URL (Slide and paper): ux.nu/BYfFi

Persistence and Snap Decision Making: Inefficient Decisions by a Reputation-Concerned Expert Tomoya Tajika IER, Hitotsubashi University

June 22, 2018 AMES @ Sogang University

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URL (Slide and paper): ux.nu/BYfFi

Introduction

• Behaving consistently is widely observed and valued phenomenon. • E.g., • Politicians who behave inconsistently are criticized. • Economists who predict economic trends that oppose their previous ones would not be trusted.

• Consistent behavior may lead to inefficient decisions.

• Consider a politician who decides to invest a public project, which is likely to make a profit. • After a while, he finds that the project is likely to make a loss. • Behaving consistency prevents him from withdrawing from the investment ○ inefficient decisions

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URL (Slide and paper): ux.nu/BYfFi

• Given that persons tend to behave consistently even it is inefficient, why they do not wait further information? ∵ Waiting information enables better choice without behaving inconsistently. ○ This study shows that “reputation concern” forces a person to “behave consistently” and make a such “snap decision.”

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URL (Slide and paper): ux.nu/BYfFi

Study overview

• There is unknown state ∈ {x , y }. • Player; a reputation-concerned expert and an evaluator. • The expert has two opportunities to recommend a choice from {x , y }. • If the last recommendation matches the realized state, the expert is (monetarily) rewarded. • In each opportunity, he receives signal ∈ {x , y }, whose accuracy is his ability. • Accuracy grows in the second opportunity. • An evaluator assesses the expert’s ability by observing the recommendations and realized state.

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URL (Slide and paper): ux.nu/BYfFi

KEY High-ability expert is less likely to receive inconsistent signal. ○ The expert has an incentive to pretend to have high-ability by behaving consistently. • This study shows that

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• Truthful recommendation is not an equilibrium. • Consistent recommendation is an equilibrium.

• Furthermore, we also consider the case that the expert can remain silent at the first opportunity.

○ Breaking a silence can be observed in an equilibrium.

KEY Assessment differs whether the expert remains silence or not.

• Remain silence: the expert assessed by the second signal. ○ assessment becomes severe. • Break silence: it leads to consistent recommendation ○ the expert assessed by the first signal. ○ assessment becomes mild (c.f. self-handicapping).

○

Incentive to break silence arises.

URL (Slide and paper): ux.nu/BYfFi

Literature

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• In social psychology, many studies find the evidences and discuss consistent behavior (Cialdini, 2006). • Falk and Zimmerman (2016) is the closest study. The differences are • knowledge/set of abilities • observation on the realized state

• Ferreira and Rezende (2007): information disclosure as a commitment for persistent behavior • Reputational herding and anti-herding: Levy (2004), Sabourian and Sibert (2009). • Belief persistence: Rabin and Schrag (1999). • Self-handicapping: Tirole (2002).

URL (Slide and paper): ux.nu/BYfFi

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Setup

• Players: expert and evaluator • The state of the world ω ∈ {x , y } • Expert recommends a choice from R1 = R2 = {x , y }. • Expert has two chances to recommend. • If the last recommendation matches to the realized state, expert is rewarded K . • Before each chance, expert receives a signal in S1 = S2 = {x , y } with accuracy θt = Prob(st = ω), t ∈ {1, 2} . • Assume

θ2 1−θ2

1 = (1 + α) 1−θ . Equiv., θ2 =

θ

1

(1+α)θ1 . 1+αθ1

URL (Slide and paper): ux.nu/BYfFi

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• θ1 ∈ D ⊆ [1/2, 1] is called ability. The expert knows θ1 . • Assume |D | ⩾ 2, inf D ∈ D and sup D ∈ D . • Let f : D → R++ be the density of θ1 (if D is discrete, f (θ) = Prob(θ1 = θ)). • By observing recommendations and state, Evaluator computes the belief about θ1 . • The expert’s vNM utility is I (r2 = ω)K + Eβ [θ1 | r1 , r2 , ω], risk neutral.

URL (Slide and paper): ux.nu/BYfFi

1st opportunity Expert observes signal s 1 ∈ S1

Expert recommends r1 ∈ R 1

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2nd opportunity

Expert observes signal s 2 ∈ S2

Expert recommends r2 ∈ R 2

Evaluator observes state ω∈X

Evaluator computes reputation Eβ [θ1 | r1 , r2 , ω]

Figure: Timeline of the model

DEFINITION 1. Recommendation r is a mapping such that r = (r1 , r2 ), r1 : S1 → R1 and r2 : R1 × S1 × S2 → R2 .

URL (Slide and paper): ux.nu/BYfFi

Truthful recommendation

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DEFINITION 2. Recommendation r is truthful if r1 (s1 ) = s1 and r2 (r1 , s1 , s2 ) = s2 . ¯ > 0 such PROPOSITION 1. For each K > 0, there exists α ¯ that for each α < α, the truthful recommendation is not a PBE.

