The Importance of Being Honest

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Nicolas Klein The Importance of Being Honest – 1

The Importance of Being Honest

Nicolas Klein University of Bonn

April 11, 2012

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Motivation

Introduction

●Introduction

●Ingredients

●Interpretation

●^Overview

●Some Stylized Facts

●^Literature Setup

Continuation Scheme Before the First Breakthrough Optimal Stopping Conclusion Appendix

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Motivation

Introduction

●Introduction

●Ingredients

●Interpretation

●^Overview

US$105 bn spent on the “War on Cancer” from 1971 till 2009.

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Motivation

Introduction

●Introduction

●Ingredients

●Interpretation

●^Overview

Too much funding devoted to low-risk, low-yield projects?

(NY Times, June 28, 2009)

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Motivation

Introduction

●Introduction

●Ingredients

●Interpretation

●^Overview

Too much funding devoted to low-risk, low-yield projects?

(NY Times, June 28, 2009)

Question: Optimal way of giving incentives so that scientists themselves would choose high-risk, high-yield projects?

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The Ingredients

Introduction

●Introduction

●Ingredients

●Interpretation

●^Overview

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The Ingredients

Introduction

●Introduction

●Ingredients

●Interpretation

●^Overview

Agent can

● kick back and relax (“shirk”),

● go low-risk (“cheat”),

● investigate an uncertain hypothesis (“be honest”).

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The Ingredients

Introduction

●Introduction

●Ingredients

●Interpretation

●^Overview

Agent can

● If honest, agent learns something about the

hypothesis/technology he is supposed to investigate.

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The Ingredients

Introduction

●Introduction

●Ingredients

●Interpretation

●^Overview

Agent can

Principal

● only observes events agent produces;

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The Ingredients

Introduction

●Introduction

●Ingredients

●Interpretation

●^Overview

Agent can

Principal

● only cares about the first honestly produced event;

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The Ingredients

Introduction

●Introduction

●Ingredients

●Interpretation

●^Overview

Agent can

Principal

● only cares about the first honestly produced event;

● can pay the agent non-negative amounts (limited liability!), conditional on the history he observes.

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Alternative Interpretation

Introduction

●Introduction

●Ingredients

●Interpretation

●^Overview

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Alternative Interpretation

Introduction

●Introduction

●Ingredients

●Interpretation

●^Overview

Scientist/worker is hired to test a new hypothesis/production method.

Yet, he could manipulate data/secretly use old method.

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Alternative Interpretation

Introduction

●Introduction

●Ingredients

●Interpretation

●^Overview

Fanelli (2009):

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Alternative Interpretation

Introduction

●Introduction

●Ingredients

●Interpretation

●^Overview

Fanelli (2009):

● One out of 7 scientists reports their colleagues have falsified data at least once.

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Alternative Interpretation

Introduction

●Introduction

●Ingredients

●Interpretation

●^Overview

Fanelli (2009):

● ^Only

< 2%

admit to having done so themselves.

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Alternative Interpretation

Introduction

●Introduction

●Ingredients

●Interpretation

●^Overview

Fanelli (2009):

● ^Only

< 2%

● One out of three (72%) admit to lesser distortion of knowledge (by their colleagues).

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Alternative Interpretation

Introduction

●Introduction

●Ingredients

●Interpretation

●^Overview

Fanelli (2009):

● ^Only

< 2%

● Note that this is all self-reported survey data!

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Alternative Interpretation

Introduction

●Introduction

●Ingredients

●Interpretation

●^Overview

Fanelli (2009):

● ^Only

< 2%

● Note that this is all self-reported survey data!

⇒

Include the option of cheating into models of incentives.

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Overview of Main Results

Introduction

●Introduction

●Ingredients

●Interpretation

●^Overview

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Overview of Main Results

Introduction

●Introduction

●Ingredients

●Interpretation

●^Overview

– Either cheating option makes implementation of honesty impossible, or it leads to no distortions at all.

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Overview of Main Results

Introduction

●Introduction

●Ingredients

●Interpretation

●^Overview

– In the latter case, only reward the

m + 1

-st observable event (

m

appropriately large).

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Overview of Main Results

Introduction

●Introduction

●Ingredients

●Interpretation

●^Overview

m + 1

m

– Only honest agent believes he can produce many future events.

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Overview of Main Results

Introduction

●Introduction

●Ingredients

●Interpretation

●^Overview

m + 1

m

– Only honest agent believes he can produce many future events.

– Hence, principal will construct a continuation phase so that agent will only ever want to enter after an honest success.

