Partial information disclosure in a contest

Contents lists available atScienceDirect

Economics Letters

journal homepage:www.elsevier.com/locate/ecolet

Partial information disclosure in a contest

Derek J. Clark

^a,∗

, Tapas Kundu

^b

aSchool of Business and Economics, UiT the Arctic University of Norway, Norway

bOslo Business School, Oslo Metropolitan University, Norway

a r t i c l e i n f o

Article history:

Received 19 March 2021

Received in revised form 21 April 2021 Accepted 14 May 2021

Available online 18 May 2021 JEL classification:

D02 D72 D82 Keywords:

Contest

Information design Bayesian persuasion

a b s t r a c t

Zhang and Zhou (2016) use the concept of Bayesian persuasion due to Kamenica and Gentzkow (2011) to analyze information disclosure in a contest with one-sided asymmetric information. They show that an effort-maximizing designer can manipulate information disclosure to increase expected efforts in the contest, based upon active contest participation by all types of the informed player. We allow some informed types to exert no effort in the contest, showing how this (i) can increase the applicability of the previous results, and (ii) in some cases, can change the type of information disclosure.

©2021 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

1. Introduction

Contests in which resources are sunk to win a prize capture competition in social, political and economic spheres. A common theme is how a designer (principal) can maximize the resources expended in the contest. RecentlyZhang and Zhou(2016) intro- duced information disclosure as an instrument at the disposal of the principal, using the Bayesian Persuasion framework of Kamenica and Gentzkow (2011). In a two-player contest,Zhang and Zhou (2016) focus on one-sided informational asymmetry, where one player has better information than the competitor and the principal. The effort-maximizing, but uninformed, principal initially commits to a set of state-conditional distributions of sig- nals before realization of the state, which is the value of the prize to the player with private information; the signals disclose all or no information at two extremes, but can also impart a particular posterior belief to the uninformed. The optimal distribution of signals raises the principal’s payoff to the concavification of the total expected effort function.

Zhang and Zhou(2016) show first that binary values for the state yields an expected effort function that is either globally convex or concave; in the former case, full information disclosure is optimal, and in the latter there is no disclosure.¹ Only when there are more than two possible valuations can partial disclosure

∗ Corresponding author.

E-mail addresses: [email protected](D.J. Clark),[email protected] (T. Kundu).

1 This followsKamenica and Gentzkow(2011), and is explained later.

appear, in which the signal reveals the true value of the prize imperfectly to the uninformed player. Zhang and Zhou (2016) consider only fully internal solutions in which all types of the informed player have an effort level above zero. Epstein and Mealem(2013) show with two types for the informed player that an equilibrium exists in which the lower value type will not exert effort in the contest. We extend the results ofZhang and Zhou (2016) by considering equilibria in which some types exert no effort, and we fully characterize optimal information disclosure in the two-type case. Furthermore, we show how these results have consequences for deriving the optimal disclosure policy when there are more types.

2. Analysis

InZhang and Zhou(2016), there are two risk-neutral players, Aand B. PlayerA’s value of winning the contest is

v

A and this is common knowledge. PlayerB’s value

v

B (the state) is private information, but it is commonly known that it takes N

≥

2 values,

v

1

< v

2

< · · · < v

N, with prior

µ

⁰

=

(

µ

⁰₁

, . . . , µ

⁰_N)

∈

P^N

=

{

(

^p1

, . . . ,

^pN

) :

p_j

≥

0

,

∑N j=₁p_j

=

1

}

. Before the state is realized, the contest designer commits to a signaling mecha- nism, which consists of a family of state-conditional distributions

{

Pr[

m_s

| v

j

]

≥

0

:

m_s

∈

S

,

∑

ms∈_SPr[ m_s

| v

j

]

=

1

} ,

^j

∈ {

1

, . . . ,

^N

}

over a finite set of messagesS. We denote the Bayesian posterior after observing messagem_s

∈

Sby

µ

^s

=

(

µ

^s₁

, . . . , µ

^s_N)

∈

P^N. We use the notation

µ ∈

P^N to represent any generic distribution over the state space.

https://doi.org/10.1016/j.econlet.2021.109915

0165-1765/©2021 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

In the posterior contest, players exert non-recoverable effort

(

^xA

,

^xB

)

, which gives playeri

∈ {

A

,

^B

}

a success probability of p_i

(

^xA

,

^xB

) =

^xi

x_A

+

x_B

.

