Bayesian Games and Auctions

Transcription

1 Bayesian Games and Auctions Mihai Manea MIT

2 Games of Incomplete Information Incomplete information: players are uncertain about the payoffs or types of others Often a player s type defined by his payoff function. More generally, types embody any private information relevant to players decision making... may include a player s beliefs about other players payoffs, his beliefs about what other players believe his beliefs are, and so on. Modeling incomplete information about higher order beliefs is intractable. Assume that each player s uncertainty is solely about payoffs. Mihai Manea (MIT) Bayesian Games and Auctions February 22, / 49

3 Bayesian Game A Bayesian game is a list B = (N, S, Θ, u, p) where N = {1, 2,..., n}: finite set of players S i : set of pure strategies of player i; S = S 1... S n Θ i : set of types of player i; Θ = Θ 1... Θ n u i : Θ S R is the payoff function of player i; u = (u 1,..., u n ) p (Θ): common prior Often assume Θ is finite and marginal p(θ i ) is positive for each type θ i. Strategies of player i in B are mappings s i : Θ i S i (measurable when Θ i is uncountable). Mihai Manea (MIT) Bayesian Games and Auctions February 22, / 49

4 First Price Auction One object is up for sale. Value θ i of player i N for the object is uniformly distributed in Θ i = [0, 1], independently across players, i.e., p(θi θ i, i N) = θ i, θ i [0, 1], i N. i N Each player i submits a bid s i S i = [0, ). The player with the highest bid wins the object (ties broken randomly) and pays his bid. Payoffs: θ i s i u i (θ, s) = {j N s i=s j } if s i s j, j N 0 otherwise. Mihai Manea (MIT) Bayesian Games and Auctions February 22, / 49

5 An Exchange Game Player i = 1, 2 receives a ticket on which there is a number from a finite set Θ i [0, 1]... prize player i may receive. The two prizes are independently distributed, with the value on i s ticket distributed according to F i. Each player is asked independently and simultaneously whether he wants to exchange his prize for the other player s prize: S i = {agree, disagree}. If both players agree then the prizes are exchanged; otherwise each player receives his own prize. Payoffs: u i (θ, s) = θ 3 i if s 1 = s 2 = agree θ i otherwise. Mihai Manea (MIT) Bayesian Games and Auctions February 22, / 49

6 Ex-Ante Representation In the ex ante representation G(B) of the Bayesian game B player i has Θ strategies (s i i(θ i )) θi Θ i S i his strategies are functions from types to strategies in B and utility function U i given by ((( U i si (θ i ) ) ) ) = E θ i Θ p (u i (θ, s 1 (θ 1 ),..., s n (θ n ))). i i N Mihai Manea (MIT) Bayesian Games and Auctions February 22, / 49

7 Interim Representation The interim representation IG(B) of the Bayesian game B has player set i Θ i. The strategy space of player θ i is S i. A strategy profile (s θi ) i N,θi Θ i yields utility for player θ i. Need p(θ i ) > 0... U θi ((s θi ) i N,θi Θ i ) = E p (u i (θ, s θ1,..., s θn ) θ i ) Mihai Manea (MIT) Bayesian Games and Auctions February 22, / 49

8 Bayesian Nash Equilibrium Definition 1 In a Bayesian game B = (N, S, Θ, u, p), a strategy profile s : Θ S is a Bayesian Nash equilibrium (BNE) if it corresponds to a Nash equilibrium of IG(B), i.e., for every i N, θ i Θ i [ ( E p( θi ) [u i (θ, s i (θ i ), s i (θ i ))] E p( θi ) ui θ, s, s i (θ i ) )], s S i. Interim rather than ex ante definition preferred since in models with a continuum of types the ex ante game has many spurious equilibria that differ on probability zero sets of types. i i Mihai Manea (MIT) Bayesian Games and Auctions February 22, / 49

