Robust Predictions in Dynamic Screening Daniel Garrett, Alessandro - - PowerPoint PPT Presentation

▶
robust predictions in dynamic screening
SMART_READER_LITE
LIVE PREVIEW

Robust Predictions in Dynamic Screening Daniel Garrett, Alessandro - - PowerPoint PPT Presentation

Apr 30, 2023 50 likes •810 views

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions Robust Predictions in Dynamic Screening Daniel Garrett, Alessandro Pavan, Juuso Toikka March 2018 Introduction Model Wedges Allocations Continuum Simple

slide-1
SLIDE 1

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Robust Predictions in Dynamic Screening

Daniel Garrett, Alessandro Pavan, Juuso Toikka March 2018

slide-2
SLIDE 2

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Long-term contracting

Benefits of long-term contracting

prices/incentives set efficiently over course of relationship agent residual claimant on joint surplus principal extracts rents through “fixed fees”

In practice: private information (at time of contracting) prevents full surplus extraction

limiting agents’ rents calls for distortions Optimal (profit maximizing) mechanisms trade off efficiency and rent extraction

Dynamics of distortions when private info evolves over time?

slide-3
SLIDE 3

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Long-term contracting

Benefits of long-term contracting

prices/incentives set efficiently over course of relationship agent residual claimant on joint surplus principal extracts rents through “fixed fees”

In practice: private information (at time of contracting) prevents full surplus extraction

limiting agents’ rents calls for distortions Optimal (profit maximizing) mechanisms trade off efficiency and rent extraction

Dynamics of distortions when private info evolves over time?

slide-4
SLIDE 4

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Long-term contracting

Benefits of long-term contracting

prices/incentives set efficiently over course of relationship agent residual claimant on joint surplus principal extracts rents through “fixed fees”

In practice: private information (at time of contracting) prevents full surplus extraction

limiting agents’ rents calls for distortions Optimal (profit maximizing) mechanisms trade off efficiency and rent extraction

Dynamics of distortions when private info evolves over time?

slide-5
SLIDE 5

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Dynamic Mechanism Design

  • revenue management (Courty and Li, 2000, Battaglini 2005,

Boleslavsky and Said, 2013, Ely, Garrett and Hinnosaar, 2014, Board and Skrzypacz, 2015, Akan, Ata, and Dana, 2015,..

  • disclosure in auctions (Eso and Szentes, 2007, Bergemann and

Wambach (2015), Li and Shi (2017)...)

  • experimentation (Bergemann and Välimäki, 2010, Pavan, Segal, and

Toikka, 2014, Fershtman and Pavan, 2017...)

  • life-cycle taxation (Farhi and Werning, 2012, Kapicka, 2013,

Stantcheva, 2014, Makris and Pavan,2017,...)

  • managerial compensation (Garrett and Pavan, 2012, 2014,...)
  • insurance (Hendel and Lizzeri, 2003, Handel, Hendel, Whinston, 2015,...)
slide-6
SLIDE 6

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Relaxed Approach

Standard approach: "relaxed program"

necessary conditions for IC ("local" constraints) ex-post verification of remaining IC constraints

Relaxed approach (when valid)

complete characterization of optimal mechanism

  • ptimal mechanism often derived in closed form

approach mimics tractability of Myerson’s (1981) auctions

slide-7
SLIDE 7

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Relaxed Approach

Standard approach: "relaxed program"

necessary conditions for IC ("local" constraints) ex-post verification of remaining IC constraints

Relaxed approach (when valid)

complete characterization of optimal mechanism

  • ptimal mechanism often derived in closed form

approach mimics tractability of Myerson’s (1981) auctions

slide-8
SLIDE 8

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Existing predictions: stringent conditions

Reverse engineering:

conditions (process and payoffs) guaranteeing policies appropriately monotone

monotone hazard rate monotone impulse responses decreasing hazard rates dynamic supermodularity

Central prediction: "vanishing distortions Prediction hinges on "relaxed approach"?

slide-9
SLIDE 9

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Existing predictions: stringent conditions

Reverse engineering:

conditions (process and payoffs) guaranteeing policies appropriately monotone

monotone hazard rate monotone impulse responses decreasing hazard rates dynamic supermodularity

Central prediction: "vanishing distortions Prediction hinges on "relaxed approach"?

slide-10
SLIDE 10

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Existing predictions: stringent conditions

Reverse engineering:

conditions (process and payoffs) guaranteeing policies appropriately monotone

monotone hazard rate monotone impulse responses decreasing hazard rates dynamic supermodularity

Central prediction: "vanishing distortions Prediction hinges on "relaxed approach"?

