Advanced Microeconomics II

Advanced Microeconomics II by Jinwoo Kim November 24, 2010 Contents II Mechanism Design 3 1 Preliminaries 3 2 Dominant Strategy Implementation 5 2.1 Gibbard-Satterthwaite Theorem........................ 6 2.2 Dominant Strategy Implementation in Quasilinear Environment...... 9 2.3 Appendix: Envelope Theorem......................... 13 3 Nash and Subgame Perfect Implementation 15 3.1 Nash Implementation.............................. 15 3.2 Subgame Perfect Implementation....................... 17 4 Bayesian Implementation 21 4.1 General Results in Quasilinear Environment................. 22 4.2 Single-Unit Auctions.............................. 27 4.2.1 Second-Price Auctions and English Auctions............. 27 4.2.2 First-Price Auctions.......................... 28 1

4.2.3 Revenue Equivalence and Optimal Auctions............. 30 4.2.4 Optimal Auction with Correlated Values............... 32 4.3 Bilateral Trade................................. 34 4.3.1 Impossibility Result........................... 34 4.3.2 Double Auction............................. 35 4.4 Dissolving Partnerships Efficiently....................... 37 5 Principal-Agent Problem: Screening 39 6 Principal-Agent Problem: Moral Hazard 42 2

Part II Mechanism Design 1 Preliminaries So far, we have fixed an economic institution/mechanism, mostly at the competitive market, and investigated whether its equilibrium gives us an efficient outcome under various circumstances. The mechanism design theory reverses this procedure, that is we here fix some social outcome we want to achieve, and ask what mechanism can implement such outcome for us. Let X denote the set of all social alternatives. As before, R : Set of all possible preferences over X; P : Set of all strict preferences. R i : Set of agent i s possible preferences. Θ i : Agent i s type space. Θ Θ 1 Θ I. For each type θ i Θ i, there is a corresponding preference in R i, denoted i (θ i ), which is represented by a utility function u i (, θ i ). Let f : Θ X denote the social choice rule (or correspondence), meaning that given type profile θ Θ, f(θ) X is the set of alternatives we want to achieve. The examples are: 1. f P E (θ) = {x X : There is no y X such that u i (y, θ i ) u i (x, θ i ), i with at least one strict inequality}, that is f P E (θ) is the set of PE alternatives. 2. f Maj (θ) = {x : #{i : u i (x, θ i ) u i (y, θ i )} I, y X}, that is f Maj (θ) is the 2 set of Condorcet winners in the pairwise majority voting. 3. f Util (θ) = {x : x arg max y X i I u i(y, θ i )}, that is f Util (θ) is the set of utilitarian welfare maximizers. 3

If f( ) is single-valued (or f(θ) is a singleton set for all θ), then it is called social choice function (SCF). Definition 1.1. f( ) is ex-post efficient (or Paretian) if for any θ Θ, we have f(θ) f P E (θ). A mechanism Γ = (S 1,, S I, g( )) is a collection of strategy sets S 1,, S I outcome function g : S 1 S I X. and Let s i (θ i ) S i denote the strategy played by agent i of type θ i and s(θ) = (s 1 (θ 1 ),, s I (θ I )) Let Eq(θ) {x X : x = g(s(θ)), where s(θ) is an equilibrium given θ}, that is the set of all equilibrium outcomes given type profile θ. Definition 1.2. A mechanism Γ implements a social choice rule f( ) if for every θ Θ, there exists an equilibrium s(θ) such that g(s(θ)) f(θ). Note: This concept of implementation is relatively weak in the following senses: (i) It only requires that some, not all, alternative belonging to f(θ) can be achieved as an equilibrium outcome; (ii) It does not rule out the possibility of equilibrium whose outcome lies outside f(θ) (so some unwanted outcome might arise in some equilibrium). Definition 1.3. A mechanism Γ fully implements a social choice rule f( ) if for every θ Θ, there exists Eq(θ) = f(θ). Suppose that f( ) is single-valued or SCF. Then, a direct revelation mechanism (DRM) is a mechanism in which S i = Θ i for all i I and g(θ) = f(θ) for all θ Θ. An interpretation of DRM is that each agent is asked to report his/her type and, given the reported type profile, the outcome is chosen following the SCF f( ). An SCF f( ) is truthfully implementable if the truth-telling is an equilibrium of the DRM. Depending on the equilibrium concept we use, there are many notions of implementation available. Among others, we will see the implementation in dominant strategy, Nash, and Bayesian Nash equilibrium in this order. 4

