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Information Acquisition and Response in Peer-Effects Networks C. Matthew Leister Monash University Conference on Econom...

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Information Acquisition and Response in Peer-Effects Networks C. Matthew Leister Monash University

Conference on Economic Networks and Finance LSE, December 11, 2015

Introduction

Setup

Equilibrium

Welfare

Individuals/firms face heterogeneous incentives to acquire and respond to information.

Conclusions

Introduction

Setup

Equilibrium

Welfare

Individuals/firms face heterogeneous incentives to acquire and respond to information. Idiosyncratic values/costs

Conclusions

Introduction

Setup

Equilibrium

Welfare

Individuals/firms face heterogeneous incentives to acquire and respond to information. Idiosyncratic values/costs Strategic position

Conclusions

Introduction

Setup

Equilibrium

Welfare

Individuals/firms face heterogeneous incentives to acquire and respond to information. Idiosyncratic values/costs Strategic position Dual role of information:

Conclusions

Introduction

Setup

Equilibrium

Welfare

Individuals/firms face heterogeneous incentives to acquire and respond to information. Idiosyncratic values/costs Strategic position Dual role of information: 1. infer the state of the world,

Conclusions

Introduction

Setup

Equilibrium

Welfare

Individuals/firms face heterogeneous incentives to acquire and respond to information. Idiosyncratic values/costs Strategic position Dual role of information: 1. infer the state of the world, 2. in equilibrium, infer the observations and subsequent actions of neighbors.

Conclusions

Introduction

Setup

Equilibrium

Welfare

Conclusions

Peer-effects networks with incomplete information



ui (x1 , . . . , xN ) = ai + ω + |

X k6=i

{z



σik xk  xi −

marginal value to xi

}

1 σii xi2 2 | {z }

O.C. to xi

Introduction

Setup

Equilibrium

Welfare

  P ui (x1 , . . . , xN ) = ai + ω + k6=i σik xk xi − 21 σii xi2 A competitive supply chain

up stream firm

+

+

− −

vert. integ. firm

− down stream firm



ω : demand for novel product xfirm : production

Conclusions

Introduction

Setup

Equilibrium

Welfare

  P ui (x1 , . . . , xN ) = ai + ω + k6=i σik xk xi − 21 σii xi2 Traders with heterogeneous funding constraints

deep pockets

− +

liquidity const.

ω : long term asset value xtrader : market order/inventory

Conclusions

Introduction

Setup

Equilibrium

Welfare

Conclusions

Basic questions

(1) How does heterogeneity in strategic positioning influence the incentives to acquire information?

Introduction

Setup

Equilibrium

Welfare

Conclusions

Basic questions

(1) How does heterogeneity in strategic positioning influence the incentives to acquire information? (2)

Who over and who under acquires information? Who gains to influence others’ beliefs?

Introduction

Setup

Equilibrium

Welfare

Positive results

EQ

Information response game −−→ value to information. Equilibrium properties: a. game on correlation-adjusted network (second stage), b. negative responses (second stage), c. multiple information acquisition equilibria (first stage).

Conclusions

Introduction

Setup

Equilibrium

Welfare

Welfare results 1. Extent of symmetry among pair-wise peer effects drives direction of two inefficiencies: a. informational externalities (network charact.: in-walks), b. strategic value to information acquisition (network charact.: closed-walks).

2. Symmetric networks (for e.g.) a. “bunching” for moderate peer effects: equilibrium information asymmetries inefficiently low, b. significant strategic substitutes: acquisition of negative responders inefficiently low, c. positive strategic distortion ∝ connectedness in network.

3. “Antisymmetric” networks: inefficiencies reverse.

Conclusions

Introduction

Setup

Equilibrium

Welfare

Policy implications

Transparency-based policy: targeted certification of information investments.

Conclusions

Introduction

Setup

Equilibrium

Welfare

Conclusions

Literature Network games with incomplete information: Calv´ o-Armengol & de Mart´ı (2007,2009), Calv´ o-Armengol, de Mart´ı, Prat (2015), de Mart´ı & Zenou (2015). Coordination games with endogenous information: Novshek & Sonnenschein (1983,1988), Vives (1988,2008), Hauk & Hurkens (2007). Morris & Shin (2002), Hellwig & Veldkamp (2009), Myatt & Wallace (2012,2013), Colombo, Femminis, & Pavan (2014).