slide-11
SLIDE 11

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

This paper: Variational Approach

Variational approach

IC-preserving perturbations of putative optimal policies

Robust predictions

convergence to efficiency

Bounds on distortions (all horizons)

slide-12
SLIDE 12

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

This paper: Variational Approach

Variational approach

IC-preserving perturbations of putative optimal policies

Robust predictions

convergence to efficiency

Bounds on distortions (all horizons)

slide-13
SLIDE 13

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Plan

Discrete types

dynamics of “wedges” (MB - MC) dynamic of allocations

Continuum of types

payoff equivalence wedges as “handicaps” convergence of expected wedges

slide-14
SLIDE 14

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Plan

Discrete types

dynamics of “wedges” (MB - MC) dynamic of allocations

Continuum of types

payoff equivalence wedges as “handicaps” convergence of expected wedges

slide-15
SLIDE 15

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

MODEL

slide-16
SLIDE 16

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Procurement environment

Canonical procurement a’ la Baron-Myerson (1982) Players:

principal: procurer agent: supplier

t = 1,2,... (many results also for finite horizon) qt ∈ (0, ¯ q): period-t output supplied by agent pt: total period-t payment from principal δ ∈ (0,1): common discount factor

slide-17
SLIDE 17

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Payoffs and information

Gross value of output to principal: B : (0, ¯ q) → R, strictly increasing, strictly concave, twice-continuously differentiable, lim

qց0B (q) = −∞.

Agent’s period-t cost C(qt,ht) with C(qt,ht) = htqt +c(qt) with c (·) : (0, ¯ q) → R+ strictly increasing, strictly convex, twice-continuously differentiable, lim

qր¯ qc (q) = +∞

Agent period-t "type": ht ∈ Θ = {θ1,...,θN}

0 < θ1 < ··· < θN ht privately observed at beginning of period t process F = (Ft(·|ht−1))

slide-18
SLIDE 18

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Payoffs and information

Gross value of output to principal: B : (0, ¯ q) → R, strictly increasing, strictly concave, twice-continuously differentiable, lim

qց0B (q) = −∞.

Agent’s period-t cost C(qt,ht) with C(qt,ht) = htqt +c(qt) with c (·) : (0, ¯ q) → R+ strictly increasing, strictly convex, twice-continuously differentiable, lim

qր¯ qc (q) = +∞

Agent period-t "type": ht ∈ Θ = {θ1,...,θN}

0 < θ1 < ··· < θN ht privately observed at beginning of period t process F = (Ft(·|ht−1))

slide-19
SLIDE 19

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Payoffs and information

Gross value of output to principal: B : (0, ¯ q) → R, strictly increasing, strictly concave, twice-continuously differentiable, lim

qց0B (q) = −∞.

Agent’s period-t cost C(qt,ht) with C(qt,ht) = htqt +c(qt) with c (·) : (0, ¯ q) → R+ strictly increasing, strictly convex, twice-continuously differentiable, lim

qր¯ qc (q) = +∞

Agent period-t "type": ht ∈ Θ = {θ1,...,θN}

0 < θ1 < ··· < θN ht privately observed at beginning of period t process F = (Ft(·|ht−1))

slide-20
SLIDE 20

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Payoffs

Principal’s intertemporal payoff UP = ∑

t≥1

δ t−1 (B (qt)−pt) Agent’s intertemporal payoff UA = ∑

t≥1

δ t−1 (pt −C (qt,ht)) Agent’s outside option: zero Principal’s payoff in case she fails to procure output: −∞

slide-21
SLIDE 21

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Mechanisms

Direct mechanism: qt (ht),pt (ht)∞

t=1 where

ht = (h1,h2,...,ht) ∈ Θt

slide-22
SLIDE 22

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Incentive compatibility

Agent payoff from t onwards under truth-telling Vt

  • ht

= E ∞

∑

s=t

δ s−t ps

  • ˜

hs −C

  • qs
  • ˜

hs ,˜ hs

  • |ht
  • IC: for all t, all ht ∈ Θt, all reporting strategies σ,

Vt

  • ht

≥ E ∞

∑

s=t

δ s−t pσ

s

  • ˜

hs −C

  • qσ

s

  • ˜

hs ,˜ hs |ht

  • IR: participation requires

V1 (h1) ≥ 0 all h1 ∈ Θ.

slide-23
SLIDE 23

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Principal’s problem

Principal chooses qt (ht),pt (ht)∞

t=1 to maximize

E

  • ∑

t≥1

δ t−1 B

  • qt
  • ˜

ht −pt

  • ˜

ht subject to IC and IR constraints. q∗

t (ht),p∗ t (ht)∞ t=1: solution to principal’s problem

Solution always exist and q∗ unique

slide-24
SLIDE 24

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Principal’s problem

Principal chooses qt (ht),pt (ht)∞

t=1 to maximize

E

  • ∑

t≥1

δ t−1 B

  • qt
  • ˜

ht −pt

  • ˜

ht subject to IC and IR constraints. q∗

t (ht),p∗ t (ht)∞ t=1: solution to principal’s problem

Solution always exist and q∗ unique

slide-25
SLIDE 25

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Principal’s problem

Principal chooses qt (ht),pt (ht)∞

t=1 to maximize

E

  • ∑

t≥1

δ t−1 B

  • qt
  • ˜

ht −pt

  • ˜

ht subject to IC and IR constraints. q∗

t (ht),p∗ t (ht)∞ t=1: solution to principal’s problem

Solution always exist and q∗ unique

slide-26
SLIDE 26

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Efficiency

Efficient policy: all t, all ht ∈ Θ, B′ qE (ht)