URL (Slide and paper): ux.nu/BYfFi

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Detail for Proposition 1. Let θTruth be the ex post expectation of θ1 . r r ω 1 2

∫

θTruth = θTruth = ∫ xxx yyy = θTruth θTruth xxy yyx

∫

θ2 θ2 f (θ)d θ θθ2 f (θ)d θ

∫

=∫

(1−θ)2

θ 1+αθ f (θ)d θ = ∫ (1−θ)2 f (θ)d θ 1+αθ

∫

= θTruth = θTruth = θTruth = ∫ θTruth xyx xyy yxx yxy

θ3 f (θ)d θ 1+αθ θ2 f (θ)d θ 1+αθ

θ2 (1−θ) f (θ)d θ 1+αθ θ(1−θ) f (θ)d θ 1+αθ

URL (Slide and paper): ux.nu/BYfFi

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Detail for Proposition 1.—Cont’d • Suppose that Expert receives s1 = x in the 1st opp. and s2 = y in the 2nd opp. pxy = Prob(ω = x | s1 = x , s2 = y ). • To the TR become PBE, the following is necessary

Truth Truth Truth Truth pxy (θxxx + K ) + (1 − pxy )θxxy ⩽ pxy θxyx + (1 − pxy )(θxyy +K)

|

{z

}

utility of r1 = r2 = x

⇐⇒

1

1+α

=

pxy

1 − pxy

|

⩽

{z

utility of r1 = x , r2 = y

θTruth − θTruth +K xyy xxy Truth − θTruth + K θxxx xyx

}

URL (Slide and paper): ux.nu/BYfFi

Consistent recommendation

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DEFINITION 3. Recommendation r is consistent if r1 (s1 ) = s1 and r2 (r1 , s1 , s2 ) = s1 . • Assume that if r1 = r2 , the evaluator believes that r2 = s2 . ¯ > 0 such PROPOSITION 2. For each K > 0, there exists α ¯ that for each α < α, a consistent recommendation is a PBE.

URL (Slide and paper): ux.nu/BYfFi

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Detail for Proposition 2. Let θCons be the ex post expectation of θ1 . r r ω 1 2

θCons xxx

=

θCons yyy

∫

= ∫ ∫

θ2 f (θ)d θ θf (θ)d θ

=

m2 μ

θ(1 − θ)f (θ)d θ μ − m2 θCons = θCons = ∫ = xxy yyx 1− μ (1 − θ)f (θ)d θ

θCons = θCons = θCons = θCons xyx xyy yxx yxy ∫ θ2 (1−θ) ∫ f (θ)d θ θ2 (1 − θ2 )f (θ)d θ 1+αθ = ∫ = ∫ θ(1−θ) θ(1 − θ2 )f (θ)d θ f (θ)d θ 1+αθ

URL (Slide and paper): ux.nu/BYfFi

Detail for Proposition 2.—Cont’d • Suppose that Expert receives s1 = x in the 1st opp. and s2 = y in the 2nd opp. • To the CR become PBE, the following is necessary (and sufficient) 1

1+α

=

pxy

1 − pxy

⩾

θCons − θCons +K xyy xxy Cons − θCons + K θxxx xyx

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URL (Slide and paper): ux.nu/BYfFi

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Example

Assume D = {1/2, 3/4, 1}, f (1/2) = a , f (1) = b ,

f (3/4) = 1 − a − b . Let V (α) := W (α) =

Truth −θTruth +K θxyy xxy Truth −θTruth +K θxxx xyx

.

Cons −θCons +K θxyy xxy

Cons −θCons +K θxxx xyx

and

1.0

0.9

0.8

W(α)

0.7

V(α)

0.6

1 1+α

0.5 0

0.2

0.4

0.6

0.8

1.0

URL (Slide and paper): ux.nu/BYfFi

Right of silence

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• Introducing a right of silence in the first opportunity. ○ R1 = {x , y , ∅}, R2 = {x , y }.

• ∅ means keeping silence.

DEFINITION 4. Recommendation r is the waiting strategy if r1 (s1 ) = ∅ and r2 (r1 , s1 , s2 ) = s2 . • Waiting strategy enables to conceal receiving inconsistent signals. • (Off-path-belief) Assume if r1 (s1 ) ̸= ∅, the evaluator believes that r1 (s1 ) = r2 (r1 , s1 , s2 ) = s1 , but not update about θ1 .

