Mechanism Design: Bayesian Incentive Compatibility

Transcription

1 May 30, 2013

2 Setup X : finite set of public alternatives X = {x 1,..., x K } Θ i : the set of possible types for player i, F i is the marginal distribution of θ i. We assume types are independently distributed. v i (x, θ i ): gross utility, or value from x for i if she is type θ i Quasi-linear utility.

3 Efficiency The efficient allocation rule α is given by α (θ) = arg max x v j (x, θ j ) j

4 The Expected Externality Mechanism Consider the efficient allocation rule α. Define, preliminarily, a subsidy rule as follows. t i (θ i ) = E θ i v j (α (θ), θ j ) j =i Now we construct the expected externality mechanism (or AGV mechanism since it was found by Arrow and d Aspremont and Gerard-Varet in 1979) The AGV mechanism has the efficient allocation rule and the following transfer rule: ti AGV (θ) = 1 N 1 t j (θ j ) t i (θ i ) j =i

5 AGV is budget-balanced i t AGV i [ (θ) = i 1 N 1 j =i Take any j. There are N 1 terms of the form 1 N 1 t j (θ j ) t j (θ j ) t i (θ i ) and 1 term of the form t j (θ j ). These cancel and this exhausts all of the terms in the summation, so ti AGV (θ) = 0 i for all θ. This is ex post budget-balance. ]

6 AGV is BIC If i announces her true type θ i, she gets [ ] U i (θ i ) = E θ i v i (α(θ), θ i ) t AGV (θ) = E θ i [v i (α (θ), θ i ) + v j (α (θ), θ j ) j =i By announcing ˆθ i she gets utility ] E θ i [v i (α ( ˆθ i, θ i ), θ i ) + v j (α ( ˆθ i, θ i ), θ j ) j =i i ] [ ] 1 E θ i N 1 t j (θ j ) j =i }{{} a constant C C Since by the definition of α, j v j (α (θ), θ j ) j v j (x, θ j ), ˆθ i, and for all x, it is optimal to announce the true type.

7 IR There is no guarantee that the AGV mechanism is individually rational, even in the interim sense: U i (θ i ) 0

8 Special Case: 2 Relevant Alternatives In many important applications, each agent views the entire set of alternatives as divided into only two distinct payoff-relevant categories. Single-Unit Auctions. (You either win or lose) Bilateral Trade. (You either have the object or you don t) Public Goods Problems. (Either the bridge is built or it isn t) In these settings an agent s type is a single (1-dimensional) parameter, such as willingness to pay.

9 Formally X {0, 1} N. x = (x 1,..., x N ) i s value depends only on x i : For example v i (x, θ i ) = 1 Bilateral trade X = {(1, 0), (0, 1)} { θ i if x i = 1 0 if x i = 0 2 Auction X = {(1, 0,..., 0), (0, 1, 0,..., 0),... (0,..., 0, 1)} 3 Public good X = {(1, 1,..., 1), (0, 0,..., 0)}

10 The Revelation Principle We will consider Bayesian Nash equilibria of arbitrary mechanisms. By the revelation principle we can restrict attention to direct-revelation mechanisms. Definition A direct-revelation mechanism is a pair Γ = (Q, t) where Q : Θ X is the allocation rule, t : Θ R N is the transfer rule.

11 Bayesian Incentive Compatibility Let Q i (θ) = Q(θ){x X : x i = 1}, i.e the probability of getting the good, and Q i (θ i ) = Q i (θ i, s i )df (s i θ i ), x X, Θ i denotes the total allocation probability for type θ i. Similarly let t i (θ i ) be type θ i s expected transfer. Definition The direct revelation mechanism (Q, t) is Bayesian (or interim) incentive compatible if for all i, and for all θ i for all ˆθ i. θ i Q i (θ i ) t i (θ i ) θ i Q i ( ˆθ i ) t i ( ˆθ i ) By the revelation principle there is no loss of generality in considering BIC direct revelation mechanisms.

12 Interim Individual Rationality Definition A mechanism (Q, t) is interim individually rational if θ i Q i (θ i ) t i 0 for all θ i, for all i.

13 Interim Indirect Utility Define the interim indirect utility function Then BIC condition becomes U i (θ i ) = Q i (θ i )θ i + t i (θ i ). U i (θ i ) U i ( ˆθ i ) + (θ i ˆθ i ) Q i ( ˆθ i ). Since it must hold for all pairs θ i and ˆθ i, it holds for the same pair with roles reversed: U i ( ˆθ i ) U i (θ i ) + ( ˆθ i θ i ) Q i (θ i ). Add these inequalities and rearrange to get the monotonicity condition (θ i ˆθ i )( Q i (θ i ) Q i ( ˆθ i )) 0.

14 Smoothness For θ i > ˆθ i, (θ i ˆθ i ) Q i (θ i ) U i (θ i ) U i ( ˆθ i ) (θ i ˆθ i ) Q i ( ˆθ i ). hence U is differentiable and U (θ i ) = lim ˆθ i θ i U i (θ i ) U i ( ˆθ i ) θ i ˆθ i = Q i (θ i )

15 Monotonicity and Envelope Proposition The direct revelation mechanism (Q, t) is Bayesian incentive compatible if and only if 1 (Monotonicity) Q i ( ) is weakly increasing for all i, 2 (Mirlees) U i (θ i ) = U i (0) + Θ i Q i (θ i )

16 Revenue Equivalence Theorem Theorem Let Q be any Bayesian Incentive allocation rule. All mechanisms which implement Q give the same indirect utility function and hence the same expected transfer rule, up to a constant.