  • = Cq
  • qE (ht),ht
  • Payment scheme implementing efficient allocation

pE (ht) = B

  • qE (ht)
  • all t, all ht ∈ Θt
slide-27
SLIDE 27

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

WEDGES

slide-28
SLIDE 28

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Convergence of expected “wedges”

Definition Process satisfies “Long-run Independence” if lim

t→∞

max

h1,h′

1,ht∈Θ

  • Pr
  • ˜

ht ≤ ht|h1

  • −Pr
  • ˜

ht ≤ ht|h′

1

  • = 0.

Proposition Suppose F satisfies “Long-run independence.” As t → +∞, E

  • B′

q∗

t

  • ˜

ht −Cq

  • q∗

t

  • ˜

ht ,˜ ht

  • → 0

Suppose distortions always of same sign. Then q∗

t (·) converge to

qE (·) in probability

slide-29
SLIDE 29

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Convergence of expected “wedges”

Definition Process satisfies “Long-run Independence” if lim

t→∞

max

h1,h′

1,ht∈Θ

  • Pr
  • ˜

ht ≤ ht|h1

  • −Pr
  • ˜

ht ≤ ht|h′

1

  • = 0.

Proposition Suppose F satisfies “Long-run independence.” As t → +∞, E

  • B′

q∗

t

  • ˜

ht −Cq

  • q∗

t

  • ˜

ht ,˜ ht

  • → 0

Suppose distortions always of same sign. Then q∗

t (·) converge to

qE (·) in probability

slide-30
SLIDE 30

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Proof Sketch

Difficulty: direction in which IC and IR binds unknown Suppose q∗

t (ht),p∗ t (ht)∞ t=1 is optimal (hence IC and IR)

then q∗

t (ht) ∈ (0, ¯

q) all t, all ht.

Idea: IC preserving perturbations

increase q∗

t (·) uniformly by small amount ν > 0

increase period-t payments by c (q∗

t (ht)+ν)−c (q∗ t (ht))

increase period-1 payments p∗

1 (·) uniformly by

δ t−1ν maxh1∈Θ E

  • ˜

ht|h1

  • New mechanism IC and IR

IC: additional quantity ν produced irrespective of reports! Each type h1 expects additional rent δ t−1ν

  • maxˆ

h1∈Θ E

  • ˜

ht|ˆ h1

  • −E
  • ˜

ht|h1

slide-31
SLIDE 31

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Proof Sketch

Difficulty: direction in which IC and IR binds unknown Suppose q∗

t (ht),p∗ t (ht)∞ t=1 is optimal (hence IC and IR)

then q∗

t (ht) ∈ (0, ¯

q) all t, all ht.

Idea: IC preserving perturbations

increase q∗

t (·) uniformly by small amount ν > 0

increase period-t payments by c (q∗

t (ht)+ν)−c (q∗ t (ht))

increase period-1 payments p∗

1 (·) uniformly by

δ t−1ν maxh1∈Θ E

  • ˜

ht|h1

  • New mechanism IC and IR

IC: additional quantity ν produced irrespective of reports! Each type h1 expects additional rent δ t−1ν

  • maxˆ

h1∈Θ E

  • ˜

ht|ˆ h1

  • −E
  • ˜

ht|h1

slide-32
SLIDE 32

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Proof Sketch

Difficulty: direction in which IC and IR binds unknown Suppose q∗

t (ht),p∗ t (ht)∞ t=1 is optimal (hence IC and IR)

then q∗

t (ht) ∈ (0, ¯

q) all t, all ht.

Idea: IC preserving perturbations

increase q∗

t (·) uniformly by small amount ν > 0

increase period-t payments by c (q∗

t (ht)+ν)−c (q∗ t (ht))

increase period-1 payments p∗

1 (·) uniformly by

δ t−1ν maxh1∈Θ E

  • ˜

ht|h1

  • New mechanism IC and IR

IC: additional quantity ν produced irrespective of reports! Each type h1 expects additional rent δ t−1ν

