Information Disclosure by Informed Intermediary in Double Auction

Sanghoon Kim

doi:10.1515/bejte-2024-0026

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Information Disclosure by Informed Intermediary in Double Auction

Sanghoon Kim

Veröffentlicht/Copyright: 12. November 2024

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Aus der Zeitschrift The B.E. Journal of Theoretical Economics Band 24 Heft 2

Abstract

In this paper, I study the role of an informed intermediary in improving market efficiency through communication with a buyer and a seller. Based on a double auction setting, I introduce an agent who is partially informed about the private information of a buyer and a seller. The agent can disclose information to either both parties, one party, or none of them. I compare the two most common incentives of the agent, maximizing either trade probability or expected price, in information disclosure. The analysis shows that the former incentive improves efficiency more than the latter; the agent with the former always discloses all relevant information, resulting in not only a higher trade probability but also a higher expected price. The buyer and the seller mostly prefer the agent disclosing information to both parties. When the agent discloses information to only one party, the buyer and the seller desire to be the one who receives the information exclusively. They sometimes prefer the agent disclosing information exclusively to the other party over no information for all. The findings of this paper have important implications for the design of compensation schemes for intermediaries who play a significant role as advisors to market participants.

Keywords: bargaining; double auction; information disclosure; intermediary

JEL Classification: C78; D44; D82; D83

Corresponding author: Sanghoon Kim, Department of Economics, University at Buffalo, The State University of New York, 421 Fronczak Hall, Buffalo, NY 14260, USA, E-mail: skim227@buffalo.edu

I thank the editor, Burkhard C. Schipper, anonymous referee, Jihye Jeon, Wooyoung Lim, Ching-to Albert Ma, Marc Rysman, and conference and seminar participants at Boston University, Chung-Ang University, Hankuk University of Foreign Studies, Korean Economic Review International Conference for their helpful comments and discussions.

Appendix

Lemma 1.

Consider intervals [ v ̲ b , v ̄ b ] ⊂ [ 0,1 ] and [ v ̲ s , v ̄ s ] ⊂ [ 0,1 ] . Suppose it is common knowledge that [ v ̲ b , v ̄ b ] includes realized v ^b and that [ v ̲ s , v ̄ s ] includes realized v ^s. Let

(1) b ̃ ( v b ) = 2 3 v b + 1 12 v ̄ b + 1 4 v ̲ s ,

and

(2) s ̃ ( v s ) = 2 3 v s + 1 4 v ̄ b + 1 12 v ̲ s .

(i) If v ̲ b < v ̄ s , then the buyer’s and seller’s equilibrium offer strategies are b ( v b ) = b ̃ ( v b ) and s ( v s ) = s ̃ ( v s ) with the following boundary conditions:

If v ̄ b > 4 3 v ̄ s − 1 3 v ̲ s , then b ( v b ) = 2 3 v ̄ s + 1 4 v ̄ b + 1 12 v ̲ s for v b > v ̄ s + 1 4 ( v ̄ b − v ̲ s )
If v ̄ b < 4 3 v ̄ s − 1 3 v ̲ s , then s(v ^s) = 1 for v s > 3 4 v ̄ b + 1 4 v ̲ s
If 4 3 v ̲ b − 1 3 v ̄ b > v ̲ s , then s ( v s ) = 2 3 v ̲ b + 1 12 v ̄ b + 1 4 v ̲ s for v s < v ̲ b − 1 4 ( v ̄ b − v ̲ s )
If 4 3 v ̲ b − 1 3 v ̄ b < v ̲ s , then b(v ^b) = 0 for v b < 3 4 v ̲ s + 1 4 v ̄ b

(ii) Otherwise, the buyer’s and seller’s equilibrium offer strategies are b ( v b ) = s ( v s ) = v ̄ s + v ̲ b 2 for all v b ∈ [ v ̲ b , v ̄ b ] and v s ∈ [ v ̲ s , v ̄ s ] .

Proof of Lemma 1.

(i) When a buyer’s valuation is v ^b, the buyer’s problem is to maximize the buyer’s expected utility by submitting an offer, B ̃ , given a seller’s strategy, s(⋅):

(3) max B ̃ ∫ v ̲ s s − 1 ( B ̃ ) v b − B ̃ + s ( v s ) 2 1 v ̄ s − v ̲ s d v s .

The first order condition of the above buyer’s problem is

(4) − 1 2 { s ̃ − 1 ( B ̃ ) − v ̲ s } + ( v b − B ̃ ) d s ̃ − 1 ( B ̃ ) d B ̃ = 0 ,

which is identical to the first order condition of the buyer’s problem in Chatterjee and Samuelson (1983) under the uniform distribution assumption of this paper.

When a seller’s valuation is v ^s, the seller’s problem is to maximize the seller’s expected utility by submitting an offer, S ̃ , given a buyer’s strategy, b(⋅):

(5) max S ̃ ∫ b − 1 ( S ̃ ) v ̄ b b ( v b ) + S ̃ 2 − v s 1 v ̄ b − v ̲ b d v b .

The first order condition of the above seller’s problem is

(6) 1 2 { v ̄ b − b ̃ − 1 ( S ̃ ) } − ( S ̃ − v s ) d b ̃ − 1 ( S ̃ ) d S ̃ = 0 ,

which is also identical to the first order condition of the seller’s problem in Chatterjee and Samuelson (1983) under the uniform distribution assumption of this paper.

b ̃ ( ⋅ ) in Equation (1) and s ̃ ( ⋅ ) in Equation (2) solve the system of Equations (4) and (6) such that ( B ̃ , s ̃ ) = ( b ̃ , s ̃ ) = ( b ̃ , S ̃ ) for each v ^b and v ^s. For the rest of the proof, see the proofs of Theorems 2 and 3 in Chatterjee and Samuelson (1983).