17 Application of Revenue Equivalence As we showed earlier in class, the first-price auction, second-price auction, Dutch auction, and English auction all yield the same allocation rule in Bayesian Nash equilibrium. By the revenue equivalence theorem, they all therefore generate the same expected revenue for the seller.

18 Bilateral Trade Recall the bilateral trade problem Consider any individually rational efficient Bayesian Incentive Compatible Mechanism. Let U i be the interim indirect utility functions. Consider the VCG mechanism with default types 0 for the buyer and 1 for the seller. Let Ui VCG be its interim indirect utility function. By payoff equivalence, for any θ b U b (θ b ) U b (0) = U VCG (θ b ). By individual rationality U b (0) 0, hence U b (θ b ) U VCG (θ b ). Since the two mechanisms have the same allocation rule, buyers must be paying less in the first mechanism. By a similar argument, for any θ s U s (θ s ) U VCG (θ s ). Implying that seller s receive more money in the first mechanism. The budget surplus is lower than in the VCG mechanism.

19 The Myerson-Satterthwaite Theorem Theorem There does not exist an efficient, Bayesian Incentive Compatible, interim Individually Rational, budget balancing mechanism in the bilateral trade problem.

20 Optimal Auction Model: N buyers, i = 1,..., N, Single indivisible object for sale, Buyer i gets value θ i [0, 1] from winning and 0 otherwise. θ i is distributed according CDF F i (θ i ) (with PDF f i (θ i )), joint distribution is independent. Seller holds the same beliefs as the players. Utility function: U i (θ) = q i (θ)θ i t i (θ), where q i is the probability of winning and t i is the transfer to the seller (even if he loses. would not affect anything if we assumed payments only when winning.). q : Θ [0, 1] N, st q i (θ) 0, i q i (θ) 1 (can be strictly less than 1, since sometimes it is optimal to keep the object). t : Θ R N. U i (θ i ) = E θ i [U i (θ)], q i (θ i ) = E θ i [q i (θ)]

21 Maximization Problem The seller chooses the mechanism (q, t) to maximize expected revenue E i t i subject to Bayesian Incentive Compatibility Interim Individual Rationality We are normalizing the seller s value for the object to zero.

22 Rewriting Expected Revenue ER = E θ N i=1 t i (θ) = E θ N i=1[q i (θ)θ i U i (θ)] = = N E θi [ q i (θ i )θ i U i (θ i )] i=1 N ER i i=1 Now we can think of the seller has choosing the functions q i and U i (θ i ).

23 Substitute the Envelope Formula Into the Objective ER i = E θi [ q i (θ i )θ i U i (θ i )] = = 1 0 q i (r)r f i (r)dr [ r 0 0 [ r ] q i (r)r U i (0) q i (s)ds f i (r)dr 0 ] q i (s)ds we will integrate the second term by parts. 1 f i (r)dr U i (0) f i (r)dr, 0 } {{ } =1

24 Integrating by Parts The second term 1 : 1 [ r ] r 1 q i (s)ds f i (r)dr = F i (r) q i (s)ds r=1 r= q i (r)f i (r)dr = 1 0 [1 F i (r)] q i (r)dr. 1 dv = f i (r)dr v = F i (r), u = r 0 q i (s)ds du = q i (r)dr

25 Back to Revenue Thus ER i = = q i (r)r f i (r)dr 0 [ q i (r) [1 F i (r)] q i (r)dr U i (0) r 1 F ] i (r) f i (r)dr U i (0) f i (r) }{{} =VS i (θ i ) (virtual surplus) = E θi [ q i (θ i )VS i (θ i )] U i (0).

26 Total Revenue [ ER = E θ q i (θ i ) θ i 1 F i (θ i ) i f i (θ i ) ] U i (0). i The seller chooses the functions q i and the constants U i (0) to maximize this subject to montonicity (we already used the other part of BIC, the envelope condition) individual rationality Clearly we set U i (0) = 0 for all i and satisfy individual rationality.

27 Ignore Monotonicity [ max E θ q i (θ i ) θ i 1 F ] i (θ i ) i f i (θ i ) If we maximize this separately for each point θ, we obtain the allocation rule: { 1 if VS i (θ i ) > VS j (θ j ) for all j = i and VS i (θ i ) > 0 q i (θ) = 0 otherwise breaking ties arbitrarily.

28 Hazard Rate Condition That allocation rule will be monotone, and hence optimal for the constrained problem if 1 F i (θ i ) f i (θ i ) is decreasing in θ i. (The monotone hazard rate condition.)

29 Properties of the Solution There is typically inefficient witholding of the object, i.e. when θ i > 0 > VS i (θ i ) If the bidders are symmetric (F i = F j ) then the allocation is efficient conditional on sale. Therefore by the revenue equivalence theorem, it can be implemented by a Vickrey auction with a reserve price.