2 Dominant Strategy Implementation Given a mechanism Γ, s (θ) = (s 1(θ 1 ),, s I (θ I)) is a (weak) dominant strategy (DS) equilibrium in state θ if for all i, u i (g(s i (θ i ), s i ), θ i ) u i (g(s i, s i ), θ i ), s i S i, s i S i. (1) As well known, the advantage of dominant strategy equilibrium is that agents do not need to know about other agents information/strategy. If agents have strict preference, then the DS outcome must be unique for any profile θ. Proof. Consider any two DS equilibria s and s given type profile θ. Since s 1 and s 1 are both dominant for agent 1, he must be indifferent between g(s 1, s 1) and g(s 1, s 1) so g(s 1, s 1) = g(s 1, s 1) (since the preference is strict). For the same reason, agent 2 must be indifferent be between g(s 1, s 2, s {1,2} ) and g(s 1, s 2, s {1,2} ) so g(s 1, s 2, s {1,2} ) = g(s 1, s 2, s {1,2} ). Repeating this way, we obtain as desired. g(s ) = g(s 1, s 1) = g(s 1, s 2, s {1,2}) = = g(s ), Definition 2.1. The mechanism Γ implements the social choice rule f in DS if for every θ Θ, there exists a DS equilibrium s (θ) such that g(s (θ)) f(θ). An SCF f( ) is truthfully implementable in DS if s i (θ i ) = θ i (or truth-telling) is a DS equilibrium for all θ i Θ i and all i I, that is u i (f(θ i, θ i ), θ i ) u i (f(θ i, θ i ), θ i ), θ i, θ i. (2) The revelation principle below greatly simplifies our analysis since it allows us to restrict our attention to DRM rather the entire set of possible mechanisms. Proposition 2.2 (Revelation Principle for DS Implementation). Suppose that there exists a mechanism Γ that implements the SCF f( ) in DS. Then, f( ) is truthfully implementable in DS. 5

Proof. If Γ implements f( ) in DS, then there exists s ( ) = (s 1( ),, s I ( )) satisfying (1), which implies u i (g(s i (θ i ), s i(θ i )), θ i ) u i (g(s i (θ i), s i(θ i )), θ i ), θ i, θ i. Thus, since g(s ( )) = f( ), we have which is precisely the condition (2). u i (f(θ i, θ i ), θ i ) u i (f(θ i, θ i ), θ i ), θ i, θ i, 2.1 Gibbard-Satterthwaite Theorem From now, we will explore the implication of DS implementation toward proving Gibbard- Satterthwaite theorem, which says that only dictatorial social choice functions are implementable in DS. Define the lower contour set of alternative x when agent i has type θ i as: L i (s, θ i ) = {y X : u i (x, θ i ) u i (y, θ i )}. Then, the following is immediate. Lemma 2.3 (Preference Reversal). The SCF f( ) is truthfully implementable in DS if and only if for all i I, all θ i Θ i and for all pairs of types θ i, θ i Θ i, we have f(θ i, θ i ) L i (f(θ i, θ i ), θ i ) and f(θ i, θ i ) L i (f(θ i, θ i ), θ i). 6

