Advanced Economic Theory Lecture 9. Bilateral Asymmetric Information. Double Auction (Chatterjee and Samuelson, 1983).

Transcription

1 Leonardo Felli 6 December, 2002 Advanced Economic Theory Lecture 9 Bilateral Asymmetric Information Double Auction (Chatterjee and Samuelson, 1983). Two players, a buyer and a seller: N = {b, s}. The seller names an asking price: p s. The buyer names an offer price: p b. The action spaces: A s = {p s 0}, A b = {p b 0}. 1

2 Advanced Economic Theory 2 The seller owns and attaches value to an indivisible unit of a good. The buyer attaches value to the unit of the good and is willing to pay up to for it. The valuations for the unit of the good of the seller and the buyer are their private information. The type spaces: T s = {0 1}, T b = {0 1}

3 Advanced Economic Theory 3 Both the buyer and the seller believe that the valuation of the opponent is uniformly distributed on [0, 1]. The believes are: µ s = 1, µ b = 1. The rules of the game are such that: If p b p s then they trade at the average price: p = (p s + p b ). 2 If p b < p s then no trade occurs.

4 Advanced Economic Theory 4 The payoffs to both the seller and the buyer are then: u s (p s, p b ;, ) = { (ps +p b ) 2 if p b p s if p b < p s and u b (p s, p b ;, ) = { vb (p s+p b ) 2 if p b p s 0 if p b < p s The strategies for this game are p s ( ) and p b ( ). We restrict attention once again to strictly monotonic and differentiable strategies for the two players.

5 Advanced Economic Theory 5 Consider now the seller s best reply. This is defined by the following maximization problem: max p s E vb {u s (p s, p b ;, ), p b ( )} consider now the seller s payoff, substituting p b ( ) we have: u s (p s, p b ;, ) = { (ps +p b ( )) 2 if p b ( ) p s if p b ( ) < p s

6 Advanced Economic Theory 6 or u s (p s, p b ;, ) = { (ps +p b ( )) 2 if p 1 b (p s ) if < p 1 b (p s ) max p s p 1 b (p s ) =0 d + 1 =p 1 b (p s ) (p s + p b ( ) d 2 Recall that: y ( ) β(y) G(x, y)dx α(y) = β(y) = G(β(y), y) β (y) G(α(y), y)α (y)+ α(y) G(x, y) dx y

7 Advanced Economic Theory 7 Therefore the first order condition of the seller s best reply maximization problem are then: dp 1 b (p s ) dp s 1 2 [ ps + p b (p 1 b (p s )) ] dp 1 b (p s ) + dp s + 1 p 1 b (p s ) 1 2 d = 0 or from p s = p b (p 1 b (p s )): ( p s ) dp 1 b (p s ) + 1 [ ] 1 vb dp s 2 p 1 b (p s ) = 0 which gives: ( p s ) dp 1 b (p s ) dp s [ 1 p 1 b (p s ) ] = 0

8 Advanced Economic Theory 8 The buyer s best reply is instead defined by the following maximization problem: max p b E vs {u b (p s, p b ;, ), p s ( )} Consider now the buyer s payoff obtained substituting p s ( ): u b (p s, p b ;, ) = { vb (p s( )+p b ) 2 if p 1 s (p b ) 0 if < p 1 s (p b ) we then get max p b p 1 s (p b ) =0 [ (p ] s( ) + p b ) d 2

9 Advanced Economic Theory 9 Therefore the first order condition of the buyer s best reply maximization problem are then: [ (p s(p 1 s (p b )) + p b ) 2 ] dp 1 s (p b ) dp b 1 2 p 1 s (p b ) =0 d = 0 or which gives: [ p b ] dp 1 s (p b ) 1 [ ] p 1 s (p b ) vs = 0 dp b 2 0 ( p b ) dp 1 s (p b ) dp b 1 2 p 1 s (p b ) = 0 To simplify notation we re-write p 1 b ( ) = q b ( ) and p 1 s ( ) = q s ( ).