(ii) Straightforward. ■

Proof of Theorem 1.

When the true state is (v ^b, v ^s) ∈ [0, 1] × [1 − y, 1], if an agent tells the true state only to a buyer, then common knowledge is that v ^b is uniformly distributed over [0, 1], and v ^s is uniformly distributed over [1 − y, 1]. Thus, by Lemma 1, the buyer’s and seller’s strategies are

(7) b 1 ( v b ) = 0 if v b ∈ 0 , 3 4 ( 1 − y ) + 1 4 2 3 v b + 1 12 + 1 4 ( 1 − y ) if v b ∈ 3 4 ( 1 − y ) + 1 4 , 1

and

(8) s ( v s ) = 2 3 v s + 1 4 + 1 12 ( 1 − y ) if v s ∈ 1 − y , 3 4 + 1 4 ( 1 − y ) 1 if v s ∈ 3 4 + 1 4 ( 1 − y ) , 1 ,

respectively. Then, the probability of trade in the equilibrium is

(9) ∫ 1 − y 3 4 + 1 4 ( 1 − y ) ∫ v s + 1 4 y 1 1 y d v b d v s = 9 y 32 ,

and the expected price is

(10) ∫ 1 − y 3 4 + 1 4 ( 1 − y ) ∫ v s + 1 4 y 1 1 2 2 3 v b + 1 12 + 1 4 ( 1 − y ) + 2 3 v s + 1 4 + 1 12 ( 1 − y ) 1 y d v b d v s = 9 ( 2 − y ) y 64 .

^[4]If an agent lies to the buyer, saying that the seller’s value is lower than 1 − y, and the buyer believes the agent’s message, then the buyer’s strategy is

(13) b 0 ( v b ) = 0 if v b ∈ 0 , 1 4 2 3 v b + 1 12 if v b ∈ 1 4 , 1

for y ∈ 0 , 1 4 , or

(14) b 0 ( v b ) = 0 if v b ∈ 0 , 1 4 2 3 v b + 1 12 if v b ∈ 1 4 , ( 1 − y ) + 1 4 2 3 ( 1 − y ) + 1 4 if v b ∈ ( 1 − y ) + 1 4 , 1

for y ∈ 1 4 , 1 by Lemma 1. Note that for y ∈ 0 , 1 4 , b 0 ( 1 ) = 2 3 + 1 12 < s ( 1 − y ) = 2 3 ( 1 − y ) + 1 4 + 1 12 ( 1 − y ) ; and for y ∈ 1 4 , 1 , b 0 ( 1 ) = 2 3 ( 1 − y ) + 1 4 < s ( 1 − y ) = 2 3 ( 1 − y ) + 1 4 + 1 12 ( 1 − y ) . That is, if the agent lies to the buyer, the buyer and seller do not trade the object at all because the buyer’s highest offer is always lower than the seller’s lowest offer. Therefore, neither a trade-maximizing agent nor a price-maximizing agent has an incentive to deviate from the equilibrium.

When the true state is (v ^b, v ^s) ∈ [0, 1] × [0, 1 − y], if an agent tells the true state only to a buyer, then common knowledge is that v ^b is uniformly distributed over [0, 1], and v ^s is uniformly distributed over [0, 1 − y]. Thus, for y ∈ 1 4 , 1 , the buyer’s and seller’s strategies are

(15) b 0 ( v b ) = 0 if v b ∈ 0 , 1 4 2 3 v b + 1 12 if v b ∈ 1 4 , ( 1 − y ) + 1 4 2 3 ( 1 − y ) + 1 4 if v b ∈ ( 1 − y ) + 1 4 , 1

and

(16) s ( v s ) = 2 3 v s + 1 4 for all v s ∈ [ 0,1 − y ] ,

respectively, by Lemma 1. Then, the probability of trade in the equilibrium is

(17) ∫ 0 1 − y ∫ ( 1 − y ) + 1 4 1 1 1 − y d v b d v s + ∫ 0 1 − y ∫ v s + 1 4 ( 1 − y ) + 1 4 1 1 − y d v b d v s = 1 4 + 1 2 y ,

and the expected price is

(18) ∫ 0 1 − y ∫ ( 1 − y ) + 1 4 1 1 2 2 3 ( 1 − y ) + 1 4 + 2 3 v s + 1 4 1 1 − y d v b d v s + ∫ 0 1 − y ∫ v s + 1 4 ( 1 − y ) + 1 4 1 2 2 3 v b + 1 12 + 2 3 v s + 1 4 1 1 − y d v b d v s = 5 + 20 y − 16 y 2 48 .

^[5]If an agent lies to the buyer, saying that the seller’s value is higher than 1 − y, and the buyer believes the agent’s message, then the buyer’s strategy is

(21) b 1 ( v b ) = 0 if v b ∈ 0 , 3 4 ( 1 − y ) + 1 4 2 3 v b + 1 12 + 1 4 ( 1 − y ) if v b ∈ 3 4 ( 1 − y ) + 1 4 , 1

by Lemma 1. Then, the probability of trade is

(22) ∫ 0 1 − y ∫ 3 4 ( 1 − y ) + 1 4 1 1 1 − y d v b d v s = 3 4 y ,

and the expected price is

(23) ∫ 0 1 − y ∫ 3 4 ( 1 − y ) + 1 4 1 1 2 2 3 v b + 1 12 + 1 4 ( 1 − y ) + 2 3 v s + 1 4 1 1 − y d v b d v s = ( 19 − 10 y ) y 32 .

Note that 1 4 + 1 2 y > 3 4 y for all y ∈ 1 4 , 1 . That is, a trade-maximizing agent has no incentive to deviate from the equilibrium.

