Designing and Selling Hard Information Nima Haghpanah r S. Nageeb - - PowerPoint PPT Presentation

Designing and Selling Hard Information

Nima Haghpanah r

• S. Nageeb Ali r

Xiao Lin r Ron Siegel Penn State September 23, 2020

1 / 18

Market for Hard Information

2 / 18

Market for Hard Information

Agent Market asset

2 / 18

Market for Hard Information

Agent Market asset Intermediary Hard info Hard info

2 / 18

Market for Hard Information

◮ Sellers of physical or financial products - Buyers ◮ Workers - Employers ◮ Entrepreneurs - Investors

Agent Market asset Intermediary Hard info Hard info

2 / 18

Market for Hard Information

◮ Sellers of physical or financial products - Buyers ◮ Workers - Employers ◮ Entrepreneurs - Investors

Intermediary’s profit-maximization problem

◮ Accurate tests or noisy tests? ◮ Upfront fees or disclosure fees? ◮ How much of the surplus can she appropriate?

Agent Market asset Intermediary Hard info Hard info

2 / 18

Agent (with asset), competitive market, intermediary

3 / 18

Agent (with asset), competitive market, intermediary

Value θ ∈ {0, 1}, prob. 0.5 each

◮ Unknown to all

3 / 18

Agent (with asset), competitive market, intermediary

Value θ ∈ {0, 1}, prob. 0.5 each

◮ Unknown to all

A test-fee structure (T, φ):

1 Test T : {0, 1} → ∆(S) ◮ Unbiased s = E[θ|s] 2 Testing fee φt

Disclosure fee φd

3 / 18

Agent (with asset), competitive market, intermediary

Value θ ∈ {0, 1}, prob. 0.5 each

◮ Unknown to all

A test-fee structure (T, φ):

1 Test T : {0, 1} → ∆(S) ◮ Unbiased s = E[θ|s] 2 Testing fee φt

Disclosure fee φd Agent

3 / 18

Agent (with asset), competitive market, intermediary

Value θ ∈ {0, 1}, prob. 0.5 each

◮ Unknown to all

A test-fee structure (T, φ):

1 Test T : {0, 1} → ∆(S) ◮ Unbiased s = E[θ|s] 2 Testing fee φt

Disclosure fee φd Agent s ∼ T(θ) φt

Test Not

3 / 18

Agent (with asset), competitive market, intermediary

Value θ ∈ {0, 1}, prob. 0.5 each

◮ Unknown to all

A test-fee structure (T, φ):

1 Test T : {0, 1} → ∆(S) ◮ Unbiased s = E[θ|s] 2 Testing fee φt

Disclosure fee φd Agent s ∼ T(θ) φt

Test Not

φd

Disclose Not

3 / 18

Agent (with asset), competitive market, intermediary

Value θ ∈ {0, 1}, prob. 0.5 each

◮ Unknown to all

A test-fee structure (T, φ):

1 Test T : {0, 1} → ∆(S) ◮ Unbiased s = E[θ|s] 2 Testing fee φt

Disclosure fee φd Agent s ∼ T(θ) φt

Test Not

φd

Disclose Not

Market observes s s

3 / 18

Agent (with asset), competitive market, intermediary

Value θ ∈ {0, 1}, prob. 0.5 each

◮ Unknown to all

A test-fee structure (T, φ):

1 Test T : {0, 1} → ∆(S) ◮ Unbiased s = E[θ|s] 2 Testing fee φt

Disclosure fee φd Agent s ∼ T(θ) φt

Test Not

φd

Disclose Not

Market observes s s “N”

• r “N”

3 / 18

Agent (with asset), competitive market, intermediary

Value θ ∈ {0, 1}, prob. 0.5 each

◮ Unknown to all

A test-fee structure (T, φ):

1 Test T : {0, 1} → ∆(S) ◮ Unbiased s = E[θ|s] 2 Testing fee φt

Disclosure fee φd Agent s ∼ T(θ) φt

Test Not

φd

Disclose Not

Market observes s s “N”