Definition 2.4. The SCF f( ) is dictatorial if there is an agent i such that for all θ Θ, f(θ) {x X : u i (x, θ i ) u i (y, θ i ), y X}, that is f( ) always chooses one of i s top-ranked alternatives. Definition 2.5. The SCF f( ) is monotonic if the following holds for any θ : If θ is such that L i (f(θ), θ i ) L i (f(θ), θ i), i I, then f(θ ) = f(θ). Lemma 2.6. If R i = P, i I, and f( ) is truthfully implementable in DS, then f( ) is monotonic. Note: It follows from the assumption R i = P, i that f( ) is single-valued or an SCF. Proof. Consider two profiles θ and θ such that L i (f(θ), θ i ) L i (f(θ), θ i), i. Let s show that f(θ ) = f(θ). Consider two profiles (θ 1, θ 1 ) and (θ 1, θ 1 ). The truth-telling being dominant for agent 1 implies f(θ 1, θ 1 ) L 1 (f(θ), θ 1 ) L 1 (f(θ), θ 1) and f(θ) L 1 (f(θ 1, θ 1 ), θ 1). Since agent 1 s preference (under θ 1) is strict, this implies f(θ 1, θ 1 ) = f(θ). The same reasoning can be applied to profiles (θ 1, θ 2, θ {1,2} ) and (θ 1, θ 2, θ {1,2} ), which yields f(θ 1, θ 2, θ {1,2} ) = f(θ 1, θ 1 ). Repeating in this fashion, we obtain f(θ) = f(θ 1, θ 1 ) = f(θ 1, θ 2, θ {1,2} ) = = f(θ ), as desired. Theorem 2.7 (Gibbard-Satterthwaite Theorem). Suppose that (i) X is finite and X 3, (ii) R i = P, i I, (iii) for all x X, there exists θ Θ such that x = f(θ). Then, the SCF f( ) is implementable in DS if and only if it is dictatorial. Proof. It is immediate that f( ) is truthfully implementable if it is dictatorial. So we focus on showing the converse. Throughout the proof, given any type θ i of agent i and any subset A X of alternatives, we will let θi A denote another type whose preference ranks 7

the alternatives in A most highly while keeping all other rankings the same as in θ. The proof is done in a few steps. Step 1 : f( ) is ex post efficient. Proof. Suppose not. Then there is θ Θ and x X such that u i (x, θ i ) > u i (f(θ), θ i ), i (since the preferences are strict by (ii)). Also, by (iii), there is ˆθ Θ such that x = f(ˆθ). Then, consider a profile ˆθ in which ˆθi = θ {x,f(θ)} i for each i I. Note that x is top-ranked according to ˆθi for all i since it is preferred to f(θ). Thus, L i (x, ˆθ i ) L i (x, ˆθi ) and L i (f(θ), θ i ) L i (f(θ), ˆθi ), i. I. Then, the monotonicity of f( ) implies f(ˆθ) = f(ˆθ) = x and also f(ˆθ) = f(θ), a contradiction because x f(θ). Before proceeding, let us define a social welfare function F : Θ R as follows: For any pair x, y X and profile θ Θ, xf p (θ)y if and only if x = f(θ {x,y} ). We verify that the social welfare function defined above satisfies all conditions of Arrow s impossibility theorem. It is clear that F ( ) is weakly Paretian since f( ) is ex post efficient. Step 2 : F ( ) satisfies IIA property. Proof. Consider two alternative x, y X and two profiles θ, ˆθ Θ such that the ranking between x and y is the same across θ i and ˆθ i for all i, which implies that L i (x, θ {x,y} i ) = L i (x, ˆθ {x,y} i ) and L i (y, θ {x,y} {x,y} i ) = L i (y, ˆθ i ), i I. (3) Suppose now that xf p (θ)y or f(θ {x,y} ) = x. Since f( ) is monotonic, due to (3), we must have f(ˆθ {x,y} ) = x so xf p (ˆθ)y. Also, for the same reasoning, if yf p (θ)x, then we must have f(ˆθ {x,y} ) = y so yf p (ˆθ)x. It remain to show that F (θ) for any θ is indeed a preference, for which it suffices to prove the transitivity of F (θ). Step 3 : For any profile θ Θ, F (θ) is transitive. 8