Note that 5 + 20 y − 16 y 2 48 ≥ ( 19 − 10 y ) y 32 for all y ∈ 1 4 , 3 41 − 17 4 , and 5 + 20 y − 16 y 2 48 < ( 19 − 10 y ) y 32 for all y ∈ 3 41 − 17 4 , 1 . That is, a price-maximizing agent has no incentive to deviate from the equilibrium for all y ∈ 1 4 , 3 41 − 17 4 .

For y ∈ 0 , 1 4 , the buyer’s and seller’s strategies are

(24) b 0 ( v b ) = 0 if v b ∈ 0 , 1 4 2 3 v b + 1 12 if v b ∈ 1 4 , 1

and

(25) s ( v s ) = 2 3 v s + 1 4 if v s ∈ 0 , 3 4 1 if v s ∈ 3 4 , 1 − y ,

respectively, by Lemma 1. Then, the probability of trade in the equilibrium is

(26) ∫ 0 3 4 ∫ v s + 1 4 1 1 1 − y d v b d v s = 9 32 ( 1 − y ) ,

and the expected price is

(27) ∫ 0 3 4 ∫ v s + 1 4 1 1 2 2 3 v b + 1 12 + 2 3 v s + 1 4 1 1 − y d v b d v s = 9 64 ( 1 − y ) .

^[6]If an agent lies to the buyer, saying that the seller’s value is higher than 1 − y, and the buyer believes the agent’s message, then the buyer’s strategy is

(30) b 1 ( v b ) = 0 if v b ∈ 0 , 3 4 ( 1 − y ) + 1 4 2 3 v b + 1 12 + 1 4 ( 1 − y ) if v b ∈ 3 4 ( 1 − y ) + 1 4 , 1

by Lemma 1. Then, the probability of trade is

(31) ∫ 0 3 4 ∫ 3 4 ( 1 − y ) + 1 4 1 1 1 − y d v b d v s = 9 y 16 ( 1 − y ) ,

and the expected price is

(32) ∫ 0 3 4 ∫ 3 4 ( 1 − y ) + 1 4 1 1 2 2 3 v b + 1 12 + 1 4 ( 1 − y ) + 2 3 v s + 1 4 1 1 − y d v b d v s = 9 ( 3 − y ) y 64 ( 1 − y ) .

Note that 9 32 ( 1 − y ) > 9 y 16 ( 1 − y ) and 9 64 ( 1 − y ) > 9 ( 3 − y ) y 64 ( 1 − y ) for all y ∈ 0 , 1 4 . That is, neither a trade-maximizing agent nor a price-maximizing agent has an incentive to deviate from the equilibrium. ■

Lemma 2.

When an informed agent provides a buyer with accurate information about a seller’s value, the trade probability and expected price are strictly higher than in the uninformative equilibrium for all y ∈ (0, 1).

Proof of Lemma 2.

When the true state is (v ^b, v ^s) ∈ [0, 1] × [0, 1 − y], in an equilibrium where an informed agent provides a buyer with accurate information about a seller’s value, the probability of trade is 9 32 ( 1 − y ) by Equation (26), and the expected price is 9 64 ( 1 − y ) by Equation (27), for y ∈ 0 , 1 4 . For y ∈ 1 4 , 1 , the probability of trade is 1 4 + 1 2 y by Equation (17), and the expected price is 5 + 20 y − 16 y 2 48 by Equation (18).

When the true state is (v ^b, v ^s) ∈ [0, 1] × [1 − y, 1], the probability of trade is 9 y 32 by Equation (9), and the expected price is 9 ( 2 − y ) y 64 by Equation (10) in the equilibrium.

Thus, for y ∈ 0 , 1 4 , the probability of trade is

(33) ( 1 − y ) 9 32 ( 1 − y ) + y 9 y 32 = 9 ( 1 + y 2 ) 32 ,

and the expected price is

(34) ( 1 − y ) 9 64 ( 1 − y ) + y 9 ( 2 − y ) y 64 = 9 ( 1 + 2 y 2 − y 3 ) 64 .

Note that in the uninformative equilibrium of Chatterjee and Samuelson (1983), the probability of trade is 9 32 and the expected price is 9 64 . 9 ( 1 + y 2 ) 32 > 9 32 and 9 ( 1 + 2 y 2 − y 3 ) 64 > 9 64 for all y ∈ 0 , 1 4 .

For y ∈ 1 4 , 1 , the probability of trade is

(35) ( 1 − y ) 1 4 + 1 2 y + y 9 y 32 = 8 + 8 y − 7 y 2 32 ,

and the expected price is

(36) ( 1 − y ) 5 + 20 y − 16 y 2 48 + y 9 ( 2 − y ) y 64 = 20 + 60 y − 90 y 2 + 37 y 3 192 .

8 + 8 y − 7 y 2 32 > 9 32 and 20 + 60 y − 90 y 2 + 37 y 3 192 > 9 64 for all y ∈ 1 4 , 1 . ■

Lemma 3.

When an informed agent provides a buyer with accurate information about a seller’s value,

the buyer’s expected payoff is strictly greater than in the uninformative equilibrium for all y ∈ (0, 1), and
the seller’s expected payoff is strictly greater than in the uninformative equilibrium for all y ∈ 0 , 7 + 13 18 ≈ 0.5892 .

Proof of Lemma 3.