  • maxˆ

h1∈Θ E

  • ˜

ht|ˆ h1

  • −E
  • ˜

ht|h1

slide-33
SLIDE 33

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Proof by contradiction

Suppose result not true. There exists increasing sequence (tn) s.t. either E

  • B′

q∗

tn

  • ˜

htn

  • −Cq
  • q∗

tn

  • ˜

htn

  • ,˜

htn

  • > ζ

for all tn in sequence, or E

  • B′

q∗

tn

  • ˜

htn

  • −Cq
  • q∗

tn

  • ˜

htn

  • ,˜

htn

  • < −ζ

for an appropriate ζ > 0. Focus on first case. Increase q∗

tn (·) uniformly at arbitrary date

tn in sequence by arbitrarily small amount νn > 0 Adjust payments as described above New mechanism IC and IR

slide-34
SLIDE 34

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Proof by contradiction

Suppose result not true. There exists increasing sequence (tn) s.t. either E

  • B′

q∗

tn

  • ˜

htn

  • −Cq
  • q∗

tn

  • ˜

htn

  • ,˜

htn

  • > ζ

for all tn in sequence, or E

  • B′

q∗

tn

  • ˜

htn

  • −Cq
  • q∗

tn

  • ˜

htn

  • ,˜

htn

  • < −ζ

for an appropriate ζ > 0. Focus on first case. Increase q∗

tn (·) uniformly at arbitrary date

tn in sequence by arbitrarily small amount νn > 0 Adjust payments as described above New mechanism IC and IR

slide-35
SLIDE 35

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Proof by contradiction

Suppose result not true. There exists increasing sequence (tn) s.t. either E

  • B′

q∗

tn

  • ˜

htn

  • −Cq
  • q∗

tn

  • ˜

htn

  • ,˜

htn

  • > ζ

for all tn in sequence, or E

  • B′

q∗

tn

  • ˜

htn

  • −Cq
  • q∗

tn

  • ˜

htn

  • ,˜

htn

  • < −ζ

for an appropriate ζ > 0. Focus on first case. Increase q∗

tn (·) uniformly at arbitrary date

tn in sequence by arbitrarily small amount νn > 0 Adjust payments as described above New mechanism IC and IR

slide-36
SLIDE 36

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Proof by contradiction

Suppose result not true. There exists increasing sequence (tn) s.t. either E

  • B′

q∗

tn

  • ˜

htn

  • −Cq
  • q∗

tn

  • ˜

htn

  • ,˜

htn

  • > ζ

for all tn in sequence, or E

  • B′

q∗

tn

  • ˜

htn

  • −Cq
  • q∗

tn

  • ˜

htn

  • ,˜

htn

  • < −ζ

for an appropriate ζ > 0. Focus on first case. Increase q∗

tn (·) uniformly at arbitrary date

tn in sequence by arbitrarily small amount νn > 0 Adjust payments as described above New mechanism IC and IR

slide-37
SLIDE 37

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Reaching a contradiction...

New mechanism increases expected surplus by at least δ tn−1ζνn

(νn small enough)

New mechanism leaves additional expected rent δ tn−1νn

  • max

ˆ h1

  • E
  • ˜

htn|ˆ h1

  • −E
  • ˜

htn|h1

  • to each initial type h1

Since, for all h1 ∈ Θ, max

ˆ h1

  • E
  • ˜

htn|ˆ h1

  • −E
  • ˜

htn|h1

  • → 0 as tn → ∞

increase in surplus dominates for tn large enough. When distortions always of same sign, convergence of wedges implies convergence of surplus and hence of policies (in probability) Q.E.D.

slide-38
SLIDE 38

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Reaching a contradiction...

New mechanism increases expected surplus by at least δ tn−1ζνn

(νn small enough)

New mechanism leaves additional expected rent δ tn−1νn

  • max

ˆ h1

  • E
  • ˜

htn|ˆ h1

  • −E
  • ˜

htn|h1

  • to each initial type h1

Since, for all h1 ∈ Θ, max

ˆ h1

  • E
  • ˜

htn|ˆ h1

  • −E
  • ˜

htn|h1

  • → 0 as tn → ∞

increase in surplus dominates for tn large enough. When distortions always of same sign, convergence of wedges implies convergence of surplus and hence of policies (in probability) Q.E.D.

slide-39
SLIDE 39

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Reaching a contradiction...

New mechanism increases expected surplus by at least δ tn−1ζνn

(νn small enough)

New mechanism leaves additional expected rent δ tn−1νn

  • max

ˆ h1

  • E
  • ˜

htn|ˆ h1

  • −E
  • ˜

htn|h1

  • to each initial type h1

Since, for all h1 ∈ Θ, max

ˆ h1

  • E
  • ˜

htn|ˆ h1

  • −E
  • ˜

htn|h1

  • → 0 as tn → ∞

increase in surplus dominates for tn large enough. When distortions always of same sign, convergence of wedges implies convergence of surplus and hence of policies (in probability) Q.E.D.

slide-40
SLIDE 40

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Reaching a contradiction...