• r “N”

Market offers price = E[θ|observation]

3 / 18

Agent (with asset), competitive market, intermediary

Value θ ∈ {0, 1}, prob. 0.5 each

◮ Unknown to all

A test-fee structure (T, φ):

1 Test T : {0, 1} → ∆(S) ◮ Unbiased s = E[θ|s] 2 Testing fee φt

Disclosure fee φd Agent s ∼ T(θ) φt

Test Not

φd

Disclose Not

Market observes s s “N”

• r “N”

Market offers price = E[θ|observation] – Observe s: price(s) = E[θ|s] = s – Observe “N”: price(N) endogeneous

3 / 18

Agent (with asset), competitive market, intermediary

Value θ ∈ {0, 1}, prob. 0.5 each

◮ Unknown to all

A test-fee structure (T, φ):

1 Test T : {0, 1} → ∆(S) ◮ Unbiased s = E[θ|s] 2 Testing fee φt

Disclosure fee φd (T, φ) may have multiple equilibria Agent s ∼ T(θ) φt

Test Not

φd

Disclose Not

Market observes s s “N”

• r “N”

Market offers price = E[θ|observation] – Observe s: price(s) = E[θ|s] = s – Observe “N”: price(N) endogeneous

3 / 18

Revenue-maximizing test-fee structures?

4 / 18

Revenue-maximizing test-fee structures?

max

test-fee structure

max

equilibria

Revenue

4 / 18

Revenue-maximizing test-fee structures?

max

test-fee structure

max

equilibria

Revenue = 0.5 = E[θ] “full surplus”

4 / 18

Revenue-maximizing test-fee structures?

max

test-fee structure

max

equilibria

Revenue = 0.5 = E[θ] “full surplus” Fully reveal, φt = 0, φd = 1

◮ Equilibrium with revenue = 0.5

◮ Agent: take test, reveal if and only if score = 1 ◮ Market: price(N) = 0 4 / 18

Revenue-maximizing test-fee structures?

max

test-fee structure

max

equilibria

Revenue = 0.5 = E[θ] “full surplus” Fully reveal, φt = 0, φd = 1

◮ Equilibrium with revenue = 0.5

◮ Agent: take test, reveal if and only if score = 1 ◮ Market: price(N) = 0

◮ Another equilibrium with revenue = 0

◮ Agent: don’t take test ◮ Market: price(N) = 0.5 4 / 18

Revenue-maximizing test-fee structures?

max

test-fee structure

max

equilibria

Revenue = 0.5 = E[θ] “full surplus” Fully reveal, φt = 0, φd = 1

◮ Equilibrium with revenue = 0.5

◮ Agent: take test, reveal if and only if score = 1 ◮ Market: price(N) = 0

◮ Another equilibrium with revenue = 0

◮ Agent: don’t take test ◮ Market: price(N) = 0.5

∀ (T, φ) with an equilibrium where revenue ≥ 0.5 − ǫ

◮ ∃ another equilibrium of (T, φ) where revenue ≤ ǫ

4 / 18

Revenue-maximizing test-fee structures?

max

test-fee structure

max

equilibria

Revenue = 0.5 = E[θ] “full surplus” Fully reveal, φt = 0, φd = 1

◮ Equilibrium with revenue = 0.5

◮ Agent: take test, reveal if and only if score = 1 ◮ Market: price(N) = 0

◮ Another equilibrium with revenue = 0

◮ Agent: don’t take test ◮ Market: price(N) = 0.5

∀ (T, φ) with an equilibrium where revenue ≥ 0.5 − ǫ

◮ ∃ another equilibrium of (T, φ) where revenue ≤ ǫ

max

test-fee structure

min

equilibria

Revenue

4 / 18

Revenue-maximizing test-fee structures?