Finance: Grossman & Stiglitz (1980), Kyle (1985,1989), Babus & Kondor (2013).

Introduction

Setup

Equilibrium

Welfare

Conclusions

Timeline of the game

each i chooses information quality ei ∈ [0, 1] at cost κi (ei )

each i observes signal θi , then chooses action xi ∈ R

state ω ∈ Ω observed, each i’s ui (x|ω) realized

t=1

t=2

t = 2+

Introduction

Setup

Equilibrium

Welfare

Model primitives: second stage (t = 2) Each i chooses xi ∈ R, yielding i’s payoffs (t = 2):   X 1 σik xk  xi − xi2 , ui (x|ω) = ω + ρ 2 k6=i

where ω ∈ Ω ⊆ R, σij ∈ R for each i, j, and ρ ∈ [0, 1],

Conclusions

Introduction

Setup

Equilibrium

Welfare

Model primitives: second stage (t = 2) Each i chooses xi ∈ R, yielding i’s payoffs (t = 2):   X 1 σik xk  xi − xi2 , ui (x|ω) = ω + ρ 2 k6=i

where ω ∈ Ω ⊆ R, σij ∈ R for each i, j, and ρ ∈ [0, 1], i observes signal θi ∈ Θ ⊆ R of quality ei ∈ [0, 1],

Conclusions

Introduction

Setup

Equilibrium

Welfare

Model primitives: second stage (t = 2) Each i chooses xi ∈ R, yielding i’s payoffs (t = 2):   X 1 σik xk  xi − xi2 , ui (x|ω) = ω + ρ 2 k6=i

where ω ∈ Ω ⊆ R, σij ∈ R for each i, j, and ρ ∈ [0, 1], i observes signal θi ∈ Θ ⊆ R of quality ei ∈ [0, 1], Pure strategy: Xi : Θ × [0, 1] → R.

Conclusions

Introduction

Setup

Equilibrium

Welfare

Model primitives: second stage (t = 2) Each i chooses xi ∈ R, yielding i’s payoffs (t = 2):   X 1 σik xk  xi − xi2 , ui (x|ω) = ω + ρ 2 k6=i

where ω ∈ Ω ⊆ R, σij ∈ R for each i, j, and ρ ∈ [0, 1], i observes signal θi ∈ Θ ⊆ R of quality ei ∈ [0, 1], Pure strategy: Xi : Θ × [0, 1] → R.

Assumption 1 (I − [sij σij ])−1 is well defined for every s ∈ [0, 1]N(N−1) .

Conclusions

Introduction

Setup

Equilibrium

Welfare

Conclusions

Model primitives: first stage (t = 1)

Each i = 1, . . . , N privately invests in information quality ei ∈ [0, 1]. i’s cost of information quality κi (·) ∈ C 2 satisfies κi (0) , κ′i (0) = 0, with non-decreasing κ′′i (ei ) ≥ 0. Assumption 2 For v0 > 0, there exists an unique ei† ∈ (0, 1) solving v0 ei† = κ′i (ei† ).

Introduction

Setup

Equilibrium

Welfare

Conclusions

Model primitives: first stage (t = 1)

Each i = 1, . . . , N privately invests in information quality ei ∈ [0, 1]. i’s cost of information quality κi (·) ∈ C 2 satisfies κi (0) , κ′i (0) = 0, with non-decreasing κ′′i (ei ) ≥ 0. Assumption 2 For v0 > 0, there exists an unique ei† ∈ (0, 1) solving v0 ei† = κ′i (ei† ). All conditions satisfied for normal state and signals case.

Introduction

Setup

Equilibrium

Welfare

Model primitives: beliefs and expectations Belief: µi (e−i ), density function over e−i ∈ [0, 1]N−1 .

Conclusions

Introduction

Setup

Equilibrium

Welfare

Model primitives: beliefs and expectations Belief: µi (e−i ), density function over e−i ∈ [0, 1]N−1 . Consistency: µi (e−i ) = 1 for t = 1 for given e−i , with µi (e′−i ) = 0 otherwise.

Conclusions

Introduction

Setup

Equilibrium

Welfare

Model primitives: beliefs and expectations Belief: µi (e−i ), density function over e−i ∈ [0, 1]N−1 . Consistency: µi (e−i ) = 1 for t = 1 for given e−i , with µi (e′−i ) = 0 otherwise. E1.