9 Connections to the Complete Information Games When i plays a best-response type by type, he also optimizes ex-ante payoffs (for any probability distribution over Θ i ). Therefore, a BNE of B is also a Nash equilibrium of the ex-ante game G (B). BNE(B): Bayesian Nash equilibria of bayesian game B NE(G): Nash equilibria of normal-form game G Proposition 1 For any Bayesian game B with a common prior p, BNE (B) NE (G (B)). If p (θ i ) > 0 for all θ i Θ i and i N, then BNE (B) = NE (G (B)). Mihai Manea (MIT) Bayesian Games and Auctions February 22, / 49

10 Business Partnership Two business partners work on a joint project. Each businessman i = 1, 2 can either exert effort (e i = 1) or shirk (e i = 0). Each face the same fixed (commonly known) cost for effort c < 1. Project succeeds if at least one partner puts in effort, fails otherwise. Players differ in how much they care about the fate of the project: i has a private, independently distributed type θ i U[0, 1] and receives 2 payoff from success. θ i Hence player i gets θ 2 2 i c from working, θ i from shirking if opponent j works, and 0 if both shirk. e 2 = 1 e 2 = 0 e 1 = 1 θ 2 1 c, θ2 2 c θ2 1 c, θ e 1 = 0 θ 1, θ2 c 0, 0 Mihai Manea (MIT) Bayesian Games and Auctions February 22, / 49

11 Equilibrium e 2 = 1 e 2 = 0 e 1 = 1 θ 2 1 c, θ2 2 c θ2 1 c, θ e 1 = 0 θ 1, θ2 c 0, 0 p j : probability that j works sufficient statistic for strategic situation faced by player i 2 Working is rational for i if θ i c pjθ 2 ( 1 p 2 i j )θ i c. Thus i must play a threshold strategy: work for θ i θ i := c. 1 p j Since p j = Prob(θ j θ ) = 1 θ j j, we get c c θ i = θ = = cθ 4 c i, j θ i so θ = 3 c. In equilibrium, i = 1, 2 works if θ i i 3 c and shirks otherwise. Mihai Manea (MIT) Bayesian Games and Auctions February 22, / 49

12 Auctions single good up for sale n buyers bidding for the good buyer i has value X i, i.i.d. with distribution F and continuous density f = F ; supp(f) = [0, ω] i knows only the realization x i of X i Mihai Manea (MIT) Bayesian Games and Auctions February 22, / 49

13 Auction Formats First-price sealed-bid auction: each buyer submits a single bid (in a sealed envelope) and the highest bidder obtains the good and pays his bid. Equivalent to descending-price (Dutch) auctions. Second-price sealed-bid auction: each buyer submits a bid and the highest bidder obtains the good and pays the second highest bid. Equivalent to open ascending-price (English) auctions. Bidding strategies: β i : [0, ω] [0, ) What are the BNEs in the two auctions? Which auction generates higher revenue? Mihai Manea (MIT) Bayesian Games and Auctions February 22, / 49

14 Second-Price Auction Each bidder i submits a bid b i, payoffs given by x i max j i b j if b i > max j i b j ui = 0 if bi < max j i b j Ties broken randomly. Proposition 2 In a second-price auction, it is a weakly dominant strategy for every player i to bid according to β II ( x i i ) = x i. Mihai Manea (MIT) Bayesian Games and Auctions February 22, / 49

15 Second-Price Auction Expected Payments Y 1 = max i 1 X i : highest value of player 1 s opponents, distributed according to G with G(y) = F(y) n 1 Expected payment by a bidder with value x is m II (x) = Prob[Win] E[2nd highest bid x is the highest bid] = Prob[Win] E[2nd highest value x is the highest value] = G(x) E[Y 1 Y 1 < x] Mihai Manea (MIT) Bayesian Games and Auctions February 22, / 49

16 First-Price Auction Each bidder i submits a bid b i, payoffs given by x i b i if b i > max j i b j ui = 0 if b i < max j i b j Ties broken randomly. Clearly, not optimal/equilibrium to bid own value. Trade-off: higher bids increase the probability of winning but decrease the gains. Symmetric equilibrium: β i = β for all buyers i. Assume β strictly increasing, differentiable. Mihai Manea (MIT) Bayesian Games and Auctions February 22, / 49