max

test-fee structure

max

equilibria

Revenue = 0.5 = E[θ] “full surplus” Fully reveal, φt = 0, φd = 1

◮ Equilibrium with revenue = 0.5

◮ Agent: take test, reveal if and only if score = 1 ◮ Market: price(N) = 0

◮ Another equilibrium with revenue = 0

◮ Agent: don’t take test ◮ Market: price(N) = 0.5

∀ (T, φ) with an equilibrium where revenue ≥ 0.5 − ǫ

◮ ∃ another equilibrium of (T, φ) where revenue ≤ ǫ

max

test-fee structure

min

equilibria

Revenue = 0.5 · (1 − 1/e) ≈ 0.31

4 / 18

“Robustly optimal” test-fee structure

5 / 18

“Robustly optimal” test-fee structure

1 Is unique (and has unique equilibrium) 2 Testing is free φt = 0 (disclosure is costly φd > 0) 3 Not fully revealing: continuum of scores 5 / 18

“Robustly optimal” test-fee structure

1 Is unique (and has unique equilibrium) 2 Testing is free φt = 0 (disclosure is costly φd > 0) 3 Not fully revealing: continuum of scores

1 1

Score distribution c.d.f

score

exponential

0.5 1/e

5 / 18

Fully reveal, free testing (φt = 0)

(Robustly) optimal disclosure fee φd?

Pr[s|θ] s = 0 s = 1 θ = 0 1 θ = 1 1

6 / 18

Fully reveal, free testing (φt = 0)

(Robustly) optimal disclosure fee φd?

Pr[s|θ] s = 0 s = 1 θ = 0 1 θ = 1 1

• Prob. of

Fee φd disclosure

6 / 18

Fully reveal, free testing (φt = 0)

(Robustly) optimal disclosure fee φd?

◮ φd ≥ 0.5: conceal any s, price(N) = 0.5

Pr[s|θ] s = 0 s = 1 θ = 0 1 θ = 1 1

• Prob. of

Fee φd disclosure 1 0.5

conceal

6 / 18

Fully reveal, free testing (φt = 0)

(Robustly) optimal disclosure fee φd?

◮ φd ≥ 0.5: conceal any s, price(N) = 0.5 ◮ φd ∈ [0, 1]: disclose iff s = 1, price(N) = 0

Pr[s|θ] s = 0 s = 1 θ = 0 1 θ = 1 1

• Prob. of

Fee φd disclosure 1 0.5 0.5

conceal disclose iff s = 1

6 / 18

Fully reveal, free testing (φt = 0)

(Robustly) optimal disclosure fee φd?

◮ φd ≥ 0.5: conceal any s, price(N) = 0.5 ◮ φd ∈ [0, 1]: disclose iff s = 1, price(N) = 0

Pr[s|θ] s = 0 s = 1 θ = 0 1 θ = 1 1

• Prob. of

Fee φd disclosure 1 0.5 0.5 1 0.5 1

conceal disclose iff s = 1

6 / 18

Fully reveal, free testing (φt = 0)

(Robustly) optimal disclosure fee φd?

◮ φd ≥ 0.5: conceal any s, price(N) = 0.5 ◮ φd ∈ [0, 1]: disclose iff s = 1, price(N) = 0

Pr[s|θ] s = 0 s = 1 θ = 0 1 θ = 1 1

• Prob. of

Fee φd disclosure 1 0.5 0.5

“robust demand curve”

1 0.5 1

conceal disclose iff s = 1

6 / 18

Fully reveal, free testing (φt = 0)

(Robustly) optimal disclosure fee φd?

◮ φd ≥ 0.5: conceal any s, price(N) = 0.5 ◮ φd ∈ [0, 1]: disclose iff s = 1, price(N) = 0

Pr[s|θ] s = 0 s = 1 θ = 0 1 θ = 1 1

Revenue

φd

• Prob. of

Fee φd disclosure 1 0.5 0.5

“robust demand curve”

1 0.5 1

conceal disclose iff s = 1

6 / 18

Fully reveal, free testing (φt = 0)

(Robustly) optimal disclosure fee φd?