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Some Stylized Facts on Innovation Performance

Introduction

●Introduction

●Ingredients

●Interpretation

●^Overview

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Some Stylized Facts on Innovation Performance

Introduction

●Introduction

●Ingredients

●Interpretation

●^Overview

⇒

Focussing incentives on the long term helps innovation.

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Some Stylized Facts on Innovation Performance

Introduction

●Introduction

●Ingredients

●Interpretation

●^Overview

⇒

Effort is a necessary condition for innovative breakthroughs.

But is it sufficient?

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Some Stylized Facts on Innovation Performance

Introduction

●Introduction

●Ingredients

●Interpretation

●^Overview

⇒

Francis, Hasan, Sharma, 2009: No impact of performance sensitivity of CEO pay on firm’s innovation performance.

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Some Stylized Facts on Innovation Performance

Introduction

●Introduction

●Ingredients

●Interpretation

●^Overview

⇒

But: Skewing incentives toward the long term does have a positive and significant impact on the number of patents and citations to patents (Francis, Hasan, Sharma, 2009, and

Lerner & Wulf, 2007).

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Some Stylized Facts on Innovation Performance

Introduction

●Introduction

●Ingredients

●Interpretation

●^Overview

⇒

But: Skewing incentives toward the long term does have a positive and significant impact on the number of patents and citations to patents (Francis, Hasan, Sharma, 2009, and

Lerner & Wulf, 2007).

Rewarding long-term performance seems crucial in spurring innovation.

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Literature Overview

Introduction

●Introduction

●Ingredients

●Interpretation

●^Overview

●^Literature

Setup

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Literature Overview

Introduction

●Introduction

●Ingredients

●Interpretation

●^Overview

●^Literature

Setup

● Holmstr ¨om & Milgrom (1991): multi-tasking agent;

different tasks can be monitored more (less) accurately;

no learning.

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Literature Overview

Introduction

●Introduction

●Ingredients

●Interpretation

●^Overview

●^Literature

Setup

no learning.

● Bergemann & Hege (1998, 2005), H ¨orner & Samuelson (2009): venture-capital financing; no “faking” of success.

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Literature Overview

Introduction

●Introduction

●Ingredients

●Interpretation

●^Overview

●^Literature

Setup

no learning.

● Fong (2009): Optimal Scoring Rules for Surgeons;

“cheating” is part of the model but learning is not.

(Surgeons know their type perfectly.)

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Literature Overview

Introduction

●Introduction

●Ingredients

●Interpretation

●^Overview

●^Literature

Setup

no learning.

● Fong (2009): Optimal Scoring Rules for Surgeons;

“cheating” is part of the model but learning is not.

(Surgeons know their type perfectly.)

● Manso (2011): Two-period binomial model.

Here: Can make additional statements on structure of incentive scheme, and optimal stopping.

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The Basic Setup

Introduction Setup

●^Setup

●^Learning

●The Principal’s Tools

●^Notation

●^Strategies

●No Surprise Continuation Scheme Before the First Breakthrough Optimal Stopping Conclusion Appendix

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The Basic Setup

Introduction Setup

●^Setup

●^Learning

●^Notation

●^Strategies

One principal, one agent (both risk neutral).

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The Basic Setup

Introduction Setup

●^Setup

●^Learning

●^Notation

●^Strategies

Time continuous; finite horizon

T < ∞

; end date if no

observed event

T < T

(at first exogenous, to be endogenized later); common discount rate

r > 0

.

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The Basic Setup

Introduction Setup

●^Setup

●^Learning

●^Notation

●^Strategies

T < ∞

; end date if no

observed event

T < T

r > 0

.

Agent operates a bandit machine, with

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The Basic Setup

Introduction Setup

●^Setup

●^Learning

●^Notation

●^Strategies

T < ∞

; end date if no

observed event

T < T

r > 0

.

● a safe arm; private benefit flow of

s > 0

;

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The Basic Setup

Introduction Setup

●^Setup

●^Learning

●^Notation

●^Strategies

T < ∞

; end date if no

observed event

T < T

r > 0

.

s > 0

;

● arm 0 (“cheating”); “successes” after exponentially

distributed times according to a known parameter

λ

⁰

k

⁰_,t;

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The Basic Setup

Introduction Setup

●^Setup

●^Learning

●^Notation

●^Strategies

T < ∞

; end date if no

observed event

T < T

r > 0

.

s > 0

;

λ

⁰

k

⁰_,t;

● arm 1 (“honesty”); successes after exponentially

distributed times according to the parameter

θλ

1

k

1,t;

λ

¹ known,

θ ∈ { 0, 1 }

initially unknown state of the world.