17 Optimal Bidding Suppose bidder 1 has value X 1 = x and considers bidding b. Clearly, b β(ω) and β(0) = 0. Bidder 1 wins the auction if max i 1 β(x i ) < b. Since β is s. increasing, 1 max i 1 β(xi) = β(max i 1 X i ) = β(y 1 ), so 1 wins if Y 1 < β (b). His expected payoff is G(β 1 (b)) (x b). FOC : G (β 1 (b)) 1 (x b) G(β (b)) = 0 β (β 1 (b)) b = β(x) G(x)β (x) + G (x)β(x) = xg(x) (G(x)β(x)) = xg(x) x 1 β(0) = 0 β(x) = yg(y)dy G(x) 0 = E[Y 1 Y 1 < x]. Mihai Manea (MIT) Bayesian Games and Auctions February 22, / 49

18 Equilibrium Proposition 3 The strategies I β (x) = E[Y 1 Y 1 < x] constitute a symmetric BNE in the first-price auction. Mihai Manea (MIT) Bayesian Games and Auctions February 22, / 49

19 Proof We only checked necessary conditions for equilibrium... Check that if all bidders follow strategy β I then it is optimal for bidder 1 to follow it. Since β I is increasing and continuous, it cannot be optimal to bid higher than β I (ω). Suppose bidder 1 with value x bids b [0, β I I (ω)]. z, β (z) = b. Since bidder 1 wins if Y 1 < z, his payoffs are Π(b, x) = G(z)[x β (z)] I = G(z)x G(z)E[Y 1 Y 1 < z] z = G(z)x yg(y)dy 0 z = G(z)x G(z)z + G(y)dy 0 z = G(z)(x z) + G(y)dy. Then I I Π(β (x), x) Π(β (z), x) = G(z)(z x) 0 z G(y)dy 0. x Mihai Manea (MIT) Bayesian Games and Auctions February 22, / 49

20 Shading x I 1 β (x) = yg(y)dy G(x) 0 x G(y) = x 0 G(x) dy x [ ] F(y) n 1 = x dy F(x) Shading, the amount by which the bid is lower than the value, is x 0 [ ] F(y) n 1 dy. 0 F(x) Depends on n, converges to 0 as n (competition). Mihai Manea (MIT) Bayesian Games and Auctions February 22, / 49

21 Example with Uniformly Distributed Values n 1 If F(x) = x for x [0, 1], then G(x) = x and I n 1 β (x) = n x. Mihai Manea (MIT) Bayesian Games and Auctions February 22, / 49

22 Example with Exponentially Distributed Values If n = 2 and F(x) = 1 exp( λx) for x [0, ) (λ > 0) then Note that E[X] = 1/λ. x I F(y) β (x) = x 0 F(x) dy = 1 λ x exp( λx) 1 exp( λx) Take λ = 1. A bidder with value $10 6 will not bid more than $1. Why? Such a bidder has a lot to lose by not bidding higher but the probability of losing is small, exp( 10 6 ). More generally, for n = 2, I β (x) = E[Y 1 Y 1 < x] E[Y 1 ] = E[X 2 ]. Mihai Manea (MIT) Bayesian Games and Auctions February 22, / 49

23 Revenue Comparison Expected payment in the first-price auction by a bidder with value x is Recall that m I (x) = Prob[Win] Amount bid = G(x) E[Y 1 Y 1 < x] m II (x) = G(x) E[Y 1 Y 1 < x], so both auctions yield the same revenue. Special case of the revenue equivalence theorem. Mihai Manea (MIT) Bayesian Games and Auctions February 22, / 49

24 Mechanism Design An auction is one of many mechanisms a seller can use to sell the good. The price is determined by the competition among buyers according to the rules set out by the seller the auction format. The seller could use other methods post different prices for each bidder, choose a winner at random ask various subsets of bidders to pay their own or others bids Options virtually unlimited... Myerson (1981): What is the optimal mechanism? Mihai Manea (MIT) Bayesian Games and Auctions February 22, / 49

25 Framework single good up for sale, worth 0 to the seller buyers: 1, 2,..., n buyers have private values, independently distributed buyer i s value X i distributed according to F i supp(f i ) = [0, ω i ] = X i, density f i = F i i knows only the realization x i of X i X = n i=1 Xi n f(x) = i=1 f i (x i ) f i( x i ) = j i fj( x j ) Mihai Manea (MIT) Bayesian Games and Auctions February 22, / 49