New mechanism increases expected surplus by at least δ tn−1ζνn

(νn small enough)

New mechanism leaves additional expected rent δ tn−1νn

  • max

ˆ h1

  • E
  • ˜

htn|ˆ h1

  • −E
  • ˜

htn|h1

  • to each initial type h1

Since, for all h1 ∈ Θ, max

ˆ h1

  • E
  • ˜

htn|ˆ h1

  • −E
  • ˜

htn|h1

  • → 0 as tn → ∞

increase in surplus dominates for tn large enough. When distortions always of same sign, convergence of wedges implies convergence of surplus and hence of policies (in probability) Q.E.D.

slide-41
SLIDE 41

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Additional predictions

Definition Process satisfies “FOSD” if ˘ ht−1 ≥ ¯ ht−1implies Ft

  • ht|˘

ht−1 ≤ Ft

  • ht|¯

ht−1 , all ht. Process satisfies “Markov” if evolution of ht governed by time-invariant irreducible transition matrix A with Aij = Pr(θi|θj) > 0 all i,j. Process satisfies “Stationary Markov” if F1 coincides with ergodic distribution

slide-42
SLIDE 42

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Additional predictions

Proposition Suppose F satisfies “FOSD”. Then, for all t, E

  • B′

q∗

t

  • ˜

ht −Cq

  • q∗

t

  • ˜

ht ,˜ ht

  • ≥ 0.

If, in addition, F satisfies “Stationary Markov”, expected wedges decrease with t. FOSD:

IR binds only for θN cut output and adjust p so that IR continues to bind for θN perturbation increases surplus and reduces rents, hence profitable

FOSD + Stationary Markov:

shift output uniformly towards later dates smaller rents due to declining persistence of initial types

slide-43
SLIDE 43

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Additional predictions

Proposition Suppose F satisfies “FOSD”. Then, for all t, E

  • B′

q∗

t

  • ˜

ht −Cq

  • q∗

t

  • ˜

ht ,˜ ht

  • ≥ 0.

If, in addition, F satisfies “Stationary Markov”, expected wedges decrease with t. FOSD:

IR binds only for θN cut output and adjust p so that IR continues to bind for θN perturbation increases surplus and reduces rents, hence profitable

FOSD + Stationary Markov:

shift output uniformly towards later dates smaller rents due to declining persistence of initial types

slide-44
SLIDE 44

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Additional predictions

Proposition Suppose F satisfies “FOSD”. Then, for all t, E

  • B′

q∗

t

  • ˜

ht −Cq

  • q∗

t

  • ˜

ht ,˜ ht

  • ≥ 0.

If, in addition, F satisfies “Stationary Markov”, expected wedges decrease with t. FOSD:

IR binds only for θN cut output and adjust p so that IR continues to bind for θN perturbation increases surplus and reduces rents, hence profitable

FOSD + Stationary Markov:

shift output uniformly towards later dates smaller rents due to declining persistence of initial types

slide-45
SLIDE 45

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

ALLOCATIONS

slide-46
SLIDE 46

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

“Sufficient patience”

Assume F is “Markov” α ≡ mini,j Aij > 0

high α: little persistence

b ≡ ∑N

i=1 θi

κ ≡ min

i,j

  • B
  • qE (θi)
  • −C
  • qE (θi),θi
  • −
  • B
  • qE (θj)
  • −c
  • qE (θj),θi

. Patience threshold: ¯ δ ≡

  • 2¯

qb−κ 2¯ qb−κ+2κα if κ < 2¯

qb 0 otherwise .

slide-47
SLIDE 47

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Vanishing distortions

Proposition Suppose T = +∞ and F is “Markov”. For any δ ∈ ¯ δ,1

  • ,

1 limt→∞ E

  • B
  • q∗

t

  • ˜

ht −C

  • q∗

t

  • ˜

ht ,˜ ht

  • = limt→∞ E
  • B
  • qE

˜ ht

  • −C
  • qE

˜ ht

  • ,˜

ht

  • .

2 For any η > 0, limt→∞ Pr

  • q∗

t

  • ˜

ht −qE ˜ ht

  • > η
  • = 0.
slide-48
SLIDE 48

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Bounds on distortions

Corollary Suppose F is Markov and let λ =

¯ qb 1−δ(1−2α). Irrespective of time

horizon and of patience, for any t,   E

  • B
  • qE

˜ ht

  • −C
  • qE

˜ ht

  • ,˜

ht

  • −E
  • B
  • q∗

t

  • ˜

ht −C

  • q∗

t

  • ˜

ht ,˜ ht

 ≤ 2λ

  • δ + κ

2λ

t−1 , with δ + κ

2λ > 1 for δ ∈

¯ δ,1

  • .

Result also provides conservative bound on rate of convergence

slide-49
SLIDE 49

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Proof: Idea

Idea: under efficient mechanism with pE (ht) = B

  • qE (ht)
  • IC slack all histories

Perturbations obtained by combining putative mechanism with efficient one guarantee slack in IC

slide-50
SLIDE 50

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Proof: approching efficiency “too fast”

Idea 1 (FAILED!): Suppose q∗

t (ht),p∗ t (ht)∞ t=1 is optimal

and convergence to efficiency does not hold.