◮ φd 0.5. Revenue 0.25 ◮ φd ≥ 0.5: conceal any s, price(N) = 0.5 ◮ φd ∈ [0, 1]: disclose iff s = 1, price(N) = 0

Pr[s|θ] s = 0 s = 1 θ = 0 1 θ = 1 1

• Prob. of

Fee φd disclosure 1 0.5 0.5

“robust demand curve”

1 0.5 1

conceal disclose iff s = 1

6 / 18

Benefit of noise (free testing φt = 0)

Pr[s|θ] s = 0 s = 3/4 s = 1 θ = 0 1 − p p θ = 1 3p 1 − 3p

7 / 18

Benefit of noise (free testing φt = 0)

Pr[s|θ] s = 0 s = 3/4 s = 1 θ = 0 1 − p p θ = 1 3p 1 − 3p

Prob. Fee 0.5 0.5

7 / 18

Benefit of noise (free testing φt = 0)

Pr[s|θ] s = 0 s = 3/4 s = 1 θ = 0 1 − p p θ = 1 3p 1 − 3p

Prob. Fee 0.5 0.5

conceal disclose s = 3/4 and s = 1 disclose s = 1

7 / 18

Benefit of noise (free testing φt = 0)

Pr[s|θ] s = 0 s = 3/4 s = 1 θ = 0 1 − p p θ = 1 3p 1 − 3p

Prob. Fee 0.5 0.5

conceal disclose s = 3/4 and s = 1 disclose s = 1

7 / 18

Benefit of noise (free testing φt = 0)

◮ φd 0.5 : Revenue > 0.25

Pr[s|θ] s = 0 s = 3/4 s = 1 θ = 0 1 − p p θ = 1 3p 1 − 3p

Prob. Fee 0.5 0.5

conceal disclose s = 3/4 and s = 1 disclose s = 1

7 / 18

Benefit of noise (free testing φt = 0)

◮ φd 0.5 : Revenue > 0.25

Increase p subject to 3 4 − φd

Disclose medium score

> 3p 1 + 3p

Nondisclosure price

Pr[s|θ] s = 0 s = 3/4 s = 1 θ = 0 1 − p p θ = 1 3p 1 − 3p

Prob. Fee 0.5 0.5

conceal disclose s = 3/4 and s = 1 disclose s = 1

7 / 18

Robustly Optimal Test (φd 0.5 & free testing φt = 0)

Prob. Fee 1 1 − 1/e 1 0.5

8 / 18

Robustly Optimal Test (φd 0.5 & free testing φt = 0)

1 1

Score distribution G

s

exponential

0.5 1/e Prob. Fee 1 1 − 1/e 1 0.5

8 / 18

Robustly Optimal Test (φd 0.5 & free testing φt = 0)

1 1

Score distribution G

s

exponential

0.5 1/e Prob. Fee 1 1 − 1/e 1 0.5

Equilibria τ: score > τ disclose score ≤ τ conceal

8 / 18

Robustly Optimal Test (φd 0.5 & free testing φt = 0)

φd = τ − E[score | score ≤ τ]

1 1

Score distribution G

s

exponential

0.5 1/e Prob. Fee 1 1 − 1/e 1 0.5

Equilibria τ: score > τ disclose score ≤ τ conceal

8 / 18

Robustly Optimal Test (φd 0.5 & free testing φt = 0)

φd = τ − E[score | score ≤ τ] =

τ

0 G(s)ds

G(τ)

1 1

Score distribution G

s

exponential

0.5 1/e Prob. Fee 1 1 − 1/e 1 0.5

Equilibria τ: score > τ disclose score ≤ τ conceal

8 / 18

Robustly Optimal Test (φd 0.5 & free testing φt = 0)

φd = τ − E[score | score ≤ τ] =

τ

0 G(s)ds

G(τ)

=

• d

dτ

• ln(

τ

0 G(s)ds)

−1

1 1

Score distribution G

s

exponential

0.5 1/e Prob. Fee 1 1 − 1/e 1 0.5

Equilibria τ: score > τ disclose score ≤ τ conceal

8 / 18

Robustly Optimal Test (φd 0.5 & free testing φt = 0)