26 Mechanisms A selling mechanism (B, π, µ) B i : set of messages (bids) for buyer i allocation rule π : B (N) payment rule µ : B R n The allocation rule determines, as a function of all n messages b, the probability π i (b) that i gets the object. Similarly the payment rule specifies a payment µ i (b) for each buyer i. Describe first- and second-price auctions as mechanisms... Every mechanism induces a game of incomplete information with strategies β i : X i B i. Mihai Manea (MIT) Bayesian Games and Auctions February 22, / 49

27 Direct Mechanisms Mechanisms can be complicated, no assumptions on the messages B i. Direct mechanism (Q, M) B i = X i, every buyer is asked to directly report a value Q : X (N) and M : X R n Q i (x): probability that i gets the object M i (x): payment by i If β i : X i X i with β i (x i ) = x i constitutes a BNE of the induced game then we say that the direct mechanism has a truthful equilibrium or is incentive compatible. Mihai Manea (MIT) Bayesian Games and Auctions February 22, / 49

28 The Revelation Principle Proposition 4 Given a mechanism and an equilibrium for that mechanism, there exists a direct mechanism in which (i) it is an equilibrium for each buyer to report his or her value truthfully and (ii) the outcomes are the same as in the given equilibrium of the original mechanism for every type realization x. Consider a mechanism (B, π, µ) with an equilibrium β. Define Q : X (N) and M : X R N as follows: Q(x) = π(β(x)) and M(x) = µ(β(x)). The direct mechanism (Q, M) asks players to report types and does the equilibrium computation for them. (ii) holds by construction. To verify (i): if buyer i finds it profitable to report z i instead of his true value x i in the direct mechanism (Q, M), then i prefers the message β i (z i ) instead of β i (x i ) in the original mechanism. Mihai Manea (MIT) Bayesian Games and Auctions February 22, / 49

29 Incentive Compatibility For a direct mechanism (Q, M), define q i (z i ) = m i (z i ) = Q i (z i, x i )f i (x i )dx i X i M i (z i, x i )f i (x i )dx i X i Expected payoff of buyer i with value x i who reports z i if other buyers report truthfully q i (z i )x i m i (z i ) (Q, M) is incentive compatible (IC) if U i is convex because U i (x i ) q i (x i )x i m i (x i ) q i (z i )x i m i (z i ), i, x i, z i U i (x i ) = max{q i (z i )x i m i (z i ) z i X i }. Mihai Manea (MIT) Bayesian Games and Auctions February 22, / 49

30 Payoff Formula Since q i (x i )z i m i (x i ) = q i (x i )x i m i (x i ) + q i (x i )(z i x i ) IC requires that = U i (x i ) + q i (x i )(z i x i ), U i (z i ) U i (x i ) + q i (x i )(z i x i ). Hence q i (z i )(z i x i ) U i (z i ) U i (x i ) q i (x i )(z i x i ). For z i > x i, U q i (z i ) i (z i ) U i (x i ) qi( x i ), zi x so q i is increasing. Since U i is convex, it is differentiable almost everywhere, U i ( x i ) = q i (x i ) U i (x i ) = U i (0) + i xi 0 q i (t i )dt i Mihai Manea (MIT) Bayesian Games and Auctions February 22, / 49

31 Monotonicity Condition IC implies monotonicity of q i. Conversely, a mechanism where U i satisfies x i U i (x i ) = U i (0) + q i (t i )dt i with q i increasing must be incentive compatible. IC condition U i (z i ) U i (x i ) q i (x i )(z i x i ) 0 boils down to zi q i (t i )dt i x i q i (x i )(z i x i ). Mihai Manea (MIT) Bayesian Games and Auctions February 22, / 49