Proof. Consider three alternatives x, y, z satisfying xf p (θ)y and yf p (θ)z. We first show that f(θ {x,y,z} ) = x. For this, note that L i (y, θ {x,y,z} i ) L i (y, θ {x,y} i ) and L i (z, θ {x,y,z} i ) L i (z, θ {y,z} i ), i I, which implies that by the monotonicity of f( ), if f(θ {x,y,z} ) = y, then f(θ {x,y} ) = y, contradicting xf p (θ)y while if f(θ {x,y,z} ) = z, then f(θ {y,z} ) = z, contradicting yf p (θ)z. Given this and the ex-post efficiency of f( ), we must have f(θ {x,y,z} ) = x. Also, since L i (x, θ {x,y,z} i ) L i (x, θ {x,z} i ), i I, we must have f(θ {x,z} ) = x by the monotonicity of f( ), meaning that xf p (θ)z, as desired. Thus, by the Arrow s theorem, we conclude that F ( ) is dictatorial, that is there is some agent h such that for any θ Θ xf p (θ)y x h (θ h )y. (4) Using this, we now show that f( ) is also dictatorial. For this, consider any profile θ and any alternative x f(θ). Then, by the monotonicity, we have f(θ {x,f(θ)} ) = f(θ) or f(θ)f p (θ)x, which implies f(θ) h (θ h )x due to (4). 2.2 Dominant Strategy Implementation in Quasilinear Environment Let us assume that the set of social alternatives is given as X = {x = (k, t 1,, t I ) : k K and t i R, i}, where k denotes a project choice and belongs to a (finite) set K while t i denotes the transfer to agent i. Each agent i s preference is given as u i (x, θ i ) = v i (k, θ i ) + t i. (5) 9

Let V denote the set of all possible functions v : K R. Example 2.8 (Single Item Auction). Suppose that there is one indivisible object to be allocated to one of I bidders. Then, k = (y 1,, y I ), y i {0, 1}, i and y i 1. Letting θ i be the value attached to the object by bidder i, we have v i (k, θ i ) = θ i y i. ` Let k : Θ K be a function satisfying v i (k (θ), θ i ) i I i I v i (k, θ i ), k K, that is an efficient project choice rule. Proposition 2.9. An SCF f( ) = (k ( ), t 1 ( ),, t I ( )) is truthfully implementable in DS if t i (θ) = t V i (θ) v j (k (θ), θ j ) + h i (θ i ), i I, θ Θ, (6) j i where h i ( ) is an arbitrary function of θ i. Proof. Suppose for a contradiction that there exist i, θ i, θ i and θ i such that v i (k (θ i, θ i ), θ i ) + t i (θ i, θ i ) > v i (k (θ i, θ i ), θ i ) + t i (θ i, θ i ). With the substitution of (6), this implies v j (k (θ i, θ i ), θ j ) > v j (k (θ i, θ i ), θ j ), j I j I which contradicts with the definition of k. Note: The DRM (k, t V 1,, t V I ) is so called Vickrey-Clark-Groves (VCG) mechanism. Proposition 2.10. Suppose that {v i (, θ) : θ i Θ i } = V, i. Then, an SCF f( ) = (k ( ), t 1 ( ),, t I ( )) is truthfully implementable in DS only if (6) holds. Proof. One can write t i (θ i, θ i ) = v j (k (θ i, θ i ), θ j ) + h i (θ i, θ i ). (7) j i 10