φd = τ − E[score | score ≤ τ] =

τ

0 G(s)ds

G(τ)

=

• d

dτ

• ln(

τ

0 G(s)ds)

−1 ⇒ G(τ) = αeτ/φd

1 1

Score distribution G

s

exponential

0.5 1/e Prob. Fee 1 1 − 1/e 1 0.5

Equilibria τ: score > τ disclose score ≤ τ conceal

8 / 18

Robustly Optimal Test (φd 0.5 & free testing φt = 0)

φd = τ − E[score | score ≤ τ] =

τ

0 G(s)ds

G(τ)

=

• d

dτ

• ln(

τ

0 G(s)ds)

−1 ⇒ G(τ) = αeτ/φd

1 1

Score distribution G

s

exponential

0.5 1/e 1 1 If value = 0 s 0.5 2/e 1 1 If value = 1 s 0.5

8 / 18

Optimal test-fee structure

Proposition

There is a unique robustly optimal test-fee structure. It consists of testing fee φ∗

t = 0, disclosure fee φ∗ d = 0.5, and test T below.

1 1

Score distribution

s

exponential

0.5 1/e 1 1 If value = 0 s 0.5 2/e 1 1 If value = 1 s 0.5

9 / 18

Optimal test-fee structure

Proposition

There is a unique robustly optimal test-fee structure. It consists of testing fee φ∗

t = 0, disclosure fee φ∗ d = 0.5, and test T below.

1 1

Score distribution

s

exponential

0.5 1/e 1 1 If value = 0 s 0.5 2/e 1 1 If value = 1 s 0.5

9 / 18

Optimal test-fee structure

Proposition

There is a unique robustly optimal test-fee structure. It consists of testing fee φ∗

t = 0, disclosure fee φ∗ d = 0.5, and test T below.

1 1

Score distribution

s

exponential

0.5 1/e 1 1 If value = 0 s 0.5 2/e 1 1 If value = 1 s 0.5

9 / 18

Optimal test-fee structure

Proposition

There is a unique robustly optimal test-fee structure. It consists of testing fee φ∗

t = 0, disclosure fee φ∗ d = 0.5, and test T below.

If testing was observed, revenue = 0

◮ Voluntary testing hurts the agent

Voluntary testing:

◮ Tempt agent with option value given zero testing fee ◮ “N” is bad news ⇒ high disclosure fee

1 1

Score distribution

s

exponential

0.5 1/e 1 1 If value = 0 s 0.5 2/e 1 1 If value = 1 s 0.5

9 / 18

Arbitrary prior over θ ∈ [0, 1]

10 / 18

Arbitrary prior over θ ∈ [0, 1]

Proposition

1 0 < Robustly optimal revenue < full surplus. 10 / 18

Arbitrary prior over θ ∈ [0, 1]

Proposition

1 0 < Robustly optimal revenue < full surplus. 2 Exists robustly optimal “step-exponential-step” test-fee structure.

CDF Score

10 / 18

Arbitrary prior over θ ∈ [0, 1]

Proposition

1 0 < Robustly optimal revenue < full surplus. 2 Exists robustly optimal “step-exponential-step” test-fee structure.

• disc. rev.

Fee Prob demand CDF Score

10 / 18

Arbitrary prior over θ ∈ [0, 1]

Proposition

1 0 < Robustly optimal revenue < full surplus. 2 Exists robustly optimal “step-exponential-step” test-fee structure.

• disc. rev.

Fee Prob demand CDF Score

10 / 18

Arbitrary prior over θ ∈ [0, 1]

Proposition

1 0 < Robustly optimal revenue < full surplus. 2 Exists robustly optimal “step-exponential-step” test-fee structure. 3 Every robustly optimal test-fee structure has disclosure fee > 0

Contrast with “maximize value and extract via testing fee”

• disc. rev.