32 Revenue Equivalence Theorem 1 U i (x i ) = U i (0) + xi q i (t i )dt i 0 If the direct mechanism (Q,M) is incentive compatible, then for all i and x i, x i m i (x i ) = m i (0) + q i (x i )x i q i (t i )dt i. The expected payments in any two incentive compatible mechanisms with the same allocation rule are equivalent up to a constant. 0 U i (x i ) = q i (x i )x i m i (x i ) = U i (0) + 0 x i q i (t i )dt i Mihai Manea (MIT) Bayesian Games and Auctions February 22, / 49

33 First-Price Auction Revisited n symmetric buyers Assuming a symmetric monotone equilibrium β in first-price auction, the highest value buyer obtains the good. Same allocation Q as in the equilibrium of the second-price auction. Buyers with value 0 bid 0, so U i (0) = 0 in both auctions. By Theorem 1, Since we obtain β(x) = E[Y 1 Y 1 < x]. m I (x) = m II (x). m I (x) = G(x) β(x) m II (x) = G(x) E[Y 1 Y 1 < x], Mihai Manea (MIT) Bayesian Games and Auctions February 22, / 49

34 All-Pay Auction n symmetric buyers. Highest bidder receives the good, but all buyers have to pay their bid (as in lobbying). Assuming a symmetric monotone equilibrium β in the all-pay auction, the highest value buyer obtains the good. Same allocation Q as in the equilibrium of the second-price auction. Buyers with value 0 bid 0, so U i (0) = 0 in both auctions. By Theorem 1, m all pay II (x) = m (x) = G(x) E[Y 1 Y 1 < x]. Since m all pay (x) = β(x), β(x) = G(x) E[Y 1 Y 1 < x]. Underbidding compared to first- and second-price auctions. Can use revenue equivalence with the second-price auction to derive equilibrium in any auction where we expect efficient allocation. Mihai Manea (MIT) Bayesian Games and Auctions February 22, / 49

35 Individual Rationality A mechanism is individually rational (IR) if U i (x i ) 0 for all x i X i. U i (x i ) = U i (0) + 0 x i q i (t i )dt i 0, x i X i U i (0) = m i (0) 0 Mihai Manea (MIT) Bayesian Games and Auctions February 22, / 49

36 Expected Revenue In a direct mechanism (Q, M), the expected revenue of the seller is E[R] = n i=1 E[m i (X i )]. Substitute the form ula for m i, ω i E[m i (X i )] = m i (x i )f i (x i )dx i 0 ωi ω i x i = m i (0) + q i (x i )x i f i (x i )dx i q i (t i )f i (x i )dt i dx i. 0 Interchanging the order of integration, ωi xi ω i ω i ω i q i (t i )f i (x i )dt i dx i = q i (t i )f i (x i )dx i dt i = (1 F i (t i ))q i (t i )dt i t i 0 ω i ( ) 1 F i (x i ) E[m i (X i )] = m i (0) + x i q i (x i )f i (x i )dx i 0 f i (x i ) ( ) 1 F i (x i ) = m i (0) + x i Q i (x)f(x)dx X f i (x i ) Mihai Manea (MIT) Bayesian Games and Auctions February 22, /

37 Optimal Mechanism The seller s objective is to maximize revenue, n n ( m i (0) + X i=1 i=1 x i ) 1 F i (x i ) Qi(x) f(x)dx fi(x i ) subject to IC and IR. IC is equivalent to q i being increasing for every i and IR to m i (0) 0. Clearly, need to set m i (0) = 0. Maximize n X i=1 1 F i (x i ) ψ i (x i )Q i (x)f(x)dx where ψ i (x i ) := x i f i (x i ) subject to q i being increasing for every i. ψ i (x i ): virtual value of player i with type x i Regularity condition: assume ψ i is s. increasing for every i Mihai Manea (MIT) Bayesian Games and Auctions February 22, / 49

38 Optimal Solution Ignoring the q i monotonicity condition, maximize for every x Set n i=1 ψ i (x i )Q i (x). Q i (x) > 0 ψ i (x i ) = max ψ j (x j ) 0. To obtain m i (x i ) = m i (0) + q i (x i )xi xi 0 j N q i (t i )dt i, define xi M i (x) = Q i (x)x i Q i (z i, x i)dz i. 0 (Q, M) is an optimal mechanism. Only need to check implied q i is increasing. If z i < x i, regularity implies ψ i (z i ) < ψ i (x i ), which means that Q i (z i, x i ) Q i (x i, x i ). E[R] = E[max(ψ 1 (X 1 ), ψ 2 (X 2 ),..., ψ n (X n ), 0)] Mihai Manea (MIT) Bayesian Games and Auctions February 22, / 49