Denote the pure strategy Bayes–Nash equilibrium by[ x^∗_A

,

^x∗_B(

v

j

)]. Observe that the effort ofBmaximizes

x_B

x_B

+

x^∗_A

v

j

−

x_B

.

The first-order condition gives x_B(

v

j

)

=

{√

v

j

√x^∗_A

−

x^∗_A for

√ v

j

−

√

x^∗_A

>

⁰

0 for

√

v

j

−

√

x^∗_A

≤

0

,

⁽¹⁾

from which it is apparent that some low

v

B types may not participate actively in the contest.

For now, fix a distribution

µ ∈

P^N of playerB types and a set of inactive types 1

, . . . ,

^{k (i.e.,x}B

(

v

j

)

=

0 for j

=

1

, . . . ,

^k), whilstk

+

1

, . . . ,

^N participate actively (i.e.,x_B(

v

j

)

>

^{0 forj}

=

k

+

1

, . . . ,

^{N); ifk}

=

0, then all playerBtypes exert effort. When k

>

^{0, player A}wins with certainty if he meets types 1

, . . . ,

^k, making his expected payoff

( _k

∑

h=₁

µ

h

+

N

∑

m=_k+₁

µ

mx_A x_A

+

x_B

(v

m

)

)

v

A

−

x_A

.

The first-order condition is

( _N

∑

m=_k+₁

µ

mx_B

(v

m

) (

^xA

+

xB

(v

m

))

²

)

v

A

=

1

.

⁽²⁾

Solving (1) and (2) gives a solution for x_A when k types are inactive as

x^∗_A

(

^k

) =

⎛

⎝

∑N m=_k+₁

(õm vm

)

v1A

+

∑N m=_k+₁

(µm vm

)

⎞

⎠

2

.

⁽³⁾

Replacingx^∗_A in(1)by(3)gives

x^∗_B(

v

j

)

= √ v

j

⎛

⎝

∑N m=k+1

(_µ

√m vm

)

v1A

+

∑N m=_k+₁

(µm vm

)

⎞

⎠

−

⎛

⎝

∑N m=_k+₁

(õm vm

)

v1A

+

∑N m=k+1

(_µ

m vm

)

⎞

⎠

2

,

^j

=

k

+

1

, . . . ,

^N

.

⁽⁴⁾

None of the inactive player Btypes will want to exert positive effort as long as

√

v

k

−

√

x^∗_A

(

^k

) ≤

0. Using(3)and(4)yields total effort withkinactive types,TE

(µ,

^k

)

^{, as}

TE

(µ,

^k

) =

x^∗_A

(

^k

) +

N

∑

m=_k+₁

µ

mx^∗_B

(v

m

) .

⁽⁵⁾

Zhang and Zhou(2016) consider an internal solution, in which casek

=

0 and the total expected effort is

TE

(µ,

⁰

) =

E_µ[

√

v

B

]E_µ [√1

vB

]

v1A

+

E_µ [1

vB

]

.

⁽⁶⁾

The expression in (1)makes it clear that low

v

B types may not find it profitable to exert effort. This implies that participation has to be checked for playerBof lowest type

v

1 first, given that the other players exert positive effort. Only if type

v

1 makes a positive contest effort do we have the internal equilibrium ofZhang and Zhou(2016); if type

v

1does not exert effort, then

active participation is checked for

v

2 given that all types with a higher valuation participate. This proceeds in sequence until two adjacent types are identified such thatx^∗_B

(v

k

) =

0

,

^x∗_B

(v

k+₁

) >

^0.

Lemma 1determines the set of active types for a given

µ ∈

P^N. Lemma 1. Consider

µ ∈

P^N. Thresholds

θ

k

(µ) >

^{0, k}

∈ {

1

, . . . ,

^N

−

1

}

exist where

θ

k

(µ) ≤ θ

k+₁

(µ)

for all k and with strict inequality ifmax

{ µ

k+₁

, . . . , µ

N

} >

0, such that

θ

k

(µ) ≤ v

A

< θ

k+₁

(µ)

^{yields x∗}_B(

v

j

)

=

0, for j

∈ {

1

, . . . ,

^k

}

and x^∗_B(

v

j

)

>

^0, for j

∈ {

k

+

1

, . . . ,

^N

}

.

Proof. Suppose that playerBtypesj

=

1

, . . . ,

^ksetx∗_B(

v

j

)

=

0.