For y ∈ 0 , 1 4 , the buyer’s expected payoff is

(37) y 9 y 2 128 + ( 1 − y ) 9 128 ( 1 − y ) = 9 ( 1 + y 3 ) 128

from Equations (11) and (28), and the seller’s expected payoff is

(38) y 9 y 2 128 + ( 1 − y ) 9 128 ( 1 − y ) = 9 ( 1 + y 3 ) 128

from Equations (12) and (29). For y ∈ 1 4 , 1 , the buyer’s expected payoff is

(39) y 9 y 2 128 + ( 1 − y ) 7 + 4 y + 16 y 2 96 = 28 − 12 y + 48 y 2 − 37 y 3 384

from Equations (11) and (19), and the seller’s expected payoff is

(40) y 9 y 2 128 + ( 1 − y ) 1 + 2 y 16 = 8 + 8 y − 16 y 2 + 9 y 3 128

from Equations (12) and (20). Note that in the uninformative equilibrium of Chatterjee and Samuelson (1983), the buyer’s and seller’s expected payoff is 9 128 each. 9 ( 1 + y 3 ) 128 > 9 128 for all y ∈ 0 , 1 4 , 28 − 12 y + 48 y 2 − 37 y 3 384 > 9 128 for all y ∈ [1/4, 1), and 8 + 8 y − 16 y 2 + 9 y 3 128 > 9 128 for all y ∈ 1 / 4 , 7 + 13 18 . ■

Proof of Theorem 2.

When the true state is (v ^b, v ^s) ∈ [0, y] × [0, 1], if an agent tells the true state only to a seller, then common knowledge is that v ^b is uniformly distributed over [0, y], and v ^s is uniformly distributed over [0, 1]. Thus, by Lemma 1, the buyer’s and seller’s strategies are

(41) b ( v b ) = 0 if v b ∈ 0 , 1 4 y 2 3 v b + 1 12 y if v b ∈ 1 4 y , y

and

(42) s 0 ( v s ) = 2 3 v s + 1 4 y if v s ∈ 0 , 3 4 y 1 if v s ∈ 3 4 y , 1 ,

respectively. Then, the probability of trade in the equilibrium is

(43) ∫ 0 3 4 y ∫ v s + 1 4 y y 1 y d v b d v s = 9 y 32 ,

and the expected price is

(44) ∫ 0 3 4 y ∫ v s + 1 4 y y 1 2 2 3 v b + 1 12 y + 2 3 v s + 1 4 y 1 y d v b d v s = 9 y 2 64 .

If an agent lies to the seller, saying that the buyer’s value is higher than y, and the seller believes the agent’s message, then the seller’s strategy is

(45) s 1 ( v s ) = 2 3 v s + 1 4 if v s ∈ 0 , 3 4 1 if v s ∈ 3 4 , 1

for y ∈ 0 , 1 4 , or

(46) s 1 ( v s ) = 2 3 y + 1 12 if v s ∈ 0 , y − 1 4 2 3 v s + 1 4 if v s ∈ y − 1 4 , 3 4 1 if v s ∈ 3 4 , 1

for y ∈ 1 4 , 1 by Lemma 1. Note that for y ∈ 0 , 1 4 , s 1 ( 0 ) = 1 4 > b ( y ) = 2 3 y + 1 12 y ; and for y ∈ 1 4 , 1 , s 1 ( 0 ) = 2 3 y + 1 12 > b ( y ) = 2 3 y + 1 12 y . That is, if an agent lies to the seller, the buyer and seller do not trade the object at all because the buyer’s highest offer is always lower than the seller’s lowest offer. Therefore, neither a trade-maximizing agent nor a price-maximizing agent has an incentive to deviate from the equilibrium.

When the true state is (v ^b, v ^s) ∈ [y, 1] × [0, 1], if an agent tells the true state only to a seller, then common knowledge is that v ^b is uniformly distributed over [y, 1], and v ^s is uniformly distributed over [0, 1]. Thus, for y ∈ 1 4 , 1 , the buyer’s and seller’s strategies are

(47) b ( v b ) = 2 3 v b + 1 12 for all v b ∈ [ y , 1 ]

and

(48) s 1 ( v s ) = 2 3 y + 1 12 if v s ∈ 0 , y − 1 4 2 3 v s + 1 4 if v s ∈ y − 1 4 , 3 4 1 if v b ∈ 3 4 , 1 ,

respectively, by Lemma 1. Then, the probability of trade in the equilibrium is

(49) ∫ 0 y − 1 4 ∫ y 1 1 1 − y d v b d v s + ∫ y − 1 4 3 4 ∫ v s + 1 4 1 1 1 − y d v b d v s = 1 4 + 1 2 y ,

and the expected price is

(50) ∫ 0 y − 1 4 ∫ y 1 1 2 2 3 v b + 1 12 + 2 3 y + 1 12 1 1 − y d v b d v s + ∫ y − 1 4 3 4 ∫ v s + 1 4 1 1 2 2 3 v b + 1 12 + 2 3 v s + 1 4 1 1 − y d v b d v s = 16 y 2 + 4 y + 7 48 .

If an agent lies to the seller, saying that the buyer’s value is lower than y, and the seller believes the agent’s message, then the seller’s strategy is

(51) s 0 ( v s ) = 2 3 v s + 1 4 y if v s ∈ 0 , 3 4 y 1 if v b ∈ 3 4 y , 1

by Lemma 1. Then, the probability of trade is

(52) ∫ 0 3 4 y ∫ y 1 1 1 − y d v b d v s = 3 4 y ,

and the expected price is

(53) ∫ 0 3 4 y ∫ y 1 1 2 2 3 v b + 1 12 + 2 3 v s + 1 4 y 1 1 − y d v b d v s = 5 y ( 1 + 2 y ) 32 .

Note that 1 4 + 1 2 y > 3 4 y and 16 y 2 + 4 y + 7 48 > 5 y ( 1 + 2 y ) 32 for all y ∈ 1 4 , 1 . That is, neither a trade-maximizing agent nor a price-maximizing agent has an incentive to deviate from the equilibrium.