39 Optimal Auction Smallest value needed for i to win against opponent types x i : y i (x i ) = inf{z i : ψ i (z i ) 0 and ψ i (z i ) ψ j (x j ), j i} In the optimal mechanism, 1 if zi > y i ( x i ) Q i (z i, x i ) =. 0 if z i < y i (x i ) Then M i (x) = Q i (x)x i 0 x i x i Q i (z i, x i )dz i = (Q i (x i, x i ) Q i (z i, x i ))dz i 0 y i (x i ) if Q i (x) = 1 =. 0 if Q i (x) = 0 The player with the highest positive virtual value wins. Only the winning player has to pay and he pays the smallest amount needed to win. Mihai Manea (MIT) Bayesian Games and Auctions February 22, / 49

40 Symmetric Case Suppose F i = F, so ψ i = ψ and In the optimal mechanism, ( ) y i (x i ) = max ψ 1 (0), max x j j i 1 if z i > y i (x i ) Q i (z i, x i ) = 0 if z i < y i (x i ) and y i (x i) if Qi(x) = 1 M i (x) = 0 if Q i (x) = 0. Proposition 5 Suppose the design problem is regular and symmetric. Then the second-price auction with a reserve price r = ψ 1 (0) is an optimal mechanism. Mihai Manea (MIT) Bayesian Games and Auctions February 22, / 49

41 Intuition for Virtual Values Why is it optimal to allocate the object based on virtual values? Consider a single buyer whose value is distributed according to F. The seller sets price p to maximize p(1 F(p)), FOC : 1 F(p) pf(p) = 0 ψ(p) = 0. Alternatively, setting the probability (or quantity) of purchase q = 1 F(p), seller obtains price p(q) = F 1 (1 q). The revenue function is with Substituting p = F 1 (1 q), R(q) = q p(q) = qf 1 (1 q) q R (q) = F 1 (1 q). F (F 1 (1 q)) 1 R F(p) (q) = p = ψ(p). f(p) The seller sets the monopoly price p where marginal revenue ψ(p) is 0, i.e., p = ψ 1 (0). Mihai Manea (MIT) Bayesian Games and Auctions February 22, / 49

42 Optimal Auction and Virtual Values When facing multiple buyers, the optimal mechanism calls for the seller to set a discriminatory reserve price r = ψ 1 ( 0) for each buyer i. If x i i i < r i for every buyer i, the seller keeps the object. Otherwise, the object is allocated to the buyer generating the highest marginal revenue. The winning buyer pays p i = y i (x i ), the smallest value needed to win. Optimal auction is inefficient: with positive probability, the object is not allocated to a buyer even though it is worth 0 to the seller and has positive values for buyers. Mihai Manea (MIT) Bayesian Games and Auctions February 22, / 49

43 Optimal Auction Favors Weak Buyers Two buyers with regular cdf s F 1 and F 2 s.t. supp F 1 = supp F 2 = [0, ω] and f 1 (x) 1 F 1 (x) < f 2 (x), x [0, ω]. 1 F 2 (x) Buyer 2 is relatively disadvantaged because his value is likely to be lower. F 1 first-oder stochastically dominates F 2, i.e., F 2 (x) F 1 (x) for x [0, ω]. Check this by integrating the inequality above for x [0, z] to obtain log(1 F 1 (z)) log(1 F 2 (z)). Reserve prices r 1 and r 2 satisfy 1 F 1 (r ) 1 ψ 2 (r 2) = 0 = ψ1(r 1 ) = r 1 f 1 (r 1 ) < r 1 1 F 2(r 1 ) = ψ f 2 (r 1 2(r 1 ). ) Then ψ 2 (r 2) < ψ 2 (r 1 ) implies r 2 < r 1. Mihai Manea (MIT) Bayesian Games and Auctions February 22, / 49