Replace payment and allocation rules with those of efficient mechanism from t onwards. Adjust payments, to ensure satisfaction of IR constraints. Such adjustment can be made s.t. (expected) increase in surplus dominates expected increase in rents (when t large enough) If IC, new mechanism improves upon q∗

t (ht),p∗ t (ht)∞ t=1

Problem: New mechanism need not be IC!

IC from date t onwards, but not necessarily at earlier dates.

slide-51
SLIDE 51

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Proof: approching efficiency “too fast”

Idea 1 (FAILED!): Suppose q∗

t (ht),p∗ t (ht)∞ t=1 is optimal

and convergence to efficiency does not hold.

Replace payment and allocation rules with those of efficient mechanism from t onwards. Adjust payments, to ensure satisfaction of IR constraints. Such adjustment can be made s.t. (expected) increase in surplus dominates expected increase in rents (when t large enough) If IC, new mechanism improves upon q∗

t (ht),p∗ t (ht)∞ t=1

Problem: New mechanism need not be IC!

IC from date t onwards, but not necessarily at earlier dates.

slide-52
SLIDE 52

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Proof: Approaching efficiency “gradually”

Any linear convex combination of q∗

t (ht)∞ t=τ and

  • qE (ht)

∞

t=τ can be implemented with payments that make IC

slack at all histories Amount of slack determined by κ and linear weights Gradual growth in weights on efficiency qnew

1

(h1) =

  • 1−α1

q∗

1 (h1)+α1qE (h1)

and, for any t ≥ 2, qnew

t

  • ht

=

  • 1−α≥2

q∗

t

  • ht

+α≥2qE (ht) with 0 < α1 ≤ α≥2 ≤ 1.

new mechanism IC if α≥2 not too much larger than α1 for fixed α≥2, mechanism IC from t = 2 onwards if α1 = α≥2, then there is slack in IC at t = 1. Hence, can decrease α1 below α≥2 by small amount and preserve IC

slide-53
SLIDE 53

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Proof: Approaching efficiency “gradually”

Any linear convex combination of q∗

t (ht)∞ t=τ and

  • qE (ht)

∞

t=τ can be implemented with payments that make IC

slack at all histories Amount of slack determined by κ and linear weights Gradual growth in weights on efficiency qnew

1

(h1) =

  • 1−α1

q∗

1 (h1)+α1qE (h1)

and, for any t ≥ 2, qnew

t

  • ht

=

  • 1−α≥2

q∗

t

  • ht

+α≥2qE (ht) with 0 < α1 ≤ α≥2 ≤ 1.

new mechanism IC if α≥2 not too much larger than α1 for fixed α≥2, mechanism IC from t = 2 onwards if α1 = α≥2, then there is slack in IC at t = 1. Hence, can decrease α1 below α≥2 by small amount and preserve IC

slide-54
SLIDE 54

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Role of discount factor

Why δ large?

Proposed "new" mechanism approaches efficiency gradually. Positive weight on efficient policy at early dates may increase information rents (by relatively large amount) When δ small, gains in surplus at later dates need not compensate for increased rents at earlier periods.

However, convergence to efficiency for all δ if process not very persistent

slide-55
SLIDE 55

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

General cost functions

Convergence results extend to general C (q,h) satisfying mild regularity conditions Deterministic mechanisms need not be optimal

arguments related to Strausz (2006) violation of integral mon. “relaxed program” need not be valid

Perturbations involve randomizations between putative optimal and efficient allocations

slide-56
SLIDE 56

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

CONTINUUM OF TYPES

slide-57
SLIDE 57

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Continuum of types

Markov chain F = (Ft); Θ =

  • θ, ¯

θ

  • ⊂ R+

F1 (abs. continuous) cdf of initial distribution (density f1) Ft (·|ht−1) cdf of ht given ht−1 ∈ Θ (ft (ht|ht−1) > 0 all ht,ht−1 ∈ Θ) Stochastic monotonicity: Ft

  • ·|h′

t−1

  • first-order stochastically

dominates Ft (·|ht−1) for h′

t−1 > ht−1

Time-invariance: Ft (·|θ) = Fs (·|θ) all t,s > 1, all θ ∈ Θ Ergodicity: ∃! invariant distribution π s.t., for all θ ∈ Θ sup

A∈B(Θ)

  • F t (A;θ)−π (A)
  • → 0 as t → ∞.
slide-58
SLIDE 58

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Continuum of types

Markov chain F = (Ft); Θ =

  • θ, ¯

θ

  • ⊂ R+

F1 (abs. continuous) cdf of initial distribution (density f1) Ft (·|ht−1) cdf of ht given ht−1 ∈ Θ (ft (ht|ht−1) > 0 all ht,ht−1 ∈ Θ) Stochastic monotonicity: Ft

  • ·|h′

t−1

  • first-order stochastically

dominates Ft (·|ht−1) for h′

t−1 > ht−1

Time-invariance: Ft (·|θ) = Fs (·|θ) all t,s > 1, all θ ∈ Θ Ergodicity: ∃! invariant distribution π s.t., for all θ ∈ Θ sup