From(1), typekwill not want to change action if

√ v

k

≤

√

x^∗_A

(

^k

)

^, i.e.,

√ v

k

≤

∑N m=_k+₁

(õm vm

)

v1A

+

∑N m=_k+₁

(_µ

m vm

), ⁽⁷⁾

which reduces to

v

A

≥

√ v

k

∑N m=_k+₁

µm

(

^{√vm−√vk}

)

vm

:= θ

k

(µ) .

⁽⁸⁾

Typekbeing inactive, it follows from(7)that playerBtypes with

v

j

< v

kwill not participate ifx^∗_B

(v

k

) =

0. By construction, player B types with

v

j

> v

kwill participate if

v

A

< θ

k+₁

(µ)

^{. To see}

θ

k

(µ) ≤ θ

k+₁

(µ)

, note that for anym

>

^k,

v

k

< v

k+₁

⇒ µ

m

(

√ v

m

− √

v

k+₁

)

v

m

√ v

k+₁

≤ µ

m

(

√ v

m

− √

v

k

)

v

m

√ v

k

.

⁽⁹⁾ Summing(9)overm

∈ {

k

+

1

, . . . ,

^N

}

,

N

∑

m=_k+₂

µ

m

(√

v

m

− √ v

k+₁

)

v

m

√ v

k+₁

≤

N

∑

m=_k+₁

µ

m

(√

v

m

− √ v

k

)

v

m

√ v

k

(10)

⇒

¹

θ

k+₁

(µ) ≤

¹

θ

k

(µ) ⇒ θ

k

(µ) ≤ θ

k+₁

(µ) .

The inequality in(10)holds strictly if max

{ µ

k+₁

, . . . , µ

N

} >

^{0, in} which case,

θ

k

(µ) < θ

k+₁

(µ)

^. □

Setting

θ

0

(µ) =

0 and

θ

N

(µ) = ∞

, byLemma 1, we can express the equilibrium total effort for a given belief

µ ∈

P^N as TE^e

(µ) =

TE

(µ,

^k

)

^if

v

A

∈

[

θ

k

(µ) , θ

k+₁

(µ)) ,

^k

=

0

,

¹

, . . . ,

^N

−

1

.

(11) Lemma 1 includes two main results: (i) it characterizes the precise condition (

v

A

< θ

1(

µ

)) under which theZhang and Zhou (2016) analysis holds in which all playerBtypes actively partic- ipate in the contest for a given belief

µ

and a set of prize values

v

1

, v

2

, . . . , v

N, (ii) it gives conditions under which a subset of types

{

1

, . . . .,

^k

}

does not exert effort in the contest, A sufficient condition for full type participation can be derived by considering belief-free thresholds; these are outlined inLemma 2.

Lemma 2. Fix k

∈ {

1

, . . . ,

^N

−

1

}

. Denote min

µ∈_P^N

θ

k

(µ)

^by

θ

_k^min. Then,

θ

k^min

=

min

vm∈

{

_vk+1,...,vN

} v

m

√ v

k

√ v

m

− √

v

k

.

⁽¹²⁾

Further,4

v

k

≤ θ

_k^min

< θ

_k^min+₁.

Proof.

θ

k

(µ)

is minimized by identifying the largest value of

(

^{√vm−√vk}

)

vm form

=

k

+

1

, . . . ,

N, and attaching belief 1 to this particular

v

mand zero to all others. To show that

θ

_k^min

< θ

_k^min+₁,

2

first note that ^vm

√vk

√vm−√

vk is increasing in

v

k. Therefore, for any m

∈ {

k

+

2

, . . . ,

^N

}

, ^vm

√vk+1

√vm−√

vk+1

>

^√v_v^m_m^√−^v√^k

vk for a common

v

m. Suppose that

v

M

∈ { v

k+₂

, . . . , v

N

}

minimizes

θ

_k^min+₁

=

^vM

√vk+1

√vM−√ vk+1. Then it is possible to choose the same

v

Mand reach a lower value of

θ

k^min. Hence

θ

k^min

< θ

k^min+1 fork

∈ {

1

, . . . ,

^N

−

1

}

. Further, note that ^vm

√vk

√vm−√

vk is decreasing in

v

mfor

v

m

<

⁴

v

kand increasing in

v

mfor

v

m

>

⁴

v

k, which gives ^vm

√vk

√vm−√

vk

≥

^vm

√vk

√vm−√

vk

|

_vm=₄vk

=

4

v

k

for any

v

m

∈ { v

k+₁

, . . . , v

N

}

, and therefore,

θ

_k^min

≥

4

v

k. □ Lemma 2 makes two important observations regarding the validity of the internal solution considered in Zhang and Zhou (2016). First, we see that for

v

A

< θ

₁^min, all player B types participate actively in the contest for any prior

µ

and the internal solution ofZhang and Zhou(2016) is valid. However, the exact value of