E2.

Ei [ω] = Ei [θi ] = 0,    v0 := Ei ω 2 ] = Ei θi2 |ei , Ei [ω|θi , ei ] = ei θi ,

E3. Ei [θj |θi , ei , ej ] = ei ej θi ,

for each ei ∈ [0, 1].

Conclusions

Introduction

Setup

Equilibrium facts

Equilibrium

Welfare

Conclusions

Theorems

1. Multiple IAE e∗ may exist even with a unique IRE β ∗ for each e.

Introduction

Setup

Equilibrium

Welfare

Equilibrium facts

Conclusions

Theorems

1. Multiple IAE e∗ may exist even with a unique IRE β ∗ for each e. 2. Significant strategic substitutes: can have βi∗ < 0.

Introduction

Setup

Equilibrium

Welfare

Equilibrium facts

Conclusions

Theorems

1. Multiple IAE e∗ may exist even with a unique IRE β ∗ for each e. 2. Significant strategic substitutes: can have βi∗ < 0. 3. Significant peer effects required for 1. or 2. to obtain.

Introduction

Setup

Equilibrium

Welfare

Equilibrium facts

Conclusions

Theorems

1. Multiple IAE e∗ may exist even with a unique IRE β ∗ for each e. 2. Significant strategic substitutes: can have βi∗ < 0. 3. Significant peer effects required for 1. or 2. to obtain. Proposition Under Assumptions 1 and 2, there exists a ρ¯ > 0 such that for ρ ∈ [0, ρ¯), a unique IAE e∗ with βi∗ > 0 for all i obtains.

Introduction

Setup

Equilibrium

Welfare

Welfare

Conclusions

Introduction

Setup

Equilibrium

Welfare

Welfare

For any e, giving X∗ : νi (X∗ |e) := Ei [ui (X∗ |θi , ei , µ∗i ) |ei , µ∗i ] − κi (ei ) .. . 1 = v0 βi∗2 − κi (ei ) . 2

Conclusions

Introduction

Setup

Equilibrium

Welfare: marginal inefficiencies Define the utilitarian problem: X νk (X∗ |e) . max e∈[0,1]N

k

Welfare

Conclusions

Introduction

Setup

Equilibrium

Welfare

Conclusions

Welfare: marginal inefficiencies Define the utilitarian problem: X νk (X∗ |e) . max e∈[0,1]N

∂ ∂ei

=

∂νi

P

k νk (X ∗ (X |e)

∂ei



k

∗ |e)

βk∗ ,k6=i

+

X ∂νi (X∗ |e) ∂β ∗ k

k6=i

∂βk∗

∂ei

+

X ∂νk (X∗ |e) ∂β ∗ k

k6=i

∂βk∗

∂ei

.

Introduction

Setup

Equilibrium

Welfare

Conclusions

Welfare: marginal inefficiencies Define the utilitarian problem: X νk (X∗ |e) . max e∈[0,1]N

∂ ∂ei

P

k

νk (X∗ |e)   X βi∗2 ∂ ∗ ′ = v0 − κ (ei ) + v0 βi∗ ei ρσik ek β + ei ∂ei k k6=i | {z } ∗ | = 0 in IAE e f.o.c. {z } k

v0

X k6=i

βk∗

∂ ∗ β . ∂ei k

= 0 in public acquisition eq. epb f.o.c.

|

{z

= 0 in planner’s solution epl f.o.c.

}

Introduction

Setup

Equilibrium

Welfare

Conclusions

Welfare: marginal inefficiencies Define the utilitarian problem: X νk (X∗ |e) . max e∈[0,1]N

∂ ∂ei

P

k

νk (X∗ |e)   X βi∗2 ∂ ∗ ′ ei ρσik ek = v0 − κ (ei ) + v0 βi∗ β + ei ∂ei k k6=i {z } | k

(marginal) strategic value

v0

X k6=i

|

∂ ∗ β . ∂ei k {z }

βk∗

(marginal) externalities

Introduction

Setup

Equilibrium

Welfare

Conclusions

Welfare: marginal inefficiencies Define the utilitarian problem: X νk (X∗ |e) . max e∈[0,1]N

∂ ∂ei

P

k

νk (X∗ |e)   X βi∗2 ∂ ∗ ′ ei ρσik ek − κ (ei ) + v0 βi∗ β + = v0 ei ∂ei k k6=i {z } | k

