MotivationNature of the Problem
FormalismResults
Proof ideasComment
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
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
Economic Motivation
Technology (computers, internet, smartphones . . . ) has maderevolution in provision/delivery of
goods to consumers: Amazon, etc.
services to consumers: Uber, TaskRabbit, etc.
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
Also making revolution in provision/delivery of services tofirms
surveys: Amazon Mechanical Turk
software: TopCoder
consulting: Capgemini, Eden McCallum
outsourcing
On-demand economy
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
Also making revolution in provision/delivery of services tofirms
surveys: Amazon Mechanical Turk
software: TopCoder
consulting: Capgemini, Eden McCallum
outsourcing
On-demand economy
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
Issues
“On-demand companies . . . have difficulties training, managingand motivating workers.” (Economist Jan 3, 2015)
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
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)
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
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
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
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)
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
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
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
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
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
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)
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
Example
W = T = [0, 1]
ω = τ = Lebesgue measure
Y (w , t) = wt supermodular
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
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
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
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
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
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
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
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
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
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
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
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
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
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
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
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
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
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?”
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
Who knows/observes what? (cont)
Workers know/observe
Task types
Own characteristics
Own history
Output distribution
Workers do not know/observe
Other workers’ characteristics
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
Essence of the Problem
Adverse selection
Firm does not observe worker characteristics
Moral hazard
Firm does not observe worker effort
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
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)
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
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
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
Model (cont)
Tasks
T = [0, 1]
uniformly distributed (without essential loss of generality)tasks identified by location in distribution
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
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)
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
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)
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
Canonical Example
OutputY (e,w , t) = ewt
Payment ruleP(y) = λy
CostC (e) = e2
Worker utility
U(e,w , t) = λewt − e2
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
Firm Objective
Maximize
profit = total output - total payment to workers
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
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
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
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).
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
Notes:
No adverse selection at equilibrium.
Wages = P(G (w)) endogenous
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
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)
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
Firm has two instruments to provide incentives
pays for current output
conditions future assignments on current output
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
Canonical example
Y (e,w , t) = ewt
C (e) = e2
P(y) = λy
U(Y (e,w , t)) = λewt − e2
Solve for G ?
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
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
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
This is an ODE that we have never seen before but . . .
- it is just a first order ODE
- how bad can it be?
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
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
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
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
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
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
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
Workaround
Solutions must be constructed
“Correct” solution must be identified
Worker utility > 0
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
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
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
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
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
Proof picture - lower bound for solution
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
Proof picture - upper bound for solution
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
Proof picture - solving ODE
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
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∗
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
From “correct” solution G∗ to SAE
Step 4 If there is a profitable deviation from G∗ thenthere is a profitable finite deviation
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
Proof picture - Finite deviation
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
From “correct” solution G∗ to SAE
Step 5 If there is a profitable finite deviation from G∗then there is a profitable finite monotone deviation
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
Proof picture - Finite Monotone deviation
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
From “correct” solution G∗ to SAE
Step 6 If there is a profitable finite monotone deviationfrom G∗ then there is a profitable stationary deviation
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
Proof picture - Stationary deviation
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
From “correct” solution G∗ to SAE
Step 7 If there is a profitable stationary deviation fromG∗ then there is a profitable infinitesimal stationarydeviation
→ this contradicts ODE
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
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]
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
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]
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
Comparisons in the Canonical Example (cont)
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)]
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
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
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
Comparisons in the Canonical Example (cont)
[(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
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
Comparisons in the Canonical Example (cont)
First three cases: optimal λ = 1/2
1/72 < 1/40 < 1/[10(4− δ)] < 1/20
Random < Myopic < Stationary Assortative EQ < 2nd best
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
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
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
What is the optimal payment schedule ?
What does “optimal” mean? What does firm know?
Linear payment schedules? (Carroll)
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
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)
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
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?
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR
MotivationNature of the Problem
FormalismResults
Proof ideasComment
Where do we go from here?
Unequal sides of the market
random arrivals
waiting, queues
Mihaela van der Schaar, Yuanzhang Xiao, William Zame CR