Mechanism Design: Bargaining

Transcription

1 Mechanism Design: Bargaining Dilip Mookherjee Boston University Ec 703b Lecture 5 (text: FT Ch 7, pp ) DM (BU) Mech Design 703b / 13

2 The Bargaining Problem Two agents: S, seller and B, a prospective buyer, of an indivisible good They know their own valuations of the good: θ s [θ s, θ s ], θ b [θ b, θ b ] Common knowledge that θ b, θ s are drawn independently according to cdf s F s, F b x: probability of sale, p price in the event of a sale Payoffs U S (p θ s )x, U B (θ b p)x Trade must be voluntary: each agent has the option not to participate (attain 0 payoff from x = 0) DM (BU) Mech Design 703b / 13

3 Negotiations and Haggling Most actual bargaining situations involve a dynamic negotiation game E.g. the seller offers to sell at an asking price, the buyer responds by saying yes, or refuses and makes a counteroffer, to which the seller responds... Suppose game ends at each round with a fixed probability q Can study the outcome of a perfect Bayesian equilibrium of this game Each agent will tend to keep negotiating for a better price, so the game may end without any sale occurring, despite the existence of gains from trade (θ b > θ s ) DM (BU) Mech Design 703b / 13

4 Chatterjee-Samuelson Bargaining Game (F-T Chapter 6, Example 6.4) Chatterjee-Samuelson (1983) studied a double auction game with one round of simultaneous offers, where both valuations are uniform on [0, 1] Buyer submits a bid θ b, seller asks for θ s ; trade occurs iff the bid exceeds the asking price, at a price equal to their average (p = θ b + θ s 2 ) A Bayesian equilibrium where bids and asks are linear in the true valuations: θ b = θ b; θ s = θ s Trade occurs iff θ b θ s 1 4 If 1 4 > θ b θ s > 0, there is no sale despite the existence of gains from trade DM (BU) Mech Design 703b / 13

5 Scope for Designing the Bargaining Game Maybe there is scope for reducing the inefficiency, by adding more rounds, or going to a sequential procedure...? Could a negotiation game be designed which always generates efficient outcomes in all possible states? Difficult to use a trial and error process to answer this question, there are infinite number of possible negotiation games Can cut through this problem, using the Revelation Principle! RP states that if there exists an efficient negotiation protocol, there must also exist a static revelation mechanism which results in efficient trade and satisfies the Partiicipation Constraint (PC) DM (BU) Mech Design 703b / 13

6 Bargaining Revelation Mechanisms In a revelation mechanism, buyer and seller simultaneously report θ s, θ b, which determines x( θ s, θ b ), t s ( θ s, θ b ), t b ( θ s, θ b ), where t s, t b denote expected transfers to (from) the seller (buyer) (if trade probability is x x( θ s, θ b ), price in event of trade is p p( θ b, θ s ) and there is no broker commission or entry fee, then t s ( θ s, θ b ) = p x = t b ( θ s, θ b )) (Interim) Payoffs: U s ( θ s ; θ s ) E θb [t s ( θ s, θ b ) θ s x( θ s, θ b )] U b ( θ b ; θ b ) E θs [θ b x(θ s, θ b ) t b (θ s, θ b )] DM (BU) Mech Design 703b / 13

7 Bargaining Revelation Mechanisms, contd. BB: t s (θ s, θ b ) + t b (θ s, θ b ) = 0 for all θ b, θ s PE: Sale occurs (does not occur) (x = 1(0)) if θ b > (<)θ s PC: U b (θ b ; θ b ) 0, U s (θ s ; θ s ) 0 for all θ b, θ s BIC: θ b = θ b maximizes U b ( θ b ; θ b ), θ s = θ s maximizes U s ( θ s ; θ s ), for all θ b, θ s The Problem: Does there exist a mechanism satisfying BB, PE, PC and BIC? DM (BU) Mech Design 703b / 13

8 Connection with the Public Good Problem We can reformulate it as a public decision problem: d x; V S = xθ S + t S, V B = xθ B + t B The ADAV Theorem states that there does exist a set of balanced budget transfers that implement the PO allocation (where truthful reporting of valuations by both agents constitutes a Bayesian equilibrium) But what about the Participation Constraint? There is no PC in the public goods problem payment of taxes is not voluntary for most people! DM (BU) Mech Design 703b / 13

9 Cases where Efficient Bargaining Mechanisms Exist Suppose there are gains from trade with probability one ( θ s < θ b ): set x 1 and p = θ s+θ b 2, t s = p θ s, t b = t s Suppose there are gains from trade with probability zero ( θ b < θ s ): set x 0 t s t b DM (BU) Mech Design 703b / 13

10 Myerson-Satterthwaite Theorem Theorem Suppose there are gains from trade with positive probability less than one ( θ s > θ b, θ b > θ s ), and F s, F b have positive densities f s, f b at every interior state (θ s, θ b ). Then there does not exist any bargaining mechanism satisfying BB, BIC, PE and PC. DM (BU) Mech Design 703b / 13

11 Proof of M-S Theorem In an efficient mechanism, x(θ b, θ s ) = 1 iff θ b > θ s (ignoring measure zero states where θ b = θ s ), hence: U b ( θ b ; θ b ) = θ b F s ( θ b ) T b ( θ b ), U s ( θ s ; θ s ) = T s ( θ s ) θ s [1 F b ( θ s )] (where T s (θ s ) E θb t s (θ s, θ b ); T b (θ b ) E θs t b (θ s, θ b )) BIC for buyer requires (using Mirrlees-Myerson characterization of IC constraint in single agent problems from L2): θb U b (θ b ; θ b ) θ b F s (θ b ) T b (θ b ) = Π b + F s ( θ b )d θ b θ b (where Π b U b (θ b ; θ b )) DM (BU) Mech Design 703b / 13

12 M-S Proof, contd. BIC implies: θb T b (θ b ) = F s (θ b ) F s ( θ b )d θ b Π b θ b T s (θ s ) = θ s [1 F b (θ s )] + θs (where Π b denotes exp payoff of seller of type θ s ) BB requires E θb T b (θ b ) = E θs T s (θ s ), or θ s [1 F b ( θ s )]d θ s + Π s E θb [F s (θ b ) θb θ b F s ( θ b )d θ b ] Π b = E θs [θ s [1 F b (θ s )] + θ s θ s [1 F b ( θ s )]d θ s ] + Π s DM (BU) Mech Design 703b / 13

13 M-S Proof, contd. E θb [F s (θ b ) θb θ b F s ( θ b )d θ b ] E θs [θ s [1 F b (θ s )] + = Π b + Π s 0 θs θ s [1 F b ( θ s )]d θ s ] (since PC requires Π b, Π s 0) On the other hand, Integrating LHS by parts, it equals (Check!) which is negative since θ s > θ b. θs F s (θ)[1 F b (θ)]dθ θ b DM (BU) Mech Design 703b / 13