URL (Slide and paper): ux.nu/BYfFi

Pooling equilibrium

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LEMMA 1. Suppose that sup D = 1 and α is sufficiently small. Suppose also that the evaluator does not update her belief about whether the expert remains silent or not. Then, the waiting strategy fails to be a PBE. • (Intuition) The evaluator’s assessment about θ1 is based on • θ2 = 1+αθ 1 if r1 = ∅ 1 • θ1 if r1 ̸= ∅ ○ Since θ2 > θ1 , the assessment becomes severe if r1 = ∅. (c.f. Self-handicapping). (1+α)θ

URL (Slide and paper): ux.nu/BYfFi

• (Re)consider consistent recommendation, that is r1 (s1 ) ̸= ∅. This is referred to as a snap decision. • (Off-path-belief) Assume if r1 (s1 ) = ∅, the evaluator does not update about θ. LEMMA 2. Suppose that inf D = 1/2 and K > 3. Suppose also that the evaluator does not update her belief about whether the expert ¯ remains silent or not. Then, there exists α ¯ the consistent such that for each α < α, recommendation fails to be a PBE.

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URL (Slide and paper): ux.nu/BYfFi

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PROPOSITION 3.

(i) The waiting strategy is a PBE. The out-of-equilibrium belief places probability 1 on θ = inf D . (ii) Suppose that α is sufficiently small. Then, the consistent recommendation is a PBE. The out-of-equilibrium belief places probability 1 on θ = inf D .

The out-of-equilibrium belief is natural in the following sense.

(i) In the equilibrium, h (θ) is minimized at θ = inf D . This type is most likely to break silence. (ii) In the equilibrium, h (θ) is maximized at θ = inf D . This type is most likely to keep silence.

URL (Slide and paper): ux.nu/BYfFi

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Keeping or breaking silence as a signaling

• Let r1 : S1 × D → R1 . • Let β be the belief system. • Let h (θ : r , β) = EU (r1 = ∅ | s1 , θ) − EU (r1 = s1 | s1 , θ). LEMMA 3. With given r , h (θ : r , β) is a concave quadratic function of θ. ○ (With sufficiently small α), Possible separation is • If θ ∈ / [θ∗ , θ∗ ]: r1 = s1 and r2 = r1 when r1 = s1 . • If θ ∈ [θ∗ , θ∗ ]: r1 = ∅ and r2 = s2 . • The evaluator believes that if r1 = ∅, the expert employs the waiting strategy and if r1 ̸= ∅, the expert employs the consistent recommendation.

URL (Slide and paper): ux.nu/BYfFi

h (θ : r , μ)

r1 = s1 1 2

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r1 = s1

r1 = ∅

θ∗

θ∗

1

θ

Figure: A tripartite equilibrium

h (θ : r , μ)

θ∗ =

1 r 2 1

=∅

r1 = s1

θ∗

1

r1 = s1 θ

1 2

h (θ : r , μ)

θ∗

Figure: Bipartite equilibria

θ ∗ r1 = ∅ θ = 1

URL (Slide and paper): ux.nu/BYfFi

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LEMMA 4. There is no bipartite separation equilibrium associated with ability such that θ∗ = 1. • Intuition: When θ = 1, the expert only concerns about his reputation. The benefit of breaking silence is only reputation. The advantage vanishes if θ = 1 expert keeps silent. • Implication: In each separating equilibrium, sufficiently high ability expert keeps silence and make a snap decision. ○ Precision of decision is not monotone in ability. • Separating equilibrium may not exist. PROPOSITION 4. Suppose that K > 3. Then, there exists ¯ such that for each α < α, ¯ no separating α equilibrium is associated with the expert’s ability.

URL (Slide and paper): ux.nu/BYfFi

Extension 1: Knowledge about θ1

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• Suppose that the expert does not know his ability θ1 but receive signal τ about θ1

COROLLARY 1. Suppose that the expert receives a signal about his ability. Then, for each K > 0, there ¯ > 0 such that for each α < α, ¯ the exists α truthful recommendation is not a PBE. COROLLARY 2. Suppose that the expert receives a signal about his ability. Then, for each K > 0, there ¯ > 0 such that for each α < α, ¯ a exists α consistent recommendation is a PBE. PROPOSITION 5. Suppose that the expert receives no information about his ability in the interim stage. Then, if K > 0, for a sufficiently small α, the waiting strategy is a PBE.

URL (Slide and paper): ux.nu/BYfFi

Extension 2: Different α

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• Assume that α depends on θ. • Assume also that

θα ′ (θ) α(θ)

∈ (−1, 0).

COROLLARY 3. Suppose that the expert receives a signal about his ability. Then, for each K > 0, there ¯ > 0 such that for each function α exists α ¯ a consistent that satisfies sup α(θ) < α, recommendation is a PBE.

URL (Slide and paper): ux.nu/BYfFi

Conclusion

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• We study the incentive of behaving consistently. • Recommending different choices hurts the expert’s reputation, which forces her to behave consistently. • Right of silence may not be a remedy for the consistent behavior. • Breaking a silence can work as a self-handicapping, which is enabled by the consistent behavior.