ξist (X∗ |e)

v0

X k6=i

|

∂ ∗ β . ∂ei k {z }

βk∗

ξiex (X∗ |e)

Introduction

Setup

Equilibrium

Welfare

Conclusions

Welfare: marginal inefficiencies Define the utilitarian problem: X νk (X∗ |e) . max e∈[0,1]N

∂ ∂ei

P

k

νk (X∗ |e)   X βi∗2 ∂ ∗ ′ = v0 ei ρσik ek − κ (ei ) + v0 βi∗ β + ei ∂ei k k6=i | {z k

v0

(marginal) public-value

X k6=i

βk∗

∂ ∗ β . ∂ei k }

Introduction

Setup

Equilibrium

Welfare

Welfare: marginal inefficiencies Theorem (marginal inefficiencies) For information qualities e, consistent beliefs µ and IRE X∗ : βi∗2 ′ 1 Ie ΣIe (I − Ie ΣIe )−1 Ie ΣIe 1i , ei∗ i β∗ ξiex (e, X∗ ) = 2v0 ∗i (β ∗ − βi∗ 1i )′ Ie ΣIe (I − Ie ΣIe )−1 1i . ei ξist (e, X∗ ) = 2v0

Conclusions

Introduction

Setup

Equilibrium

Welfare

Welfare: marginal inefficiencies Theorem (marginal inefficiencies) For information qualities e, consistent beliefs µ and IRE X∗ : βi∗2 ′ 1 Ie ΣIe (I − Ie ΣIe )−1 Ie ΣIe 1i , ei∗ i β∗ ξiex (e, X∗ ) = 2v0 ∗i (β ∗ − βi∗ 1i )′ Ie ΣIe (I − Ie ΣIe )−1 1i . ei ξist (e, X∗ ) = 2v0

ξist (e, X∗ ) ∝ 1′i

∞ X ([ei ej ρσij ]i6=j )τ τ =2

!

1i :

summation of closed walks on [ei ej ρσij ]i6=j beginning and ending on i.

Conclusions

Introduction

Setup

Equilibrium

Welfare

Conclusions

Welfare: marginal inefficiencies Theorem (marginal inefficiencies) For information qualities e, consistent beliefs µ and IRE X∗ : βi∗2 ′ 1 Ie ΣIe (I − Ie ΣIe )−1 Ie ΣIe 1i , ei∗ i β∗ ξiex (e, X∗ ) = 2v0 ∗i (β ∗ − βi∗ 1i )′ Ie ΣIe (I − Ie ΣIe )−1 1i . ei ξist (e, X∗ ) = 2v0

ξiex (e, X∗ ) ∝ (β ∗ − βi∗ 1i )′

∞ X τ =1

([ei ej ρσij ]i6=j )τ

!

1i :

summation of walks on [ei ej ρσij ]i6=j beginning with j and ending on i, weighted by βj and aggregate over j 6= i.

Introduction

Setup

Equilibrium

Welfare

Conclusions

Example: three-player symmetric network, common κ ei .80

2

+⊗

.75

-.1 .1

1

b

.70

+

.65



+⊗

e†

b

b

eipl +eipb ei∗ ⊗

.60

-.1 3

b

.55 1

2

3

i

Introduction

Setup

Equilibrium

Welfare

Conclusions

Example: three-player symmetric network, common κ ei .80

2

.75





+

+

b

e†

b

-1/3 .70

.1

1

.65 eipl +eipb ei∗ ⊗

-1/3 3

.60 .55

+

b

b

1 e1pl = 0

2

3

i

Introduction

Setup

Equilibrium

Welfare

Conclusions

Example: three-player symmetric network, common κ ei



+

+

b

.80

2



b

.75

e†

-1/3 .70

1/3

1

.65 eipl +eipb ei∗ ⊗

-1/3 3

.60

b

.55 1

2

e1pl = e1pb = e1∗ = 0

3

i

Introduction

Setup

Equilibrium

Welfare

Conclusions

Example: two-player antisymmetric network, common κ

ei .75

b

+ ⊗

e† ⊗

1/3 1

2

.65

-1/3

b

+

eipl +eipb ei∗ ⊗

b

.55 1

2

i

Introduction

Setup

Equilibrium

Welfare

Welfare and policy design

Conclusions

Introduction

Setup

Equilibrium

Welfare

Welfare and the neutral player symmetric networks ξipl (II ) ei†

(III )

(I )

βi∗

antisymmetric networks ξipl (VI )

(IV )

(V )

ei†

βi∗

Conclusions

Introduction

Setup

Equilibrium

Welfare

Conclusions

Market efficiency in liquidity crises

conclusion

Introduction

Setup

Equilibrium

Welfare

Market efficiency in liquidity crises

N = 8 traders comprise non-trivial share of market.