A∈B(Θ)

  • F t (A;θ)−π (A)
  • → 0 as t → ∞.
slide-59
SLIDE 59

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Continuum of types

Markov chain F = (Ft); Θ =

  • θ, ¯

θ

  • ⊂ R+

F1 (abs. continuous) cdf of initial distribution (density f1) Ft (·|ht−1) cdf of ht given ht−1 ∈ Θ (ft (ht|ht−1) > 0 all ht,ht−1 ∈ Θ) Stochastic monotonicity: Ft

  • ·|h′

t−1

  • first-order stochastically

dominates Ft (·|ht−1) for h′

t−1 > ht−1

Time-invariance: Ft (·|θ) = Fs (·|θ) all t,s > 1, all θ ∈ Θ Ergodicity: ∃! invariant distribution π s.t., for all θ ∈ Θ sup

A∈B(Θ)

  • F t (A;θ)−π (A)
  • → 0 as t → ∞.
slide-60
SLIDE 60

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Continuum of types

Markov chain F = (Ft); Θ =

  • θ, ¯

θ

  • ⊂ R+

F1 (abs. continuous) cdf of initial distribution (density f1) Ft (·|ht−1) cdf of ht given ht−1 ∈ Θ (ft (ht|ht−1) > 0 all ht,ht−1 ∈ Θ) Stochastic monotonicity: Ft

  • ·|h′

t−1

  • first-order stochastically

dominates Ft (·|ht−1) for h′

t−1 > ht−1

Time-invariance: Ft (·|θ) = Fs (·|θ) all t,s > 1, all θ ∈ Θ Ergodicity: ∃! invariant distribution π s.t., for all θ ∈ Θ sup

A∈B(Θ)

  • F t (A;θ)−π (A)
  • → 0 as t → ∞.
slide-61
SLIDE 61

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Continuum of types

Markov chain F = (Ft); Θ =

  • θ, ¯

θ

  • ⊂ R+

F1 (abs. continuous) cdf of initial distribution (density f1) Ft (·|ht−1) cdf of ht given ht−1 ∈ Θ (ft (ht|ht−1) > 0 all ht,ht−1 ∈ Θ) Stochastic monotonicity: Ft

  • ·|h′

t−1

  • first-order stochastically

dominates Ft (·|ht−1) for h′

t−1 > ht−1

Time-invariance: Ft (·|θ) = Fs (·|θ) all t,s > 1, all θ ∈ Θ Ergodicity: ∃! invariant distribution π s.t., for all θ ∈ Θ sup

A∈B(Θ)

  • F t (A;θ)−π (A)
  • → 0 as t → ∞.
slide-62
SLIDE 62

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Continuum of types

Markov chain F = (Ft); Θ =

  • θ, ¯

θ

  • ⊂ R+

F1 (abs. continuous) cdf of initial distribution (density f1) Ft (·|ht−1) cdf of ht given ht−1 ∈ Θ (ft (ht|ht−1) > 0 all ht,ht−1 ∈ Θ) Stochastic monotonicity: Ft

  • ·|h′

t−1

  • first-order stochastically

dominates Ft (·|ht−1) for h′

t−1 > ht−1

Time-invariance: Ft (·|θ) = Fs (·|θ) all t,s > 1, all θ ∈ Θ Ergodicity: ∃! invariant distribution π s.t., for all θ ∈ Θ sup

A∈B(Θ)

  • F t (A;θ)−π (A)
  • → 0 as t → ∞.
slide-63
SLIDE 63

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Continuous types: auxiliary shocks

Stochastic process can be represented by "auxiliary shocks" independent of initial private information

e.g., Eso, Szentes (2007), Pavan, Segal, Toikka (2014)

ht = z (ht−1,εt), where ε = (εt) are i.i.d. random variables E.g., εt drawn from U(0,1) and z (ht−1,εt) = F −1(εt|ht−1) with F −1(εt|ht−1) ≡ inf{θt : F(θt|ht−1) ≥ εt}

"probability integral transform"

Regularity: ∂z(ht−1,εt)

∂ht−1

exists, continuous and bounded

slide-64
SLIDE 64

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Continuous types: auxiliary shocks

Let (Zτ,t)t≥τ be a collection of functions s.t. ht = Zτ,t (hτ,ε) for t ≥ τ Impulse responses: Iτ→t

  • ht

= ∂Zt,τ(hτ,ε) ∂hτ (where vector ε derived from ht using function z(·)) AR(1) example (violates full support): ht = γht−1 +εt = Zτ,t (hτ,ε) = γt−τhτ +γt−τ−1ετ+1 +···+γεt−1 +εt → Iτ→t

  • ht

= γt−τ.

slide-65
SLIDE 65

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Characterization of IC

Theorem Suppose F satisfies “Markov”, “FOSD,” and “regularity”. Mechanism qt (ht),pt (ht)∞

t=1 IC iff, for all t ≥ 0, all ht−1, Vt (ht)