For y ∈ 0 , 1 4 , the buyer’s and seller’s strategies are

(54) b ( v b ) = 0 if v b ∈ 0 , 1 4 2 3 v b + 1 12 if v b ∈ 1 4 , 1

and

(55) s 1 ( v s ) = 2 3 v s + 1 4 if v s ∈ 0 , 3 4 1 if v s ∈ 3 4 , 1 ,

respectively, by Lemma 1. Then, the probability of trade in the equilibrium is

(56) ∫ 0 3 4 ∫ v s + 1 4 1 1 1 − y d v b d v s = 9 32 ( 1 − y ) ,

and the expected price is

(57) ∫ 0 3 4 ∫ v s + 1 4 1 1 2 2 3 v b + 1 12 + 2 3 v s + 1 4 1 1 − y d v b d v s = 9 64 ( 1 − y ) .

If an agent lies to the seller, saying that the buyer’s value is lower than y, and the seller believes the agent’s message, then the seller’s strategy is

(58) s 0 ( v s ) = 2 3 v s + 1 4 y if v s ∈ 0 , 3 4 y 1 if v b ∈ 3 4 y , 1

by Lemma 1. Then, the probability of trade is

(59) ∫ 0 3 4 y ∫ y 1 1 1 − y d v b d v s = 3 y 4 ,

and the expected price is

(60) ∫ 0 3 4 y ∫ y 1 1 2 2 3 v b + 1 12 + 2 3 v s + 1 4 y 1 1 − y d v b d v s = 5 y ( 1 + 2 y ) 32 .

Note that 9 32 ( 1 − y ) > 3 y 4 and 9 64 ( 1 − y ) > 5 y ( 1 + 2 y ) 32 for all y ∈ 0 , 1 4 . That is, neither a trade-maximizing agent nor a price-maximizing agent has an incentive to deviate from the equilibrium. ■

Lemma 4.

When an informed agent provides a seller with accurate information about a buyer’s valuation, the trade probability and expected price are strictly higher than in the uninformative equilibrium for all y ∈ (0, 1).

Lemma 5.

When an informed agent provides a seller with accurate information about a buyer’s value,

the seller’s expected payoff is strictly greater than in the uninformative equilibrium for all y ∈ (0, 1), and
the buyer’s expected payoff is strictly greater than in the uninformative equilibrium for all y ∈ 0 , 7 + 13 18 ≈ 0.5892 .

Proofs of Lemmas 4 and 5.

I omit the proofs because they are symmetrical to the proofs of Lemmas 2 and 3. ■

Proof of Theorem 3.

When the true state is (v ^b, v ^s) ∈ [y, 1] × [0, 1 − y], if an agent tells the true state to a buyer and a seller, then common knowledge is that v ^b is uniformly distributed over [y, 1], and v ^s is uniformly distributed over [0, 1 − y].

For y ∈ 1 2 , 1 , the buyer’s and seller’s strategies are b 0 ( v b ) = s 1 ( v s ) = 1 2 for all v ^b ∈ [y, 1] and v ^s ∈ [0, 1 − y]. Thus, the buyer and seller trade the object with certainty, and the expected price is 1 2 . Note that a trade-maximizing agent has no incentive to deviate from the equilibrium since the probability of trade equals to 1.

Note that the possible deviations of an agent are lying to the seller, saying that the buyer’s value is lower than y, or lying to the buyer, saying that the seller’s value is higher than 1 − y, or both. Because of the symmetric setting of the model, lying to the seller and lying to the buyer have the same effect on the change in the probability of trade. Moreover, it is easy to check that when an agent lies to the seller and the buyer at the same time, the trade probability is lower than when the agent deceives only one of them.

Also, note that lying to the seller that the buyer’s value is lower than y makes the seller submit a lower offer, and lying to the buyer that the seller’s value is higher than 1 − y makes the buyer submit a higher offer by Lemma 1. Therefore, lying to the buyer is the most profitable deviation of a price-maximizing agent among the possible deviations. Thus, I only need to check the case where an agent deceives the buyer.

If an agent lies to the buyer, saying that the seller’s value is higher than 1 − y, and the buyer believes the agent’s message, then the buyer’s strategy is

(61) b 1 ( v b ) = 0 if v b ∈ y , 3 4 ( 1 − y ) + 1 4 2 3 v b + 1 12 + 1 4 ( 1 − y ) if v b ∈ 3 4 ( 1 − y ) + 1 4 , 1

for y ∈ 1 2 , 4 7 , or for y ∈ 4 7 , 1 ,

(62) b 1 ( v b ) = 2 3 v b + 1 12 + 1 4 ( 1 − y ) for all v b ∈ [ y , 1 ]

by Lemma 1. Thus, the expected price is

(63) ∫ 0 1 − y ∫ 3 4 ( 1 − y ) + 1 4 1 1 2 2 3 v b + 1 12 + 1 4 ( 1 − y ) + 1 2 1 ( 1 − y ) 2 d v b d v s = 3 y ( 3 − y ) 16 ( 1 − y )

for y ∈ 1 2 , 4 7 , or

(64) ∫ 0 1 − y ∫ y 1 1 2 2 3 v b + 1 12 + 1 4 ( 1 − y ) + 1 2 1 ( 1 − y ) 2 d v b d v s = 14 + y 24

for y ∈ 4 7 , 1 . Note that 14 + y 24 > 1 2 for all y ∈ 4 7 , 1 , 3 y ( 3 − y ) 16 ( 1 − y ) ≤ 1 2 for all y ∈ 1 2 , 17 − 193 6 , and 3 y ( 3 − y ) 16 ( 1 − y ) > 1 2 for all y ∈ 17 − 193 6 , 4 7 . That is, a price-maximizing agent has no incentive to deviate from the equilibrium for all y ∈ 1 2 , 17 − 193 6 .