θ

1^min depends on the parameters

v

2

, . . . , v

N. Lemma 2 further implies that if

v

A

≤

4

v

1, then

v

A

< θ

1^min for any

v

2

, . . . , v

N and the internal solution remains valid. This links to the analysis ofZhang and Zhou(2016, footnote 5) who state that a sufficient condition for the interior equilibrium is

v

A

≤

4

v

1. Our statement of the sufficient condition extends the parameter range for whichZhang and Zhou(2016) is valid.

FollowingKamenica and Gentzkow(2011), we can determine the optimal information disclosure from the concave closure of TE^e

(µ)

. The principal increases her expected payoff to the con- cavification of TE^e

(µ)

by optimally choosing a distribution of Bayes-plausibleposteriors generated from the signal distributions

{

Pr[

m_s

| v

j

]

,

^ms

∈

S

} ,

^j

∈ {

1

, . . . ,

^N

}

. If TE^e

(µ)

is globally concave (convex), then no- (full-) information disclosure yields the principal a payoff equal to the concavification ofTE^e

(µ)

^{. The} principal’s preferred signaling mechanism can partially disclose information only ifTE^e

(µ)

has both concave or convex parts. To highlight the role of information disclosure in the case ofk

=

0 (all types participate actively), andk

>

0 (some inactive types), we first present the binary-type case and then look at the case of more types.

2.1. N

=

2

Consider a posterior

µ = (µ

1

, µ

2

) ∈

P² over playerBtypes

(v

1

, v

2

)

^{. SinceN}

=

2, the posterior

µ

can be identified with a scalar

µ

2

=

Pr [

v

B

= v

2]

∈

[0

,

1]. Both types exert effort in the contest for any

µ

2 if

v

A

< θ

₁^min

=

^v2

√v1

(

^{√v2−√v1}

)

^.Zhang and Zhou (2016, Lemma 1 and Proposition 3) show that the total effort TE

(µ,

⁰

)

with both player B types active is strictly concave in

µ

2

∈

[0

,

^{1] for}

v

A

< √

v

2

v

1and therefore no disclosure is optimal;

and TE

(µ,

⁰

)

is strictly convex in

µ

2

∈

[0

,

^{1] for}

v

A

> √ v

2

v

1

and therefore full disclosure is optimal.²Note that

θ

₁^min

> √ v

2

v

1, and so the full-information disclosure finding ofZhang and Zhou (2016) holds for

√

v

2

v

1

< v

A

< θ

₁^min.

Fact 1(Zhang and Zhou(2016, Proposition 3, modified)). For N

=

2, consider

v

A

< θ

₁^min. Then, both types of player B exert non-zero effort in the contest under asymmetric information for any posterior

µ

^. Further, for

v

A

< √

v

2

v

1, no disclosure is optimal and for

√ v

2

v

1

<

v

A

< θ

₁^min, full disclosure is optimal.

This is an important result sinceZhang and Zhou(2016) show that the general case withN

>

2 can be reduced to that ofN

=

2.

For our extended parameter space, even the caseN

=

2 is not so 2 _{Unlike us,}Zhang and Zhou(2016) describe the concavity/convexity property of(6)in terms ofµ1=Pr [vB=v1]. However, the findings are comparable since the second-order derivatives ofTE^ewith respect toµ1andµ2=(¹−µ1)^have the same sign.

clear cut; we show below that partial information disclosure can be optimal.

Consider

v

A

≥ θ

₁^{min. By Lemma 1} and the fact that

θ

1

(µ

2

)

is decreasing in

µ

2, there exists a unique ˜

µ

2 satisfying

v

A

= θ

1

(˜ µ

2

)

such that both types exert effort for

µ

2

∈ [

0

,

˜

µ

2). Direct calculation gives

˜

µ

2

= v

2

√ v

1

v

A

(√

v

2

− √ v

1

)

.