10 Advanced Economic Theory 10 The two differential equations that define the best reply of the seller and the buyer are then: [q s (p s ) p s ]q b(p s ) [1 q b(p s )] = 0 [q b (p b ) p b ]q s(p b ) 1 2 q s(p b ) = 0 Solving the second equation for q b (p b ) we obtain: q b (p b ) = p b + q s(p b ) 2 q s(p b ) differentiating yields: q b(p b ) = 1 2 [3 q s(p b )q s(p ] b ) [q s(p b )] 2 Substituting both formulae into the first differential equation we get: [q s (p s ) p s ] [3 q s(p s )q s(p s ) [q s(p s )] 2 ] [ + 1 p s q ] s(p s ) 2 q s(p s ) = 0

11 Advanced Economic Theory 11 This is a second-order differential equation in q s ( ) that has a two parameter family of solutions. The simplest family of solution takes the form: q s (p s ) = α p s + β solves the second- Then the values α = 3 2 and β = 3 8 order differential equation. Therefore substituting into q b (p b ) we obtain: q s (p s ) = 3 2 p s 3 8, q b(p b ) = 3 2 p b 1 8 Given that = q s (p s ) and = q b (p b ) we conclude that: p s = p b =

12 Advanced Economic Theory 12 This is the unique Bayesian Nash equilibrium of this game. Notice now that it is efficient to trade whenever: However in this double auction game trade occurs whenever: p b p s or

13 In other words: Advanced Economic Theory Trade = = (0, 0) 1

14 Advanced Economic Theory 14 Consider now a general bilateral trade model (Myerson and Satterthwaite 1983). The key question is whether the inefficiency we found in the previous example can be eliminated by choosing a different trading mechanism. The revelation principle provides the right tool to get an answer to the question above. Obviously, there is no principal, but the two parties at an ex-ante stage, before they learn their private information act as the mechanism designer. Assume further that this is a pure bilateral contract transfers cannot involve a third party. In jargon the mechanism has to be budget balancing.

15 Advanced Economic Theory 15 A seller and a buyer trade a single unit of a good. The seller s valuation for the good is and it is the seller s private information: P s ( ), [, ] The buyer s valuation for the good is and it is the buyer s private information: P b ( ), [, ] A contract in this environment is a pair (φ, t) where φ is the probability of trade, t is the transfer from the buyer to the seller. Revelation principle implies that wlog we can restrict attention to truth-telling direct mechanisms (φ(ˆ, ˆ ), t(ˆ, ˆ ))

16 Advanced Economic Theory 16 The seller s indirect utility is then: U s (ˆ, ˆ ) = ( +) t(ˆ, ˆ ) φ(ˆ, ˆ ) The buyer s indirect utility is instead: U b (ˆ, ˆ ) = φ(ˆ, ˆ ) t(ˆ, ˆ ) Denote: U s (ˆ ) = E vb [t(, ˆ ) φ(, ˆ ) ] = t s (ˆ ) φ s (ˆ ) U b (ˆ ) = E vs [φ(ˆ, ) t(ˆ, )] = φ b (ˆ ) t b (ˆ ) The following, by now standard, result, helps us to have (IC) and (IR) constraints in a manageable form.

17 Advanced Economic Theory 17 Result: (Myerson and Satterthwaite 1983) For any probability φ(, ) there exists a transfer function t(, ) that satisfies (IR) and (IC) if and only if: E vb, [φ(, ) (J b ( ) J s ( ))] 0 (1) where and J b ( ) = J s ( ) = dφ s ( ) d 0, ( 1 P ) b( ) p b ( ) ( + P ) s( ) p s ( ) (2) (3) dφ b ( ) d 0 (4) Proof: The standard transformation we used twice before gives us that the FOC and SOC of the truthtelling constraints imply:

18 Advanced Economic Theory 18 and Therefore U b ( ) = φ b ( ) U s ( ) = φ s ( ) 2 U b ( ) ˆ = dφ b( ) d 0, 2 U s ( ) ˆ = dφ s( ) d 0 and U b ( ) = U b ( ) + φ b (ν)dν U s ( ) = U s ( ) + vs φ s (γ)dγ