Similarly, for y ∈ 1 4 , 1 2 , the buyer’s and seller’s strategies are

(65) b 0 ( v b ) = 2 3 v b + 1 12 if v b ∈ y , ( 1 − y ) + 1 4 2 3 ( 1 − y ) + 1 4 if v b ∈ ( 1 − y ) + 1 4 , 1

and

(66) s 1 ( v s ) = 2 3 y + 1 12 if v s ∈ 0 , y − 1 4 2 3 v s + 1 4 if v s ∈ y − 1 4 , 1 − y ,

respectively, by Lemma 1. Then, the probability of trade in equilibrium is

(67) ∫ 0 y − 1 4 ∫ y ( 1 − y ) + 1 4 1 ( 1 − y ) 2 d v b d v s + ∫ 0 y − 1 4 ∫ ( 1 − y ) + 1 4 1 1 ( 1 − y ) 2 d v b d v s + ∫ y − 1 4 1 − y ∫ v s + 1 4 ( 1 − y ) + 1 4 1 ( 1 − y ) 2 d v b d v s + ∫ y − 1 4 1 − y ∫ ( 1 − y ) + 1 4 1 1 ( 1 − y ) 2 d v b d v s = 7 + 16 y − 32 y 2 32 ( 1 − y ) 2 ,

and the expected price is

(68) ∫ 0 y − 1 4 ∫ y ( 1 − y ) + 1 4 1 2 2 3 v b + 1 12 + 2 3 y + 1 12 1 ( 1 − y ) 2 d v b d v s + ∫ 0 y − 1 4 ∫ ( 1 − y ) + 1 4 1 1 2 2 3 ( 1 − y ) + 1 4 + 2 3 y + 1 12 1 ( 1 − y ) 2 d v b d v s + ∫ y − 1 4 1 − y ∫ v s + 1 4 ( 1 − y ) + 1 4 1 2 2 3 v b + 1 12 + 2 3 v s + 1 4 1 ( 1 − y ) 2 d v b d v s + ∫ y − 1 4 1 − y ∫ ( 1 − y ) + 1 4 1 1 2 2 3 ( 1 − y ) + 1 4 + 2 3 v s + 1 4 1 ( 1 − y ) 2 d v b d v s = 7 + 16 y − 32 y 2 64 ( 1 − y ) 2 .

If an agent lies to the buyer, saying that the seller’s value is higher than 1 − y, and the buyer believes the agent’s message, then the buyer’s strategy is

(69) b 1 ( v b ) = 0 if v b ∈ y , 3 4 ( 1 − y ) + 1 4 2 3 v b + 1 12 + 1 4 ( 1 − y ) if v b ∈ 3 4 ( 1 − y ) + 1 4 , 1

by Lemma 1. Then, the probability of trade is

(70) ∫ 0 1 − y ∫ 3 4 ( 1 − y ) + 1 4 1 1 ( 1 − y ) 2 d v b d v s = 3 y 4 ( 1 − y ) ,

and the expected price is

(71) ∫ 0 y − 1 4 ∫ 3 4 ( 1 − y ) + 1 4 1 1 2 2 3 v b + 1 12 + 1 4 ( 1 − y ) + 2 3 y + 1 12 1 ( 1 − y ) 2 d v b d v s + ∫ y − 1 4 1 − y ∫ 3 4 ( 1 − y ) + 1 4 1 1 2 2 3 v b + 1 12 + 1 4 ( 1 − y ) + 2 3 v s + 1 4 1 ( 1 − y ) 2 d v b d v s = y ( 77 − 124 y + 56 y 2 ) 128 ( 1 − y ) 2 .

Note that 7 + 16 y − 32 y 2 32 ( 1 − y ) 2 > 3 y 4 ( 1 − y ) for all y ∈ 1 4 , 1 2 . That is, a trade-maximizing agent has no incentive to deviate from the equilibrium.

Note that 7 + 16 y − 32 y 2 64 ( 1 − y ) 2 ≥ y ( 77 − 124 y + 56 y 2 ) 128 ( 1 − y ) 2 for all y ∈ 1 4 , 1 28 { 10 − 5 × 2 2 2 / 3 ( 27 + 7 71 ) 1 / 3 + ( 594 + 154 71 ) 1 / 3 } , and 7 + 16 y − 32 y 2 64 ( 1 − y ) 2 < y ( 77 − 124 y + 56 y 2 ) 128 ( 1 − y ) 2 for all y ∈ 1 28 { 10 − 5 × 2 2 2 / 3 ( 27 + 7 71 ) 1 / 3 + ( 594 + 154 71 ) 1 / 3 } , 1 2 . That is, a price-maximizing agent has no incentive to deviate from the equilibrium for all y ∈ 1 4 , 1 28 { 10 − 5 × 2 2 2 / 3 ( 27 + 7 71 ) 1 / 3 + ( 594 + 154 71 ) 1 / 3 } .

For y ∈ 0 , 1 4 , the buyer’s and seller’s strategies are

(72) b 0 ( v b ) = 0 if v b ∈ y , 1 4 2 3 v b + 1 12 if v b ∈ 1 4 , 1

and

(73) s 1 ( v s ) = 2 3 v s + 1 4 if v s ∈ 0 , 3 4 1 if v s ∈ 3 4 , 1 − y ,

respectively, by Lemma 1. Then, the probability of trade in the equilibrium is

(74) ∫ 0 3 4 ∫ v s + 1 4 1 1 ( 1 − y ) 2 d v b d v s = 9 32 ( 1 − y ) 2 ,

and the expected price is

(75) ∫ 0 3 4 ∫ v s + 1 4 1 1 2 2 3 v b + 1 12 + 2 3 v s + 1 4 1 ( 1 − y ) 2 d v b d v s = 9 64 ( 1 − y ) 2 .