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The Basic Setup

Introduction Setup

●^Setup

●^Learning

●^Notation

●^Strategies

T < ∞

; end date if no

observed event

T < T

r > 0

.

s > 0

;

λ

⁰

k

⁰_,t;

● arm 1 (“honesty”); successes after exponentially

distributed times according to the parameter

θλ

1

k

1,t;

λ

¹ known,

θ ∈ { 0, 1 }

initially unknown state of the world.

Principal wants to implement use of arm 1 until the first breakthrough (possibly all the way up to

T

).

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Learning

Introduction Setup

●^Setup

●^Learning

●^Notation

●^Strategies

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Learning

Introduction Setup

●^Setup

●^Learning

●^Notation

●^Strategies

Whenever using arm 1, agent updates his private belief according to Bayes’ rule (conditional on no success):

ˆ

p

_t

= p

0

e

^−λ¹^∫⁰^t^k¹^,τ ^dτ

p

0

e

^−λ¹^∫⁰^t^k¹^,τ ^dτ

+ 1 − p

0

,

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Learning

Introduction Setup

●^Setup

●^Learning

●^Notation

●^Strategies

ˆ

p

_t

= p

0

e

^−λ¹^∫⁰^t^k¹^,τ ^dτ

p

0

e

^−λ¹^∫⁰^t^k¹^,τ ^dτ

+ 1 − p

0

,

or, equivalently,

p ˙ˆ

_t

= −λ

¹

k

¹_,t

p ˆ

_t

( 1 − p ˆ

_t

)

^.

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Learning

Introduction Setup

●^Setup

●^Learning

●^Notation

●^Strategies

ˆ

p

_t

= p

0

e

^−λ¹^∫⁰^t^k¹^,τ ^dτ

p

0

e

^−λ¹^∫⁰^t^k¹^,τ ^dτ

+ 1 − p

0

,

or, equivalently,

p ˙ˆ

_t

= −λ

¹

k

¹_,t

p ˆ

_t

( 1 − p ˆ

_t

)

^.

In equilibrium, the principal knows the agent’s belief:

p

_t

= p ˆ

_t

= p

0

e

^−λ¹^t

p

⁰

e

^−λ¹^t

+ 1 − p

⁰

.

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Learning

Introduction Setup

●^Setup

●^Learning

●^Notation

●^Strategies

ˆ

p

_t

= p

0

e

^−λ¹^∫⁰^t^k¹^,τ ^dτ

p

0

e

^−λ¹^∫⁰^t^k¹^,τ ^dτ

+ 1 − p

0

,

or, equivalently,

p ˙ˆ

_t

= −λ

¹

k

¹_,t

p ˆ

_t

( 1 − p ˆ

_t

)

^.

In equilibrium, the principal knows the agent’s belief:

p

_t

= p ˆ

_t

= p

0

e

^−λ¹^t

p

⁰

e

^−λ¹^t

+ 1 − p

⁰

.

After the first success,

p

_t

= 1

.

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The Principal’s Instruments

Introduction Setup

●^Setup

●^Learning

●^Notation

●^Strategies

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The Principal’s Instruments

Introduction Setup

●^Setup

●^Learning

●^Notation

●^Strategies

The principal has two tools: Payments and choosing the end date

T ˇ ( t ) ∈ [ t, T )

conditional on the first event having

occurred at time

t

. Formally,

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The Principal’s Instruments

Introduction Setup

●^Setup

●^Learning

●^Notation

●^Strategies

T ˇ ( t ) ∈ [ t, T )

occurred at time

t

. Formally,

● Point processes

N

_t¹ and

N

_t⁰: number of events produced on arm 1 (0) up to, and including, time

t

.

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The Principal’s Instruments

Introduction Setup

●^Setup

●^Learning

●^Notation

●^Strategies

T ˇ ( t ) ∈ [ t, T )

occurred at time

t

. Formally,

● Point processes

N

_t¹ and

N

t

.

●

( N

_t⁰

, N

_t¹

)

⁰≤t≤T induces filtration

F ∶= { F

_t

}

⁰≤t≤T.

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The Principal’s Instruments

Introduction Setup

●^Setup

●^Learning

●^Notation

●^Strategies

T ˇ ( t ) ∈ [ t, T )

occurred at time

t

. Formally,

● Point processes

N

_t¹ and

N

t

.

●

( N

_t⁰

, N

_t¹

)

F ∶= { F

_t

}

⁰≤t≤T.

●

{ N

_t

}

⁰≤t≤T (with

N

_t

∶= N

_t⁰

+ N

_t¹) induces filtration

F

^N

∶= { F

^N_t

}

⁰_≤t≤T^.