We need to show that the function h i should not depend on θ i in order for f( ) to be truthfully implementable in DS. Let us consider two type profiles (θ i, θ i ) and (θ i, θ i ). If k (θ i, θ i ) = k (θ i, θ i ), then by (7), we must have h i (θ i, θ i ) = h i (θ i, θ i ) as desired. So assume that k (θ i, θ i ) k (θ i, θ i ). For convenience, denote k k (θ i, θ i ) k k (θ i, θ i ). Suppose for a contradiction that h i (θ i, θ i ) > h(θ i, θ i ). (The case where h i (θ i, θ i ) < h(θ i, θ i ) can be dealt with analogously). We can find a type θ ϵ i such that for small ϵ > 0, j i v j (k, θ j ) if k = k v i (k, θi) ϵ = j i v j (k, θ j ) + ϵ if k = k. otherwise Thus, k (θ ϵ i, θ i ) = k and h i (θ ϵ i, θ i ) = h i (θ i, θ i ). Also, we must have v i (k, θ ϵ i) + t i (θ ϵ i, θ i ) v i (k, θ ϵ i) + t i (θ i, θ i ), which implies ϵ + h i (θ ϵ i, θ i ) h i (θ i, θ i ). Since ϵ can be arbitrarily small, this means h i (θ i, θ i ) = h i (θ ϵ i, θ i ) h i (θ i, θ i ), a contradiction. Among many VCG mechanisms, the pivotal mechanism is defined as [ ] [ ] t P i (θ) v j (k (θ), θ j ) v j (k i(θ i ), θ j ), (8) where j i k i(θ i ) arg max k K j i v j (k, θ j ). j i Apply (8) to the auction environment to obtain max t P j i θ j if θ i > max j i θ j i (θ) = 0 otherwise, which corresponds to the second-price auction that makes the bidder with the highest valuation pay the second highest valuation. We say that an SCF (k( ), t 1 ( ),, t I ( )) is (ex-post) budget-balanced if t i (θ) = 0, θ Θ. i I 11

Sometimes ex-post budget-balancing is considered as part of ex-post efficiency since it requires none of the numeraire should be wasted. As a weaker notion of budget balancedness, the SCF is is said to be ex-ante budget balanced if [ ] E t i (θ) = 0. We say that the SCF satisfies no ex-ante budget deficit if [ ] E t i (θ) 0. Proposition 2.11. Suppose that K = {0, 1} and {v i (, θ) : θ i Θ i } = V, i. Then, there is no SCF f( ) = (k ( ), t 1 ( ),, t I ( )) that is truthfully implementable in DS and budget-balanced. Proof. Suppose, to the contrary, that such a mechanism exists. Without loss of generality, we can normalize v i (0, θ i ) = 0, θ i, i. Pick θ 1, θ 1, and θ 2 such that v 1 (1, θ 1 ) + v 2 (1, θ 2 ) > 0 and v 1 (1, θ 1 ) + v 2 (1, θ 2 ) < 0. (9) Then, by applying Proposition 2.10, we have that v 1 (1, θ 1 ) + v 2 (1, θ 2 ) + h 1 (θ 2 ) + h 2 (θ 1 ) = t i (θ 1, θ 2 ) = 0 i I i I i=1,2 h 1 (θ 2 ) + h 2 ( θ 1 ) = t i ( θ 1, θ 2 ) = 0, which can be subtracted side-by-side to yield v 1 (1, θ 1 ) + h 2 (θ 1 ) h 2 ( θ 1 ) = v 2 (1, θ 2 ). Here the left-hand side is independent of θ 2 while the right-hand side is not. So this equality cannot hold with the varying values of θ 2 within the set satisfying (9). For the case of I 3, refer to Green and Laffont (1979). i=1,2 12