For

µ

2

∈

[˜

µ

2

,

1], type 1 is inactive andTE^e

(µ

2

) =

TE

(µ

2

,

¹

)

^{. We} can calculate the derivatives as

∂

^TE

(µ

2

,

¹

)

∂µ

2

=

²

µ

2

v

A

v

₂²

(v

A

+ v

2

)

(µ

2

v

A

+ v

2

)

³

>

⁰

,

⁽¹³⁾

∂

^2TE

(µ

2

,

¹

)

∂µ

²₂

=

²

v

A

v

²₂

(v

2

−

2

v

A

µ

2

)

(µ

2

v

A

+ v

2

)

⁴

.

⁽¹⁴⁾

Defineˆ

µ

2

:=

^v2

2vA. From(13)and(14), it follows thatTE

(µ

2

,

¹

)

is always increasing in

µ

2, strictly concave (convex) for

µ

2

>

(

<

⁾ˆ

µ

2. When ˆ

µ

2

≥

1, which occurs if

v

A

≤

^v₂², the total expected effort is piecewise convex in

µ

2.Lemma 3shows that full information disclosure remains optimal.

Lemma 3. Suppose

θ

₁^min

<

^v₂² and consider

v

A

∈

[

θ

₁^min

,

^v₂²]

. Then, full information disclosure is optimal.

Proof. Note thatTE^e

(µ

2

)

is given byTE

(µ

2

,

⁰

)

^for

µ

2

∈ [

0

,˜ µ

2), and TE

(µ

2

,

¹

)

otherwise; both functions are convex in

µ

2 and TE^e

(µ

2

)

is continuous at˜

µ

2. Therefore,TE^e

(µ

2

)

is continuous and piecewise convex in

µ

2

∈

[0

,

1]. Further,

TE^e

(˜ µ

2

) =

TE

(˜ µ

2

,

^k

=

0

) ≤ (

¹

−

˜

µ

2

)

^TE

(µ

2

=

0

,

^k

=

0

) +

˜

µ

2TE

(µ

2

=

1

,

^k

=

0

)

= (

¹

−

˜

µ

2

)

^TE

(µ

2

=

0

,

^k

=

0

) +

˜

µ

2TE

(µ

2

=

1

,

^k

=

1

)

= (

¹

−

˜

µ

2

)

^TEe

(

⁰

) +

˜

µ

2TE^e

(

¹

) ,

which follows from convexity of TE

(µ

2

,

⁰

)

and the fact that TE

(µ

2

=

1

,

^k

=

0

) =

TE

(µ

2

=

1

,

^k

=

1

) =

_v^vAv2

A+v2. Therefore, the graph of TE^e

(µ

2

)

will always be lower than the straight line joining TE^e

(

⁰

)

^{and TEe}

(

¹

)

, implying that full disclosure is optimal. □

Whenˆ

µ

2

<

1, which occurs if

v

A

>

^v₂², total expected effort is concave for

µ

2

≥

max

{

ˆ

µ

2

,˜ µ

2

}

and either convex or piecewise convex for

µ

2

<

^max

{

ˆ

µ

2

,˜ µ

2

}

.Proposition 1shows that partial information disclosure is optimal for sufficiently large values of

v

A.

Proposition 1. Consider

v

A

>

^max{

θ

1^min

,

^v₂²}

. Then, there exists

v

A

>

^max{

θ

₁^min

,

^v₂²}

such that max{

θ

₁^min

,

^v₂²}

< v

A

< v

A, full information disclosure is optimal and for

v

A

≤ v

A, partial information disclosure is optimal.

Proof. TE^e

(µ

2

)

is given by TE

(µ

2

,

⁰

)

^for

µ

2

∈ [

0

,˜ µ

2), and TE

(µ

2

,

¹

)

^for

µ

2

∈

[˜

µ

2

,

1]; the former is convex, whilst the latter is either concave for

µ

2

∈

[˜

µ

2

,

^{1] if}ˆ

µ

2

≤

˜

µ

2, or, first convex for

µ

2

∈

[˜

µ

2

,

ˆ

µ

2] and then concave for

µ

2

∈

[ˆ

µ

2

,

^{1] if}˜

µ

2

<

ˆ

µ

2. Full information disclosure is optimal if

(

¹

− µ

2

)

^TEe

(

⁰

) + µ

2TE^e

(

¹

) = (

¹

− µ

2

)

_v^v_A^A+^vv¹1

+ µ

2 vAv2

vA+v2

>

^TEe

(µ

2

)

