not know/observe

Characteristics of workers

Effort

Firm cannot disentangle“what happened?” from “why did it happen?”



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Who knows/observes what? (cont)

Workers know/observe

Task types

Own characteristics

Own history

Output distribution

Workers do not know/observe

Other workers’ characteristics



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Essence of the Problem

Adverse selection

Firm does not observe worker characteristics

Moral hazard

Firm does not observe worker effort



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Repeated interaction

Endogenous matchingMatch better producers to better tasks

assortative matching → optimal matchingeliminate adverse selection

mitigate moral hazard

profit comparisons

random matching, fixed payment scheduleour solutionincentive optimum with assortative matching (2nd best)



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Model

WorkersW = [0, 1]uniformly distributed (without essential loss of generality)workers identified by location in distributionCost C (e,w) : [0,∞)×W → [0,∞) smooth

C (0,w) = 0

C2 ≤ 0

C1 > 0 if e > 0,w > 0

C1 = 0 if e = 0

C11 > 0

C12 ≤ 0Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR


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Model (cont)

Tasks

T = [0, 1]

uniformly distributed (without essential loss of generality)tasks identified by location in distribution



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Model (cont)

Output Y smooth

Y (e,w , t) : [0,∞)×W × T → [0,∞)

ewt = 0 ⇒ Y (e,w , t) = 0

ewt 6= 0 ⇒ Y1,Y2,Y3 > 0

Y (e,w , t)→∞ as e →∞ if (w , t) > (0, 0)

Y is strictly supermodular in each pair of variables

Y11 ≤ 0

messy technical conditions(multiplicatively separable in task enough)



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Model (cont)

Payment schedule P : [0,∞)→ [0,∞) smooth

P(y) ≤ y

P ′(y) > 0

P ′′(y) ≤ 0

Utility of worker w matched to task t, exerts effort e:

U(e,w , t) = P[Y (e,w , t)]− C (e,w)



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Canonical Example

OutputY (e,w , t) = ewt

Payment ruleP(y) = λy

CostC (e) = e2

Worker utility

U(e,w , t) = λewt − e2



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Firm Objective

Maximize

profit = total output - total payment to workers



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Stationary Assortative Equilibrium

Equilibrium notion:Stationary (steady-state)Assortative: match better workers to better tasksMatching rule: rank in output distributionWorker strategy: effort g , output G

g : W × T × history → [0,∞)

G : W × T × history → [0,∞)

Workers discount future utility at constant rate δ ∈ (0, 1]Workers optimize (given that others play equilibrium )Stationarity → G depends only on w on equilibrium path

G : W → [0,∞)

G completely determines equilibriumMihaela van der Schaar, Yuanzhang Xiao, William Zame CR


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Main Results - 1

Theorem 1

Stationary Assortative Equilibrium exists and is unique.

The output distribution G is continuously differentiable.

Better workers are matched with better tasks, producehigher output, receive higher utility (but may not exertgreater effort).



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Notes:

No adverse selection at equilibrium.

Wages = P(G (w)) endogenous



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Main Results - 2

Theorem 2 Payment schedule linear ⇒Rankings of firm profit (output net of payments)

Profit(random matching, workers exert optimaleffort given fixed payment schedule)

< Profit(stationary assortative equilibrium,fixed payment schedule)

< Profit(assortative matching, firm usesincentive-optimal payment schedule)



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Firm has two instruments to provide incentives

pays for current output

conditions future assignments on current output



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Canonical example

Y (e,w , t) = ewt

C (e) = e2

P(y) = λy

U(Y (e,w , t)) = λewt − e2

Solve for G ?



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Stationary deviation: worker w pretends to be worker w

today: w matched to task w , produces output G (w)

future: w matched to task w , produces output G (w)

U(w |w) = λG (w)− [G (w)/w 2]2

+δ/(1− δ){λG (w)− [G (w)/ww ]2]

}Equilibrium → optimum occurs when w = wFOC for equilibrium: dU/dw = 0 when w = w

G ′ =2δG 2

[2G − λw 4]w= Φ(w ,G )

Initial condition: G (0) = 0Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR


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This is an ODE that we have never seen before but . . .

- it is just a first order ODE

- how bad can it be?



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Guess G (w) = Aw 4

Plug in and equate

4Aw 3 =2δA2w 8

w [2Aw 4 − λw 4]

Eliminate w

4A =2δA2

[2A− λ]

Two (!) solutions

A =2λ

4− δ; A = 0

Both solutions satisfy the initial condition G (0) = 0



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Guess G (w) = Aw 4

Plug in and equate

4Aw 3 =2δA2w 8

w [2Aw 4 − λw 4]

Eliminate w

4A =2δA2

[2A− λ]

Two (!) solutions

A =2λ

4− δ; A = 0

Both solutions satisfy the initial condition G (0) = 0Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR


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OOPS

Why not inconsistent with existence/uniqueness for ODE’s?