The Differentiable Case Assumption D. For all i I, Θ i = [ θ i, θ i ], and v i (k, θ i ) is a C 1 function for all k K. Given this assumption, we obtain a chacracterization of DS implementability similar to the Proposition 2.10: Theorem 2.12. Under Assumption D, an SCF f( ) = (k ( ), t 1 ( ),, t I ( )) is truthfully implementable in DS only if (6) holds. Proof. Note first that the VCG mechanism f( ) = (k ( ), t V 1 ( ),, t V I ( )) defined in (6) is truthfully implementable in DS. Now consider any DS implementable SCF f( ) = (k ( ), t 1 ( ),, t I ( )) and then we must have that for any given θ i, θ i arg max θ i v i (k (θ i, θ i ), θ i ) + t i (θ i, θ i ), which implies by the envelope theorem (in the Appendix of this Section) that or v i (k (θ), θ i ) + t i (θ) = v i (k ( θ i, θ i ), θ i ) + t i ( θ i, θ i ) + t i (θ) = v i (k ( θ i, θ i ), θ i ) + t i ( θ i, θ i ) + θi θi θi θi v i (k (s, θ i ), s) ds θ i v i (k (s, θ i ), s) θ i ds v i (k (θ), θ i ). Since this holds true for any DS implementable SCF and the VCG mechanism is DS implementable, t i (θ) t V i (θ) = t i ( θ i, θ i ) t V i ( θ i, θ i ), from which the desired conclusion follows because t i ( θ i, θ i ) t V i ( θ i, θ i ) is independent of θ i. Green and Laffont (1977) shows that an impossibility result similar to the Proposition 2.11 obtains in the differentiable case also. 2.3 Appendix: Envelope Theorem The envelope theorem gives us a formula about how the value of a parameterized optimization problem responds to the marginal change in the parameter. 13

Letting X be the choice set, t [0, 1] the relevant parameter, and f : X [0, 1] R, let us define V (t) = sup f(x, t) x X X (t) = {x X : f(x, t) = V (t)}. Theorem (Milgrom and Segal). Suppose that f(x, ) is absolutely continuous for all x X. Suppose also that there exists an integrable function b : [0, 1] R + such that f t (x, t) b(t) for all x X and almost all t [0, 1].Then V is absolutely continuous. Suppose, in addition, that f(x, ) is differentiable for all x X, and X (t) almost everywhere on [0, 1]. Then, for any selection x (t) X (t), V (t) = V (0) + t 0 f t (x (s), s)ds. 14

3 Nash and Subgame Perfect Implementation We have seen that not many SCF s can be implemented via dominant strategy equilibrium. This is related to the fact that only few games admit a dominant strategy equilibrium. So requiring a weaker equilibrium concept, whose existence is guaranteed in most circumstances, is one way to fix the problem. In this section, we explore how a weaker equilibrium concept, such as Nash or subgame perfect equilibrium, can expand the set of implementable outcomes. 3.1 Nash Implementation In many circumstances, agents share a great deal of knowledge among themselves while an outsider (or mechanism designer) does not. It is interesting to know how a mechanism designer can utilize such knowledge to achieve the desired outcome. Thus we here assume that all agents are informed of θ = (θ 1,, θ I ) Θ while the designer is uninformed, which is so called complete information environment. The right concept of equilibirium in this setup is Nash equilibrium (NE) so we define: Definition 3.1. A mechanism Γ = (S 1,, S I, g) (fully or strongly) implements an SCF f in Nash equilibrium if for each θ Θ, (i) there exists a Nash equilibrium s (θ) = (s 1(θ),, s I (θ)) such that g(s (θ)) = f(θ) and (ii) every Nash equilibrium results in outcome f(θ). A crucial condition for the Nash implementation is the monotonicity defined in the Definition 2.5. Theorem 3.2. If f is implementable in Nash equilibrium, then it is monotonic. Proof. Suppose that f is not monotonic. Then, there exists θ and θ such that L i (f(θ), θ i ) L i (f(θ), θ i), i but f(θ ) f(θ). Consider a NE s (θ) with g(s (θ)) = f(θ) and then it is also Nash equilibrium in state θ since g(ŝ i, s i(θ i )) L i (g(s (θ)), θ i ) L i (g(s (θ)), θ i ), ŝ i S i, i, which contradicts with f being implementable in NE. 15