^{for all}

µ

2

∈ (

⁰

,

¹

)

; necessary and sufficient for this is that the slope of the straight line is greater than the slope ofTE^e

(µ

2

)

measured at

µ

2

=

1, which requires

v

A

v

2

v

A

+ v

2

− v

A

v

1

v

A

+ v

1

>

²

v

A

v

₂²

(v

A

+ v

2

)

²

3

⇔ v

_A²

(v

2

− v

1

) − v

A

v

2

(v

1

+ v

2

) −

2

v

1

v

²₂

<

⁰

⇔ v

A

< v

A

,

where

v

A

=

^v2

2

[

v1+v2+

√v²2+10v1v2−7v1² v2−v1

]

. When

v

A

> v

A, define

µ

2that solves ^TE

(

^µ2,1

)

^−TE(0,0)

µ2

=

^∂TE_∂µ^(µ2,1)

2

| µ

2. The concavification of TE^e

(µ

2

)

consists of the line

(µ2−µ2 µ2

)

TE

(

⁰

,

⁰

) +

^µ_µ²

2TE

(µ

2

,

¹

)

for

µ

2

∈

[0

, µ

2] and TE

(µ

2

,

¹

)

^for

µ

2

∈

[

µ

2

,

1]. Then the principal uses partial information disclosure for

µ

2

∈

[0

, µ

2] and no disclosure otherwise. □

Example 1illustrates the relationship between our results and those ofZhang and Zhou(2016).

Example 1. ConsiderN

=

2,

v

1

=

1,

v

2

=

4. In this case,

θ

1^min

=

v2

√v1

√v2−√

v1

=

4, and

v

A

=

8. CombiningFact 1andProposition 1 gives the optimal policy for information disclosure:

Optimal disclosure

=

⎧

⎪⎪

⎨

⎪⎪

⎩

no disclosure (ND) if

v

A

<

² full disclosure (FD) if 2

< v

A

<

⁴ full disclosure (FD) if 4

≤ v

A

<

⁸ partial disclosure (PD) if 8

≤ v

A

(15) The first two lines in(15)reflect the results ofZhang and Zhou (2016), and the last two are our extension.³Thus, we extend the parameter range for which full disclosure is the optimal policy, and after this the principal implements partial disclosure. To see how this is implemented, suppose that

v

A

=

16, and calculate ˆ

µ

2

=

^v2

2vA

=

0

.

¹²⁵

<

˜

µ

2

=

^v2

√v1

vA

(

^{√v2−√v1}

) =

0

.

25. Therefore, for

µ

2

<

˜

µ

2, both types are active andTE^e

(µ)

is convex; For

µ

2

≥

˜

µ

2, only type

v

2 is active and TE^e

(µ)

is concave.Fig. 1 plots TE^e

(µ)

^against

µ

2

∈

[0

,

^{1]. For}

µ

2

=

0

.

3, the principal’s payoffs from no disclosure and from full disclosure are 1.4876 and 1.61882, respectively. Consider a distribution of Bayes-plausible posteriors:

µ

¹

= (

¹

,

⁰

)

^,

µ

²

= (

⁰

.

⁴

,

⁰

.

⁶

)

with probabilities

β

1

=

1

/

^2,

β

2

=

1

/

2, which can be generated with two messagesm₁ andm₂and the signal distributions matrix:

S

=

[ 5

/

^{7 2/7}

0 1

]

,

where S_(ij) denotes Pr[

m_j

| v

i

]

,

ⁱ

∈ {

1

,

²

} ,

^j

∈ {

1

,

²

}

. FromKa- menica and Gentzkow(2011), we know that the principal’s pay- off from partial disclosure of the above kind is

β

1TE^e(

µ

¹)

+ β

2TE^e(

µ

²)

=

1

.

71626, which is higher than her payoffs from full or no disclosure.

2.2. N

≥

3

ForN

≥

3,Zhang and Zhou(2016, Corollary 2) show that full disclosure is optimal for sufficiently high

v

A(i.e.,

v

A

≥ √

v

N−₁

v

N), and partial disclosure can arise otherwise. For our extended pa- rameter space, partial disclosure can be optimal even for high values of

v

A. To understand why, recall the underlying mechanism inZhang and Zhou(2016): For

µ ∈

int(

P^N)

, there always exists a direction along which TE

(µ,

⁰

)

is convex, and therefore, the principal can obtain a higher expected payoff from a distribution over two Bayes-plausible posteriors onEdge(

P^N)

where the di- rectional vector intersectsEdge(

P^N)

. This reduces the dimension of the problem by one, and gradually optimal posteriors can be found on pairwise edges. The analysis of theN

=

2 case shows 3 _WhenvA=2, total expected effort is independent of information disclosure.