Lipschitz continuous in ht with ∂Vt (ht) ∂ht = −E

  • ∑

s≥t

δ s−tIt→s

  • ˜

hs qs

  • ˜

hs |ht

  • a.e. ht,

and, for all ht−1, ht, ˆ ht,

ht

ˆ ht

  • Dt
  • ht−1,x
  • ;x
  • −Dt
  • ht−1,x
  • ;ˆ

ht

  • dx ≥ 0

where Dt (ht;y) ≡ −E

  • ∑s≥t δ s−tIt→s
  • ˜

hs qs

  • ˜

hs

−t,y

  • |ht

.

slide-66
SLIDE 66

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Dynamic virtual surplus

Previous result implies principal’s payoff equals "dynamic virtual surplus" E   ∑

t≥1

δ t−1    B

  • qt
  • ˜

ht −C

  • qt
  • ˜

ht ,˜ ht

  • −
  • F1(˜

h1) f1(˜ h1) I1→t

  • ˜

ht qt

  • ˜

ht       −V1 ¯ θ

  • (FOSD ensures IC binds at ¯

θ, so, at optimum, V1 ¯ θ

  • = 0).
slide-67
SLIDE 67

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Relaxed approach

"Relaxed/First-order approach": Pointwise maximization B′ q∗

t

  • ht

= Cq

  • q∗

t

  • ht

,ht

  • + F1 (h1)

f1 (h1) I1→t

  • ht

FOSD (I1→t ≥ 0) ⇒ downward distortions

Distortions driven by impulse responses Validity of FOA: above policies must satisfy "integral monotonicity" constraints

Condition for convergence to efficiency (point-wise)

vanishing impulse responses

slide-68
SLIDE 68

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Beyond FOA: Convergence of wedges

Definition “Policies eventually interior” if ∃T and (bt) and (¯ bt), with 0 < bt < ¯ bt < ¯ q, s.t., for all t ≥ T, q∗

t (ht) ∈ [bt, ¯

bt]. Theorem Assume F satisfies “Markov”, “FOSD” and “Regularity”. If optimal policies “eventually interior”, E

  • B′

q∗

t

  • ˜

ht −Cq

  • q∗

t

  • ˜

ht ,˜ ht

  • = E

  F1

  • ˜

h1

  • f1
  • ˜

h1 It(˜ ht)   If, in addition, F satisfies “ergodicity”, then E

  • F1(θ1)

f1(θ1) It(θ t)

  • → 0 as t → ∞ and convergence from above and

monotone in t.

slide-69
SLIDE 69

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Proof Sketch

Observe that E

  • It
  • ˜

ht |h1

  • =

d dh1 E

  • ˜

ht|h1

  • .

Thus, E   F1

  • ˜

h1

  • f1
  • ˜

h1 It

  • ˜

ht   = E   F1

  • ˜

h1

  • f1
  • ˜

h1 E

  • It
  • ˜

ht |˜ h1

 =

¯

θ θ F1(h1)E

  • It
  • ˜

ht |h1

  • dh1

= F1(θ1)E[˜ ht | h1]

  • h1=¯

θ h1=θ

+

θ

θ f1(h1)E[˜

ht | h1]dh1 = E[˜ ht | ¯ θ]−E[˜ ht ] → 0 by ergodicity.

slide-70
SLIDE 70

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Proof Sketch

If F monotone (FOSD), E[˜ ht | ¯ θ]−E[˜ ht ] ≥ 0 implying that convergence is from above. If, in addition, F1 = π, then E

  • F1(˜

h1) f1(˜ h1) I1→t

  • ˜

ht −E

  • F1(˜

h1) f1(˜ h1) I1→s

  • ˜

hs = E[˜ ht | ¯ θ]−E[˜ hs | ¯ θ] ≤ 0 for t > s, implying convergence is monotone in time.

slide-71
SLIDE 71

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

SIMPLE MECHANISMS

slide-72
SLIDE 72

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Simple mechanisms

Results apply to settings where mechanisms constrained to be “simple” provided above perturbations are admissible (i.e., preserve simplicity) “Simple” might mean

continuity restrictions on allocations measurability restrictions

e.g. allocations depend only on last few reported types

slide-73
SLIDE 73

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

CONCLUSIONS

slide-74
SLIDE 74

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Summary

DMD literature− → “relaxed” approach (as in Myerson) This paper: variational approach

IC-preserving perturbations Idea related to Rogerson (1985) for moral hazard settings

slide-75
SLIDE 75

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

Summary

Convergence of wedges (in expectation)

fairly robust property (long-run independence)

Convergence of allocations (in probability)

enough patience

Additional “economic” properties (e.g., FOSD and stationarity)

convergence from above and monotone in time

Results apply to environments where we don’t know which IC and IR constraints bind

slide-76
SLIDE 76

Introduction Model Wedges Allocations Continuum Simple Mechanisms Conclusions

THANK YOU!!!