If an agent lies to the buyer, saying that the seller’s value is higher than 1 − y, and the buyer believes the agent’s message, then the buyer’s strategy is

(76) b 1 ( v b ) = 0 if v b ∈ y , 3 4 ( 1 − y ) + 1 4 2 3 v b + 1 12 + 1 4 ( 1 − y ) if v b ∈ 3 4 ( 1 − y ) + 1 4 , 1

by Lemma 1. Then, the probability of trade is

(77) ∫ 0 3 4 ∫ 3 4 ( 1 − y ) + 1 4 1 1 ( 1 − y ) 2 d v b d v s = 9 y 16 ( 1 − y ) 2 ,

and the expected price is

(78) ∫ 0 3 4 ∫ 3 4 ( 1 − y ) + 1 4 1 1 2 2 3 v b + 1 12 + 1 4 ( 1 − y ) + 2 3 v s + 1 4 1 ( 1 − y ) 2 d v b d v s = 9 ( 3 − y ) y 64 ( 1 − y ) 2 .

Note that 9 32 ( 1 − y ) 2 > 9 y 16 ( 1 − y ) 2 and 9 64 ( 1 − y ) 2 > 9 ( 3 − y ) y 64 ( 1 − y ) 2 for all y ∈ 0 , 1 4 . That is, neither a trade-maximizing agent nor a price-maximizing agent has an incentive to deviate from the equilibrium.

Similarly, it is easy to check the following. When the true state is (v ^b, v ^s) ∈ [0, y] × [0, 1 − y], a trade-maximizing agent has no incentive to deviate from the equilibrium, and a price-maximizing agent has no incentive to deviate from the equilibrium for all y ∈ 0 , 25 + 3 41 64 ≈ 0.6908 . When the true state is (v ^b, v ^s) ∈ [y, 1] × [1 − y, 1] or (v ^b, v ^s) ∈ [0, y] × [1 − y, 1], neither a trade-maximizing agent nor a price-maximizing agent has an incentive to deviate. ■

Lemma 6.

When an informed agent provides both a buyer and a seller with accurate information about the other party’s valuation, the trade probability and expected price are strictly higher than in the uninformative equilibrium for all y ∈ (0, 1).

Lemma 7.

When an informed agent provides both a buyer and a seller with accurate information about the other party’s valuation, the buyer’s and seller’s expected payoffs are strictly greater than in the uninformative equilibrium for all y ∈ (0, 1).

Proofs of Lemmas 6 and 7.

I omit the proofs because they are similar to the proofs of Lemmas 2 and 3. ■

Lemma 8.

When an informed agent provides a buyer with accurate information about a seller’s valuation, the trade probability is

the same as when an informed agent provides a seller with accurate information about a buyer’s valuation.
strictly higher than in the uninformative equilibrium.
strictly lower than when an informed agent provides both a buyer and a seller with accurate information about the other party’s valuation.

Proof of Lemma 8.

It is straightforward to calculate the trade probability in each equilibrium. The trade probability in the uninformative equilibrium is 9 32 . In both equilibria in which an informed agent only advises either a buyer or a seller truthfully, the trade probability is 9 ( 1 + y 2 ) 32 for y ∈ 0 , 1 4 and 8 + 8 y − 7 y 2 32 for y ∈ 1 4 , 1 . The trade probability in the equilibrium in which an informed agent truthfully advises both sides of a buyer and a seller is 9 ( 1 + 2 y 2 ) 32 for y ∈ 0 , 1 4 , 7 + 16 y − 14 y 2 32 for y ∈ 1 4 , 1 2 , 41 − 100 y + 86 y 2 32 for y ∈ 1 2 , 4 7 , and 3 ( 3 + 4 y − 4 y 2 ) 32 for y ∈ 4 7 , 1 .

9 ( 1 + y 2 ) 32 > 9 32 for all y ∈ 0 , 1 4 , and 8 + 8 y − 7 y 2 32 > 9 32 for all y ∈ 1 4 , 1 .

9 ( 1 + 2 y 2 ) 32 > 9 ( 1 + y 2 ) 32 for all y ∈ 0 , 1 4 , 7 + 16 y − 14 y 2 32 > 8 + 8 y − 7 y 2 32 for all y ∈ 1 4 , 1 2 , 41 − 100 y + 86 y 2 32 > 8 + 8 y − 7 y 2 32 for all y ∈ 1 2 , 4 7 , and 3 ( 3 + 4 y − 4 y 2 ) 32 > 8 + 8 y − 7 y 2 32 for all y ∈ 4 7 , 1 . ■

Lemma 9.

(i) When an informed agent provides a buyer with accurate information about a seller’s valuation, the expected price is higher than when an informed agent provides a seller with accurate information about a buyer’s valuation for y ∈ 0 , 2 ( 8 + 3 3 ) 37 ≈ 0.7133 ; otherwise, the opposite.

(ii) The expected price in the uninformative equilibrium is strictly lower than all other equilibria.

(iii) When an informed agent provides both a buyer and a seller with accurate information about the other party’s valuation, the expected price is strictly higher than all other equilibria.

Proof of Lemma 9.

It is straightforward to calculate the expected price in each equilibrium. The expected price in the uninformative equilibrium is 9 64 . The expected price in the equilibrium in which an informed agent only advises a buyer truthfully is 9 ( 1 + 2 y 2 − y 3 ) 64 for y ∈ 0 , 1 4 and 20 + 60 y − 90 y 2 + 37 y 3 192 for y ∈ 1 4 , 1 . The expected price in the equilibrium in which an informed agent only advises a seller truthfully is 9 ( 1 + y 3 ) 64 for y ∈ 0 , 1 4 and 28 − 12 y + 48 y 2 − 37 y 3 192 for y ∈ 1 4 , 1 . The expected price in the equilibrium in which an informed agent truthfully advises both sides of a buyer and a seller is 9 ( 1 + 2 y 2 ) 64 for y ∈ 0 , 1 4 , 7 + 16 y − 14 y 2 64 for y ∈ 1 4 , 1 2 , 41 − 100 y + 86 y 2 64 for y ∈ 1 2 , 4 7 , and 3 ( 3 + 4 y − 4 y 2 ) 64 for y ∈ 4 7 , 1 .