The monotonicity is also sufficient for the NE impelementability when coupled with an additional property called no veto power Definition 3.3. f satisfies no veto power (NVP) if no agent i has a veto power in the sense that if there exists some x X such that then f(θ) = x. u j (x, θ j ) u j (y, θ j ), y X, j i, Theorem 3.4. If I 3 and f is monotonic and satisfies NVP, then f is implementable in NE. Proof. We aim to construct a mechanism that implements f in NE. Let S i = Θ X {0, 1, 2, 3, } and s i = (θ i, x i, m i ) S i, where θ i = (θ1, i, θi i ) is a preference profile (preferences of all agents) reported by i, x i is an alternative, and m i is a number that i chooses. Then a mechanism Γ = (S 1,, S I, g) is defined as follows: Case I: If for all i I, (θ i, x i ) = ( θ, x) for some ( θ, x) satisfying x = f( θ), then g(s 1,, s I ) = f( θ). Case II: If for all j i, (θ j, x j ) = ( θ, x) for some ( θ, x) satisfying x = f( θ), then x i if u i ( x, θ i ) u i (x i, θ i ) g(s 1,, s I ) = x otherwise. (10) Case III: For all other strategies, let g(s 1,, s I ) = x i, where i = arg max i I m i. The proof that Γ implements f in NE consists of 4 steps. Step 1 : If (θ i, x i ) = (θ, x), i satisfying x = f(θ), then (s 1,, s I ) forms a NE in θ and g(s 1,, s I ) = x. Proof. If agent i deviates to report some (θ, x ) (θ, x), then x is chosen over x only when the first condition of (10) holds, so the deviation is unprofitable. Step 2 : If (θ i x i ) = ( θ, x), i satisfying x = f( θ), and (s 1,, s I ) is a NE in θ, then x = f(θ). 16

Proof. Consider any y X such that u i ( x, θ i ) u i (y, θ i ). If there is some i such that u i ( x, θ i ) < u i (y, θ i ), then he has a profitable deviation of reporting (y, ) in state θ to have y chosen over x. So we must have u i ( x, θ i ) u i (y, θ i ) whenever u i ( x, θ i ) u i (y, θ i ), which, by monotonicity, implies x = f(θ). Step 3 : Suppose that for all j i, (θ j, x j ) = ( θ, x) satisfying x = f( θ), and (θ i, x i ) ( θ, x). If (s 1,, s I ) is a NE in state θ, then g(s 1,, s I ) = f(θ). Proof. Since any j i could deviate and get his favorite alternative, we must have u j (g(s 1,, s I ), θ j ) u j (y, θ j ), y X, j i, which, by NVP, implies g(s 1,, s I ) = f(θ). Step 4 : Suppose that (s 1,, s I ) is a NE for θ and that 2 or more agents are playing different strategies. Then, g(s 1,, s I ) = f(θ). Proof. Clearly, for all i, u i (g(s 1,, s I ), θ i ) u i (y, θ i ), y X since otherwise agent i could deviate to ensure his favorite alternative. So, by NPV, g(s 1,, s I ) = f(θ). The Step 1-4 tells us that any NE (s 1,, s I ) in state θ must result in g(s 1,, s I ) = f(θ), as desired. 3.2 Subgame Perfect Implementation We first present an example to illustrate the limitation of Nash implementation and a possibility of implementation through (i) a mechanism with multiple stages and (ii) the equilibrium concept accordingly changed to the subgame perfect equilibrium. Example 3.5. Consider an Edgeworth box economy where there are two states: C where two agents have Cobb-Douglas preference; L where two agents have Leontief preference. 17

.f(c).f(l).y.x. Here f violates the monotonicity and thus cannot be implemented in NE. Nontheless, f can be implemented as the unique subgame perfect equilibrium outcome of the following three-stage mechanism..choose x.x.challenge.a1.announce C.A2.Choose y.y..a1.agree.f(c).announce L.f(L) State C : At the last stage, agent 1 should choose x over y. Anticipating this, agent 2 should agree at the second stage. Then, the optimal reponse of agent 1 is to announce C at the first stage. 18