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The Principal’s Instruments

Introduction Setup

●^Setup

●^Learning

●^Notation

●^Strategies

T ˇ ( t ) ∈ [ t, T )

occurred at time

t

. Formally,

● Point processes

N

_t¹ and

N

t

.

●

( N

_t⁰

, N

_t¹

)

F ∶= { F

_t

}

⁰≤t≤T.

●

{ N

_t

}

⁰≤t≤T (with

N

_t

∶= N

_t⁰

+ N

_t¹) induces filtration

F

^N

∶= { F

^N_t

}

⁰_≤t≤T^.

● ^At

t = 0

, the principal commits to a non-decreasing, non-negative,

F

^N_-adapted process of payments

{W

^t

}

⁰_≤t≤T, and to a schedule of end dates

{ T ˇ ( t )}

⁰^≤t≤T^,

with

W

^t: time-0 value of cumulated payments to agent up to time

t

.

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A Simplification

Introduction Setup

●^Setup

●^Learning

●^Notation

●^Strategies

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A Simplification

Introduction Setup

●^Setup

●^Learning

●^Notation

●^Strategies

Clearly not a good idea to pay the agent in the absence of a event. Can simplify notation!

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A Simplification

Introduction Setup

●^Setup

●^Learning

●^Notation

●^Strategies

h

_t

∶= e

^rt

(W

^t

− lim

_τ_↑t

W

^τ

)

: immediate reward for first event;

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A Simplification

Introduction Setup

●^Setup

●^Learning

●^Notation

●^Strategies

h

_t

∶= e

^rt

(W

^t

− lim

_τ_↑t

W

^τ

)

w

_t: time

t

expected continuation value of an agent who has had a breakthrough on arm 1 at time

t

;

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A Simplification

Introduction Setup

●^Setup

●^Learning

●^Notation

●^Strategies

h

_t

∶= e

^rt

(W

^t

− lim

_τ_↑t

W

^τ

)

w

_t: time

t

;

ω

_t

( p ˆ

_t

)

^{: time}

t

expected continuation value of an agent who has had a “breakthrough” on arm 0 at time

t

while holding the (private) belief

p ˆ

_t.

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A Simplification

Introduction Setup

●^Setup

●^Learning

●^Notation

●^Strategies

h

_t

∶= e

^rt

(W

^t

− lim

_τ_↑t

W

^τ

)

w

_t: time

t

;

ω

_t

( p ˆ

_t

)

^{: time}

t

expected continuation value of an agent who has had a “breakthrough” on arm 0 at time

t

while holding the (private) belief

p ˆ

_t.

The principal’s objective is to minimize

∫

₀ ^T

e

^−rt−λ¹^∫⁰^t^p^τ ^dτ

p

_t

λ

¹

( h

_t

+ w

_t

) dt

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The Agent’s Strategies & Objective

Introduction Setup

●^Setup

●^Learning

●^Notation

●^Strategies

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The Agent’s Strategies & Objective

Introduction Setup

●^Setup

●^Learning

●^Notation

●^Strategies

By choosing a strategy, the agent influences the distribution over

( N

_t⁰

, N

_t¹

)

^t ^and

( N

_t

)

^t^.

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The Agent’s Strategies & Objective

Introduction Setup

●^Setup

●^Learning

●^Notation

●^Strategies

( N

_t⁰

, N

_t¹

)

^t ^and

( N

_t

)

^t^.

The agent chooses a process

( k

0,t

, k

1,t

)

_t ^{that is}

F

-predictable, where

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The Agent’s Strategies & Objective

Introduction Setup

●^Setup

●^Learning

●^Notation

●^Strategies

( N

_t⁰

, N

_t¹

)

^t ^and

( N

_t

)

^t^.

( k

0,t

, k

1,t

)

_t ^{that is}

F

-predictable, where

●

k

⁰_,t

≥ 0

,

k

¹_,t

≥ 0

, and

k

⁰_,t

+ k

¹_,t

≤ 1

;

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The Agent’s Strategies & Objective

Introduction Setup

●^Setup

●^Learning

●^Notation

●^Strategies

( N

_t⁰

, N

_t¹

)

^t ^and

( N

_t

)

^t^.

( k

0,t

, k

1,t

)

_t ^{that is}

F

-predictable, where

●

k

⁰_,t

≥ 0

,

k

¹_,t

≥ 0

, and

k

⁰_,t

+ k

¹_,t

≤ 1

;

●

k

_i,t: fraction devoted to arm

i

at time

t

;

●

1 − k

⁰_,t

− k

¹_,t is devoted to the safe arm at time

t

.