G ′ =2δG 2

[2G − λw 4]w= Φ(w ,G )

Φ is not well-behaved: denominator can be 0For ODE’s of this type:

uniqueness does not obtain

existence is in doubt



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Workaround

Solutions must be constructed

“Correct” solution must be identified

Worker utility > 0



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Constructing a Solution

Key Observations

Denominator 2G − λw 4 . . . and

2G − λw 4 = 0

is the FOC for optimality when workers are myopic δ = 0

One period utility is

(λw 4 − G )(G/w 4)

worker utility increasing ⇒ worker utility > 0



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U(w |w) = λG (w)− [G (w)/w 2]2

dU(w |w)

dw= λG ′ − 2GG ′/w 4 + 5G 2/w 5

= λG ′λw 4 − 2G

w 4+ 5G 2/w 5

=

(− 2δG 2

[2G − λw 4]w

)(λw 4 − 2G

w 4

)+ 5G 2/w 5

=−2δG 2

w 5+ 5G 2/w 5

> 0



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Proof picture - lower bound for solution



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Proof picture - upper bound for solution



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Proof picture - solving ODE



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Steps in proof of Theorem 1

Step 1 Equilibrium → no deviations→ no stationary deviations

Step 2 No stationary deviations → FOC → ODE

G ′ = Φ(w ,G )

Step 3 ODE has unique “correct” solution G∗



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From “correct” solution G∗ to SAE

Step 4 If there is a profitable deviation from G∗ thenthere is a profitable finite deviation



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Proof picture - Finite deviation



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Step 5 If there is a profitable finite deviation from G∗then there is a profitable finite monotone deviation



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Proof picture - Finite Monotone deviation



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Step 6 If there is a profitable finite monotone deviationfrom G∗ then there is a profitable stationary deviation



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Proof picture - Stationary deviation



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Step 7 If there is a profitable stationary deviation fromG∗ then there is a profitable infinitesimal stationarydeviation

→ this contradicts ODE



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Comparisons in the Canonical Example

Random matching, worker optimization

Worker w matched with task t exerts effort to maximize

λewt − e2

Effort e = λwt/2, output = λw 2t2/2Total net of payments∫ 1

0

∫ 1

0

(1− λ)[λw 2t2/2]dtdw = [(1− λ)λ][1/18]



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Comparisons in the Canonical Example (cont)

Assortative matching, myopic worker optimization

Worker w matched with task w exerts effort to maximize

λeww − e2

Effort e = λw 2/2, output = λw 4/2Total net of payments∫ 1

0

(1− λ)[λw 4/2]dw = [(1− λ)λ][1/10]



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Steady-State Assortative Equilibrium

Worker w matched with task w produces

G (w) = [2λ/(4− δ)]w 4

Total net of payments∫ 1

0

(1− λ)(2λ/(4− δ)]w 4

)dw = [(1− λ)λ][2/5(4− δ)]



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Comparisons in the Canonical Example

2nd best: assortative + incentive-optimal payment

Worker w matched with task w is paid p to maximize

ew 2 − p subject to p ≥ e2

Effort e = w 2/2, output = w 4/2, net = w 4/4Total net of payments∫ 1

0

[w 4/4]dw = 1/20



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[(1−λ)λ][1/18] < [(1−λ)λ][1/10] < [(1−λ)λ][2/5(4−δ)] < 1/20

Random < Myopic < Stationary Assortative EQ < 2nd best



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First three cases: optimal λ = 1/2

1/72 < 1/40 < 1/[10(4− δ)] < 1/20

Random < Myopic < Stationary Assortative EQ < 2nd best



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Wages

Payment schedule is chosen exogenously.Wages are determined endogenously.Wages are highly convex in worker type

Better workers matched to better tasks

To maintain high ranking, better workers must producemore output



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What is the optimal payment schedule ?

What does “optimal” mean? What does firm know?

Linear payment schedules? (Carroll)



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Setting I: many clients, each with task

Payment schedules are task-specific

Who sets payment schedules?

Payment schedules determined as part of equilibrium

Competition determines payment schedules?(Gretsky, Ostroy & Zame)



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Many competing platforms

Uber, Lyft, Carmel . . .

Issues

exclusive dealing by clients?exclusive dealing by workers?who knows reputation of workers?who knows reputation of platforms?



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Unequal sides of the market

random arrivals

waiting, queues


endogenous matching: adverse selection & moral...

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