Fee Prob demand CDF Score

10 / 18

Arbitrary prior over θ ∈ [0, 1]

Proposition

1 0 < Robustly optimal revenue < full surplus. 2 Exists robustly optimal “step-exponential-step” test-fee structure. 3 Every robustly optimal test-fee structure has disclosure fee > 0

Contrast with “maximize value and extract via testing fee” Testing fee?

◮ Positive for log-concave priors ◮ May be zero (e.g., for binary prior)

• disc. rev.

Fee Prob demand CDF Score

10 / 18

The Testing Stage

How to preclude no-testing equilibria

11 / 18

The Testing Stage

How to preclude no-testing equilibria The answer will be useful to explain:

1 disclosure fee > 0 2 If φd = 0 then full revelation is optimal 3 If ∃ eq with revenue = FS, then ∃ eq with revenue = 0 11 / 18

Precluding no-testing equilibria

12 / 18

Precluding no-testing equilibria

price(N) = µ < 1 max{µ, s − φd}

• Option Value

dG − φt,

12 / 18

Precluding no-testing equilibria

price(N) = µ < 1 max{µ, s − φd}

• Option Value

dG − φt, Rearranging: φt < 1

µ+φd

[s − (µ + φd)]dG, (P)

12 / 18

Precluding no-testing equilibria

price(N) = µ < 1 max{µ, s − φd}

• Option Value

dG − φt, Rearranging: φt < 1

µ+φd

[s − (µ + φd)]dG, (P)

Lemma

1 If !(P), ∃ equilibrium: agent tests asset with probability 0 2 If (P), ∀ equilibria:

agent tests asset with probability 1

12 / 18

Precluding no-testing equilibria

price(N) = µ < 1 max{µ, s − φd}

• Option Value

dG − φt, Rearranging: φt < 1

µ+φd

[s − (µ + φd)]dG, (P)

Lemma

1 If !(P), ∃ equilibrium: agent tests asset with probability 0 2 If (P), ∀ equilibria:

agent tests asset with probability 1 In robustly optimal test-fee structure,

◮ Agent tests asset with probability 1 in all equilibria

12 / 18

The Testing Stage

How to preclude no-testing equilibria: Constraint P

13 / 18

The Testing Stage

How to preclude no-testing equilibria: Constraint P The answer will be useful to explain:

1 disclosure fee > 0 2 If φd = 0 then full revelation is optimal 3 If ∃ eq with revenue = FS, then ∃ eq with revenue = 0 13 / 18

Optimality of positive disclosure fee

Recall P: φt < 1

µ+φd[s − (µ + φd)]dG =

1

µ+φd(1 − G(s))ds

14 / 18

Optimality of positive disclosure fee

Recall P: φt < 1

µ+φd[s − (µ + φd)]dG =

1

µ+φd(1 − G(s))ds

revenue = φt

Score distribution G

s 1

µ (1 − G(s))ds

µ

14 / 18

Optimality of positive disclosure fee

Recall P: φt < 1

µ+φd[s − (µ + φd)]dG =

1

µ+φd(1 − G(s))ds

revenue = φt

Score distribution G

s 1

µ (1 − G(s))ds

µ revenue = φ′

t+φd(1 − G(τ))

≥ φ′

t+φd(1 − G(µ))

14 / 18

Optimality of positive disclosure fee

Recall P: φt < 1

µ+φd[s − (µ + φd)]dG =

1

µ+φd(1 − G(s))ds

revenue = φt

Score distribution G

s 1

µ (1 − G(s))ds

µ 1

µ+φd(1 − G(s))ds

Score distribution G

s µ + φd revenue = φ′

t+φd(1 − G(τ))

≥ φ′

t+φd(1 − G(µ))