9 ( 1 + 2 y 2 − y 3 ) 64 > 9 ( 1 + y 3 ) 64 > 9 64 for all y ∈ 0 , 1 4 , 20 + 60 y − 90 y 2 + 37 y 3 192 > 28 − 12 y + 48 y 2 − 37 y 3 192 > 9 64 for all y ∈ 1 4 , 2 ( 8 + 3 3 ) 37 , and 28 − 12 y + 48 y 2 − 37 y 3 192 > 20 + 60 y − 90 y 2 + 37 y 3 192 > 9 64 for all y ∈ 2 ( 8 + 3 3 ) 37 , 1 .^[7]

9 ( 1 + 2 y 2 ) 64 > 9 ( 1 + 2 y 2 − y 3 ) 64 for all y ∈ 0 , 1 4 , 7 + 16 y − 14 y 2 64 > 20 + 60 y − 90 y 2 + 37 y 3 192 for all y ∈ 1 4 , 1 2 , 41 − 100 y + 86 y 2 64 > 20 + 60 y − 90 y 2 + 37 y 3 192 for all y ∈ 1 2 , 4 7 , 3 ( 3 + 4 y − 4 y 2 ) 64 > 20 + 60 y − 90 y 2 + 37 y 3 192 for all y ∈ 4 7 , 2 ( 8 + 3 3 ) 37 , and 3 ( 3 + 4 y − 4 y 2 ) 64 > 28 − 12 y + 48 y 2 − 37 y 3 192 for all y ∈ 2 ( 8 + 3 3 ) 37 , 1 . ■

Lemma 10.

The following is the relationship between the expected payoffs of a buyer and a seller in each equilibrium.

For y ∈ 0 , 1 4 , Advice for both > Advice only for them = Advice only for the other > Uninformative.
For y ∈ 1 4 , 7 + 13 18 ≈ 0.5892 , Advice for both > Advice only for them > Advice only for the other > Uninformative.
For y ∈ 7 + 13 18 , 5 − 1 2 ≈ 0.6180 , Advice for both > Advice only for them > Uninformative > Advice only for the other.
For y ∈ 5 − 1 2 , 1 , Advice only for them > Advice for both > Uninformative > Advice only for the other.

Proof of Lemma 10.

It is straightforward to calculate the expected payoff of a buyer or a seller in each equilibrium. The expected payoff in the uninformative equilibrium is 9 128 for each buyer and seller. The expected payoff of each buyer and seller in the equilibrium in which an informed agent only advises them truthfully is 9 ( 1 + y 3 ) 128 for y ∈ 0 , 1 4 and 7 − 3 y + 12 y 2 − 16 y 3 96 + 9 y 3 128 for y ∈ 1 4 , 1 . The expected payoff of each buyer and seller in the equilibrium in which an informed agent only advises the other truthfully is 9 ( 1 + y 3 ) 128 for y ∈ 0 , 1 4 and 1 + y − 2 y 2 16 + 9 y 3 128 for y ∈ 1 4 , 1 . The expected payoff of each buyer and seller in the equilibrium in which an informed agent truthfully advises both sides of a buyer and a seller is 9 ( 1 + 2 y 3 ) 128 for y ∈ 0 , 1 4 , 25 + 12 y − 10 y 3 384 for y ∈ 1 4 , 1 2 , 154 y 3 − 236 y 2 + 118 y − 9 128 for y ∈ 1 2 , 4 7 , and 37 − 30 y + 48 y 2 − 28 y 3 384 for y ∈ 4 7 , 1 .

9 ( 1 + 2 y 3 ) 128 > 9 ( 1 + y 3 ) 128 > 9 128 for all y ∈ 0 , 1 4 .
25 + 12 y − 10 y 3 384 > 7 − 3 y + 12 y 2 − 16 y 3 96 + 9 y 3 128 > 1 + y − 2 y 2 16 + 9 y 3 128 > 9 128 for all y ∈ 1 4 , 1 2 , 154 y 3 − 236 y 2 + 118 y − 9 128 > 7 − 3 y + 12 y 2 − 16 y 3 96 + 9 y 3 128 > 1 + y − 2 y 2 16 + 9 y 3 128 > 9 128 for all y ∈ 1 2 , 4 7 , and 37 − 30 y + 48 y 2 − 28 y 3 384 > 7 − 3 y + 12 y 2 − 16 y 3 96 + 9 y 3 128 > 1 + y − 2 y 2 16 + 9 y 3 128 > 9 128 for all y ∈ 4 7 , 7 + 13 18 .
37 − 30 y + 48 y 2 − 28 y 3 384 > 7 − 3 y + 12 y 2 − 16 y 3 96 + 9 y 3 128 > 9 128 > 1 + y − 2 y 2 16 + 9 y 3 128 for all y ∈ 7 + 13 18 , 5 − 1 2 .^[8]
7 − 3 y + 12 y 2 − 16 y 3 96 + 9 y 3 128 > 37 − 30 y + 48 y 2 − 28 y 3 384 > 9 128 > 1 + y − 2 y 2 16 + 9 y 3 128 for all y ∈ 5 − 1 2 , 1 .^[9] ■

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Received: 2024-02-29

Accepted: 2024-10-19

Published Online: 2024-11-12

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https://doi.org/10.1515/bejte-2024-0026

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bargaining; double auction; information disclosure; intermediary