19 Advanced Economic Theory 19 Budget balancing implies that in equilibrium E vb [t b ( )] = E vs [t s ( )] Since we have t s ( ) = U s ( ) + φ s ( ) and t b ( ) = φ b ( ) U b ( ) We obtain: 0 = E vs [t s ( )] E vb [t b ( )] = = U s ( ) + U b ( ) + vs ( vs ) + φ s ( ) + φ s (γ)dγ p s ( ) d + + ( ) φ b (ν)dν φ( ) p b ( )d

20 Advanced Economic Theory 20 Integrating by parts we get: U s ( ) + U b ( ) = ( = 1 P ) b( ) φ p b ( ) b ( ) p b ( )d + vs ( + P ) s( ) φ p s ( ) s ( ) p s ( ) d (5) or U s ( ) + U b ( ) = = E vb, [φ(, ) (J b ( ) J s ( ))] (6) Since (IR) constraints are implies by U s ( ) 0 and U b ( ) 0 together with the monotonicity conditions we then have: E vb, [φ(, ) (J b ( ) J s ( ))] 0

21 Advanced Economic Theory 21 Sufficiency is a bit more complex to prove it requires us to solve the partial differential equation that is represented by the FOC of the (IC) constraints. The parties ex-ante problem is now: max φ i E vb, [φ(, ) ( )] s.t. E vb, [φ(, ) (J b ( ) J s ( ))] 0 dφ s ( ) d 0, dφ b ( ) d 0 Ignoring monotonicity conditions and denoting µ the lagrange multiplier of the remaining constraint we get a lagrangian function that is linear in φ i : [ E vb, {φ(, ) ( ) (1+µ) + µ ( 1 Pb ( ) p b ( ) + P ) s( ) ]} p s ( )

22 Advanced Economic Theory 22 The solution is to set φ = 1 if and only if the term in brackets is strictly positive. In other words trade occurs if and only if: µ ( ) 1 Pb ( ) + µ ( ) Ps ( ) 1 + µ p b ( ) 1 + µ p s ( ) where µ 0. This φ(c, v) is weakly monotonic in: µ ( ) 1 Pb ( ) 1 + µ p b ( ) and + µ 1 + µ ( ) Ps ( ) p s ( )

23 Advanced Economic Theory 23 Then MHRP implies that both monotonicity conditions are satisfied and hence local and global (IC) holds. Clearly if µ > 0 there will be inefficiencies in trade. Result: (Myerson and Satterthwaite 1983) Let > and > then necessarily µ > 0 Proof: We consider the case < < < the proof in the other cases is similar in structure.

24 Advanced Economic Theory 24 From equation (5) above we have that: U s ( ) + U b ( ) = vs [ p b ( ) 1 + P b ( )] φ(, ) p s ( )d d vs [ p s ( ) + P s ( )] φ(, ) p b ( ) d d Consider now the first best probability of trade: φ (, ) = { 1 if vb 0 if < If we substitute φ (, ) into the formula for U s ( ) + U b ( ) we obtain that the first integral can be written as: min{vb, } p s ( )d [ p b ( ) 1 + P b ( )] d

25 Advanced Economic Theory 25 or P s ( ) [ p b ( ) 1 + P b ( )] d while the second integral is: ( vb min{vb, } ) [ p s ( ) + P s ( )] d p b ( )d or min{ P s ( ), } p b ( ) d Notice that P s ( ) = 1 for, which implies that: min{ P s ( ), } = { vb P s ( ) if if >

26 Advanced Economic Theory 26 We can therefore re-write U s ( ) + U b ( ) as: U s ( ) + U b ( ) = P s ( ) p b ( ) d + P s ( ) [1 P b ( )] d + vs P s ( ) p b ( ) d + p b ( ) d In other words: U s ( ) + U b ( ) = ( ) p b ( ) d + P s ( ) [1 P b ( )] d

27 Advanced Economic Theory 27 Integrating by parts the first of the two integrals above we get: U s ( ) + U b ( ) = [1 P b ( )] d + P s ( ) [1 P b ( )] d Recall now that P s ( ) = 1 if > we obtain that U s ( ) + U b ( ) = = vs P s ( ) [1 P b ( )] d Clearly this last expression is strictly negative provided that, as we assumed, <. In other words, efficient trading φ (, ) does not satisfies constraint (5). This an important result: with bilateral asymmetric information even the optimal mechanism will lead to inefficient outcomes with strictly positive probability.