G(µ) µ

14 / 18

φd = 0 ⇒ Full Revelation is Robustly Optimal

Recall P: φt < 1

µ+φd(1 − G(s))ds

15 / 18

φd = 0 ⇒ Full Revelation is Robustly Optimal

Recall P: φt < 1

µ+φd(1 − G(s))ds

revenue = φt ≤ 1

µ (1 − G(s))ds

Score distribution G

s 1

µ (1 − G(s))ds

µ

15 / 18

φd = 0 ⇒ Full Revelation is Robustly Optimal

Recall P: φt < 1

µ+φd(1 − G(s))ds

revenue = φt ≤ 1

µ (1 − G(s))ds

Score distribution G

s 1

µ (1 − G(s))ds

µ G mean-preserving contraction of F: 1

µ (1 − G(s))ds ≤

1

µ (1 − F(s))ds

G = F is opt

15 / 18

∃ eq with revenue = FS, ∃ eq with revenue = 0

Recall P: φt < 1

µ+φd(1 − G(s))ds

revenue = φt + φd(1 − G(τ)) ≤ φt + φd(1 − G(φd)) ≤ 1 (1 − G(s))ds = µ CDF Score G µ + φd 1 1 φd

16 / 18

∃ eq with revenue = FS, ∃ eq with revenue = 0

Recall P: φt < 1

µ+φd(1 − G(s))ds

revenue = φt + φd(1 − G(τ)) ≤ φt + φd(1 − G(φd)) ≤ 1 (1 − G(s))ds = µ CDF Score G µ + φd 1 1 φd CDF Score G µ + φd 1 1 φd

16 / 18

∃ eq with revenue = FS, ∃ eq with revenue = 0

Recall P: φt < 1

µ+φd(1 − G(s))ds

revenue = φt + φd(1 − G(τ)) ≤ φt + φd(1 − G(φd)) ≤ 1 (1 − G(s))ds = µ Binary scores {0, φd}, φt = 0, φd > 0

◮ Equilibrium: test asset, reveal score iff s = φd, price(N) = 0 ◮ Other equilibrium: don’t test asset, price(N) = µ

CDF Score G µ + φd 1 1 φd CDF Score G µ + φd 1 1 φd

16 / 18

Related Work

Revenue-maximizing information intermediaries:

◮ Lizzeri (1999). Informed agent, mandatory disclosure:

◮ Signaling vs. voluntary disclosure

◮ DeMarzo, Kremer, Skrzypacz (2019). “favorable” selection

17 / 18

Related Work

Revenue-maximizing information intermediaries:

◮ Lizzeri (1999). Informed agent, mandatory disclosure:

◮ Signaling vs. voluntary disclosure

◮ DeMarzo, Kremer, Skrzypacz (2019). “favorable” selection

Adversarial equilibrium selection in information/mechanism design:

◮ Dworczak and Pavan (2020), Halac, Kremer, Winter (2020), Halac,

Lipnowski, Rappoport (2020), ...

17 / 18

Related Work

Revenue-maximizing information intermediaries:

◮ Lizzeri (1999). Informed agent, mandatory disclosure:

◮ Signaling vs. voluntary disclosure

◮ DeMarzo, Kremer, Skrzypacz (2019). “favorable” selection

Adversarial equilibrium selection in information/mechanism design:

◮ Dworczak and Pavan (2020), Halac, Kremer, Winter (2020), Halac,

Lipnowski, Rappoport (2020), ... Information design and unit-elastic distributions:

◮ Roesler and Szentes (2017), Ortner and Chassang (2018), Condorelli

and Szentes (2020), ...

◮ Indifference condition vs. cross-equilibrium constraint 17 / 18

Extensions

1 Technological constraints: Certifier has a set of feasible tests ◮ Assumption: feasible to garble a feasible test ◮ Step-exponential-step is optimal, φd > 0 2 Designing multiple pieces of evidence ◮ Single evidence optimal 18 / 18

Extensions

1 Technological constraints: Certifier has a set of feasible tests ◮ Assumption: feasible to garble a feasible test ◮ Step-exponential-step is optimal, φd > 0 2 Designing multiple pieces of evidence ◮ Single evidence optimal

Revenue

Demand Fee

Prob.

Score distribution

s Thanks!

18 / 18