44 More on Discrimination and Inefficiency When both buyers have the same value x > r 1, buyer 2 obtains the good in the optimal mechanism because 1 F1(x) 0 < ψ 1 ( x) = x f 1 (x) < x 1 F 2(x) = ψ2 (x). f 2 (x) For small ε > 0, ψ 1 (x) < ψ 2 (x ε) so buyer 2 gets the good even if x 2 = x ε when x 1 = x. Second type of inefficiency: object not allocated to the highest value buyer Mihai Manea (MIT) Bayesian Games and Auctions February 22, / 49

45 Application to Bilateral Trade Myerson and Satterthwaite (1983) seller with privately known cost C; cdf F s, density f s > 0, supp [c, c] buyer with privately known value V; cdf F b, density f b > 0, supp [v, v] c < v < c < v A direct mechanism (Q, M) specifies the probability of trade Q(c, v) and the transfer M(c, v) from the buyer to the seller for every reported profile (c, v). Is there any efficient mechanism (Q(c, v) = 1 if c < v and Q(c, v) = 0 if c > v) that is individually rational and incentive compatible? Mihai Manea (MIT) Bayesian Games and Auctions February 22, / 49

46 More General Mechanisms Useful to allow for mechanisms (Q, M s, M b ) where M s denotes the transfer to the seller and M b the transfer from the buyer. (Q, M) special case with M b = M s. Alternative question: is there an efficient mechanism (Q, M s, M b ) that is individually rational and incentive compatible, which does not run a budget deficit, i.e., c v c v (M b (c, v) M s (c, v))f b (v)f s (c)dvdc 0? If the answer is negative, then the answer to the initial question is negative. Mihai Manea (MIT) Bayesian Games and Auctions February 22, / 49

47 Revenue Equivalence q b (v) = c c Q(c, v)f s (c)dc & m b (v) = Incentive compatibility for the buyer requires As before, c c M b (c, v)f s (c)dc U b (v) q b (v)v m b (v) q b (v )v m b (v ), v [v, v]. v U b (v) = U b (v) + q b (v )dv, v which implies that m b (v) = U b (v) + f(v, Q). Similarly, U s (c) = m s (c) q s (c)c = U s (c) + c c q s(c )dc and m s (c) = U s (c) + g(c, Q). For every incentive compatible mechanism (Q, M s, M b ), c v c v (M b (c, v) M s (c, v))f b (v)f s (c)dvdc = U b (v) U s (c) + h(q). Mihai Manea (MIT) Bayesian Games and Auctions February 22, / 49

48 The Vickrey-Clarke-Groves (VCG) Mechanism Fix efficient allocation Q and consider the following payments. If v > c then set M b (c, v) = max(c, v) and M s (c, v) = min(v, c). Otherwise, M b (c, v) = M s (c, v) = 0. (Q, M s, M b ) is incentive compatible (similar argument to the second-price auction with reserve). Since U b (v) = U s (c) = 0, h(q) = = = = c v c c v c c c v c c c (M b (c, v) M s (c, v))f b (v)f s (c)dvdc v (max(c, v) min(v, c))f b (v)f s (c)dvdc c (max(c, v) + max( v, c))f b (v)f s (c)dvdc v (max(c v, v v, c c, v c))f b (v)f s (c)dvdc < 0. c Mihai Manea (MIT) Bayesian Games and Auctions February 22, / 49

49 Negative Result Suppose (Q, M s, M b ) is an efficient mechanism that is individually rational and incentive compatible. In light of the VCG mechanism, efficiency and incentive compatibility imply h(q) < 0. Individual rationality requires U b (v), U s (c) 0. Then c v c v (M b (c, v) M s (c, v))f b (v)f s (c)dvdc = U b (v) U s (c) + h(q) < 0. Every efficient, individually rational, and incentive compatible mechanism must run a budget deficit. Theorem 2 If c < v < c < v, there exists no efficient bilateral trade mechanism (Q, M b, M s ) with M b = M s that is individually rational and incentive compatible. Mihai Manea (MIT) Bayesian Games and Auctions February 22, / 49

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