Conclusions

Introduction

Setup

Equilibrium

Welfare

Market efficiency in liquidity crises

N = 8 traders comprise non-trivial share of market. xi : i’sPinventory/market order (e.g. Kyle (1985)); x¯ := 8i=1 xi .

Conclusions

Introduction

Setup

Equilibrium

Welfare

Market efficiency in liquidity crises

N = 8 traders comprise non-trivial share of market. xi : i’sPinventory/market order (e.g. Kyle (1985)); x¯ := 8i=1 xi . t = 2 market price φ(¯ x ) = A + B x¯, B > 0.

Conclusions

Introduction

Setup

Equilibrium

Welfare

Market efficiency in liquidity crises

N = 8 traders comprise non-trivial share of market. xi : i’sPinventory/market order (e.g. Kyle (1985)); x¯ := 8i=1 xi . t = 2 market price φ(¯ x ) = A + B x¯, B > 0. ω: risky asset’s long term value.

Conclusions

Introduction

Setup

Equilibrium

Welfare

Market efficiency in liquidity crises

N = 8 traders comprise non-trivial share of market. xi : i’sPinventory/market order (e.g. Kyle (1985)); x¯ := 8i=1 xi . t = 2 market price φ(¯ x ) = A + B x¯, B > 0. ω: risky asset’s long term value. t = 2 payoffs: ui (x|ω) = (ω + pi φ(¯ x )) xi − xi2   X xk  xi − (1 − pi B) xi2 . = ω + pi A + pi B k6=i

Conclusions

Introduction

Setup

Equilibrium

Market efficiency in liquidity crises

Liquidity flush market: pi < 0 for each unconstrained i.

Welfare

Conclusions

Introduction

Setup

Equilibrium

Market efficiency in liquidity crises

Liquidity flush market: pi < 0 for each unconstrained i. Market crowding in information acquisition.

Welfare

Conclusions

Introduction

Setup

Equilibrium

Welfare

Market efficiency in liquidity crises

Liquidity flush market: pi < 0 for each unconstrained i. Market crowding in information acquisition. Traders set ei∗ , βi∗ < e † (region (II)): over-acquire; over exertion in informationally inefficient markets.

Conclusions

Introduction

Setup

Equilibrium

Welfare

Market efficiency in liquidity crises

Liquidity crises: Liquidity spirals `a la Brunnermeier and Pedersen (2009) → upward sloping demand.

Conclusions

Introduction

Setup

Equilibrium

Welfare

Market efficiency in liquidity crises

Liquidity crises: Liquidity spirals `a la Brunnermeier and Pedersen (2009) → upward sloping demand. pi > 0 for liquidity-constrained trader i.

Conclusions

Introduction

Setup

Equilibrium

Welfare

Market efficiency in liquidity crises Market structure: pi B < 0 pi B > 0

− +

+

− −

constrained traders

unconstrained traders

Conclusions

Introduction

Setup

Equilibrium

Welfare

Market efficiency in liquidity crises

Liquidity crisis paradigm shift: Constrained traders set ei∗ , βi∗ > e † 1. Flush market: antisymmetric relationships → over-acquire. 2. Crisis: symmetric relationships → under-acquire.

Conclusions

Introduction

Setup

Equilibrium

Welfare

Conclusions

Market efficiency in liquidity crises

ei

antisymmetric peer effects

symmetric peer effects

1.0

pl ecnst.

0.96

bc bc bc

bc

bc

bc

bc

bc bc

bc bc

bc

bc

bc bc

∗ ecnst.

e†

0.92 bc

0.88

0

1

2

3

4

5

6

7

8

# cnst.

Introduction

Setup

Equilibrium

Welfare

Market efficiency in liquidity crises

Liquidity crisis paradigm shift: Constrained traders set ei∗ , βi∗ > e † 1. Flush market: antisymmetric relationships → over-acquire. 2. Crisis: symmetric relationships → under-acquire.