Fig. 1. TE^eagainstµ2,N=2.

Fig. 2. TE^eagainst(µ2, µ3)^,N=3.

that these edges are fully convex (concave) for high (low) values of

v

A when only interior solutions are considered. However, as we have shown, the possibility of a corner solution implies that pairwise edges will not always be convex for high

v

A, because of which the findings ofZhang and Zhou (2016) will not hold.⁴ Example 2illustrates how partial disclosure can dominate full or no disclosure.

Example 2. Consider N

=

3,

v

1

=

1,

v

2

=

4,

v

3

=

9, and

v

A

=

16. We have

θ

₁^min

=

min

{ _v

2

√v1

√v2−√

v1

,

^√v_v³₃^√−√^v1 v1

}

=

4,

θ

2^min

=

^v3

√v2

√v3−√

v2

=

18, and

θ

1^min

< v

A

< θ

2^min. Further,

θ

1

(µ) =

³⁶

9µ2+₈µ3 and

v

A

< θ

1

(µ) ⇔

36

µ

2

+

32

µ

3

<

^9.

4 In addition, we conjecture that the finding thatTE^e is convex alongsome directional vector forµ∈int(

P^N)

, which holds when all types are active, is not robust when some types choose to remain inactive. Therefore, the optimal posteriors may not necessarily be found onEdge(

P^N) . 4

Therefore, for prior

µ

^{0 with 36}

µ

⁰₂

+

32

µ

⁰₃

<

9, all three types are active and for

µ

^{0 with 36}

µ

⁰₂

+

32

µ

⁰₃

≥

9, type

v

1 will be inactive. Fig. 2 plots TE^e against

(µ

2

, µ

3

) ,

⁰

≤ µ

2

+ µ

3

≤

1.

TE^e is neither globally concave or convex. The darker region at the top of the graph represents the area where the principal’s payoffs from no disclosure is higher than that from full disclosure.

For

µ

⁰

= (

⁰

.

³

,

⁰

.

³

,

⁰

.

⁴

)

, her payoffs from full and no disclosure are TE_FD(

µ

⁰)

=

3

.

^{54635 and TEe}(

µ

⁰)

=

3

.

53056, respec- tively. Consider a distribution of Bayes-plausible posteriors:

µ

¹

= (

⁰

.

⁵

,

⁰

.

⁴

,

⁰

.

¹

)

^,

µ

²

= (

⁰

.

²

,

⁰

.

⁷

,

⁰

.

¹

)

^,

µ

³

= (

⁰

.

²

,

⁰

,

⁰

.

⁸

)

^with probabilities

β

1

=

1

/

^3,

β

2

=

5

/

^21,

β

3

=

3

/

7, which can be generated with three messages m₁, m₂, m₃, and the signal distributions matrix:

S

=

[ ₅

/

^{9 10}

/

^{63 2}

/

⁷ 4

/

^{9 5}

/

^{9 0} 1

/

^{12 5}

/

^{84 6}

/

⁷

]

,

where S_(i,j)

=

Pr[

m_j

| v

i

]. The principal’s payoff from partial disclosure is

β

1TE^e(

µ

¹)

+ β

2TE^e(

µ

²)

+ β

3TE^e(

µ

³)

=

3

.

^60892,

which is higher than her payoffs from full or no disclosure.

Although the posteriors considered here are not necessarily op- timal, the exercise shows that the payoff from partial disclosure can dominate that from full or no disclosure.

Acknowledgments

We would like to thank the editor Joseph E. Harrington and one anonymous referee for insightful comments. Remaining er- rors are our own.

References

Epstein, G.S., Mealem, Y., 2013. Who gains from information asymmetry? Theory Decis. 75 (3), 305–337.

Kamenica, E., Gentzkow, M., 2011. Bayesian persuasion. Amer. Econ. Rev. 101 (6), 2590–2615.

Zhang, J., Zhou, J., 2016. Information disclosure in contests: A Bayesian persuasion approach. Econ. J. 126 (597), 2197–2217.

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