Unconstrained traders set ei∗ , βi∗ < e † 1. Flush market: symmetric relationships → over-acquire. 2. Crisis: antisymmetric relationships → under-acquire. 3. Extreme crisis: few unconstrained traders set ei∗ , βi∗ < 0.

Conclusions

Introduction

Setup

Equilibrium

Welfare

Conclusions

Market efficiency in liquidity crises ei

symmetric peer effects

antisymmetric peer effects

1.0 bc

bc bc

0.8

e†

bc bc

bc bc

bc bc

pl eunc.

bc

bc

∗ eunc. bc

4

5

0.6

negative response bc

0.4 0.2 0.0

0

1

2

3

bc

bc

6

7

8

# cnst.

Introduction

Setup

Equilibrium

Welfare

Conclusions

Policy suggestion in liquidity crises

Constrained traders impose symmetric, positive informational externalities on each other: under acquire, with positive strategic values...

Introduction

Setup

Equilibrium

Welfare

Conclusions

Policy suggestion in liquidity crises

Constrained traders impose symmetric, positive informational externalities on each other: under acquire, with positive strategic values... Couple stress-tests with certification of information investments of constrained traders.

Introduction

Setup

Equilibrium

Welfare

Conclusions

Conclusions 1. Introduce problem of costly information acquisition into new context: general network of peer effects. 2. Symmetric networks: a. Equilibrium information inefficiently symmetric. b. Players moving against their information do so too little. c. Strategic values to information are positive.

3. Direction of welfare and strategic motives determined by network “position” and extent of symmetry in relationships: direction of inefficiencies reverse in antisymmetric networks. 4. Information externalities and “position”: βi∗ w.r.t. ei† and origin, Strategic values and “position”: connectedness.

Introduction

Setup

Equilibrium

Welfare

Conclusions

Conclusions II

1. Liquidity crisis paradigm shift: over acquisition of information in liquid markets, under acquisition in constrained markets. 2. Unconstrained “shorters” in crisis: inefficient. 3. Transparency-based policy intervention: stress test with information investment certification.

Introduction

Setup

Equilibrium

Welfare

Equilibrium characterization

Conclusions

Introduction

Setup

Equilibrium

Welfare

Equilibrium characterization Theorem (t = 2 information-response equilibrium (IRE)) Under Assumption 1, for any e and consistent µ there exists a unique linear IRE of the form: X∗ = [Xi∗ (θi |ei )] = [βi∗ θi ] , P where each βi∗ solves βi∗ = ei + k6=i ei ek ρσik βk∗ : β ∗ := (I − [ei ej ρσij ]i6=j )−1 e ∞ X ([ei ej ρσij ]i6=j )τ e. = τ =0

Conclusions

Introduction

Setup

Equilibrium

Welfare

Conclusions

Equilibrium characterization Theorem (t = 2 information-response equilibrium (IRE)) Under Assumption 1, for any e and consistent µ there exists a unique linear IRE of the form: X∗ = [Xi∗ (θi |ei )] = [βi∗ θi ] , P where each βi∗ solves βi∗ = ei + k6=i ei ek ρσik βk∗ : β ∗ := (I − [ei ej ρσij ]i6=j )−1 e ∞ X ([ei ej ρσij ]i6=j )τ e. = τ =0

βi∗ : i’s “informational centrality” (weighted Bonacich centrality).

Introduction

Setup

Equilibrium

Welfare

Equilibrium characterization

Conclusions

Back

Theorem (t = 1 information-acquisition equilibrium (IAE)) Under Assumption 1, for IRE X∗ and consistent beliefs µ there exists a (generically unique∗ ) IAE e∗ . For any such IAE, and ∀ i with ei∗ ∈ (0, 1): β ∗2 v0 i∗ = κ′i (ei∗ ) . ei

Introduction

Setup

Equilibrium

Welfare

Conclusions

Equilibrium characterization

Back

Theorem (t = 1 information-acquisition equilibrium (IAE)) Under Assumption 1, for IRE X∗ and consistent beliefs µ there exists a (generically unique∗ ) IAE e∗ . For any such IAE, and ∀ i with ei∗ ∈ (0, 1): β ∗2 v0 i∗ = κ′i (ei∗ ) . ei

ei∗ 2 -.1 .1

.80 1

.75 .70

b

b

2

3

b

-.1 3

.65

1

i