State L : Agent 1 should choose y over x if the last stage were reached. Given this, agent 2 should get y chosen by challenging at the stage 2 if agent 1 announces C at the first stage. So, agent 1 should announce L at the first stage in order to avoid y getting chosen. Moore and Repullo (1988) (partially) characterize the set of SCF s that are implementable in SPE in general environment. In particular, almost any SCF can be implemented in SPE in economic environments where there is at least one private good. Quasi-linear Environment Let us focus on the environment in which agents have quasilinear preferences. In this environment also, many desirable SCF s fail to be monotonic and thus are not implementable in NE. For instance, assume two agents and define f( ) = (k( ), t 1 ( ), t 2 ( )) such that k(θ) arg max k K v i (k, θ i ) and i=1,2 t i (θ i, θ i ) < t i (θ i, θ i ) if v i (k, θ i ) > v i (k, θ i), k. (11) Applied to the public good problem, (11) requires an agent to contribute more if he has a higher WTP for public goods. This SCF is not implementable in NE. To see it, given any type profile θ = (θ 1, θ 2 ), consider another profile θ = (θ 1, θ 2 ) satisfying v 1 (k, θ 1) = v 1 (k, θ 1 ) + C, k K, for some constant C > 0. The monotonicity requires f(θ ) = f(θ), which contradicts with (11). Using the subgame perfect implementation, however, virtually all SCF s are implementable in the quali-linear environment: Theorem 3.6. Suppose that there are two agents and consider an SCF f( ) = (k( ), t 1 ( ), t 2 ( )) such that the utility of each agent i from f( ) is uniformly bounded, that is, Then, f is implementable in SPE. sup v i (k(θ), θ i ) + t i (θ) < M for some M. (12) θ Θ Proof. We construct the following multi-stage game. Stage 1 (eliciting agent 1 s preference): 19

.Choose k = x and.t 1 = t x t.(x, t x t, t x t).a1..announce θ 1.A2.Challenge and.announce ϕ 1 θ 1.A1.Choose k = y and.t 1 = t y t.(y, t y t, t y + t).agree.go to Stage 2, where t is a large positive number, and 1 v 1 (x, θ 1 ) + t x > v 1 (y, θ 1 ) + t y and (13) v 1 (x, ϕ 1 ) + t x < v 1 (y, ϕ 1 ) + t y ; (14) Stage 2 (eliciting agent 2 s preference): Same as Stage 1, except that the roles of agents 1 and 2 are switched; If both agents agree on θ = (θ 1, θ 2 ), then implement (k(θ), t 1 (θ), t 2 (θ)). It is straightforward to argue that each agent has an incentive to tell the truth. For instance, if agent 1 lies, then agent 2 can challenge him with the truth ϕ 1 = θ 1, whereafter agent 1 will optimally choose (y, t y t) due to (14). This is worse for agent 1 than telling the truth while it is better for agent 2 than agreeing, if t is set sufficiently large, given the assumption (12). On the other hand, if agent 1 tells the truth, then agent 2 will not (falsely) challenge, since agent would then choose (x, t x t) due to (13), which yields a large penalty t for agent 2. 1 Note that one can always find x, y, t x, and t y satisfying (13) and (14). Otherwise we must have v 1 (x, θ 1 ) v 1 (y, θ 1 ) = v 1 (x, ϕ 1 ) v 1 (y, ϕ 1 ), x, y, meaning that two types θ 1 and ϕ 1 have the same preference. 20

4 Bayesian Implementation Assume that each agent i only knows about his own preference θ i, but not about others. Let p(θ) denote the probability with which θ = (θ 1,, θ I ) is drawn. Thus, the belief of each agent i with type θ i over others types is given by the conditional probability p(θ i θ i ). Given this setup, the Bayesian Nash equilibrium is an appropriate equilibrium notion for implementation. Definition 4.1. Given a mechanism Γ = (S 1,, S I, g( )), a strategy profile s ( ) = (s 1 ( ),, s I ( )) is a Bayesian Nash equilibrium (BNE) if i, θ i, [ E θ i ui (g(s i (θ i ), s i(θ i )), θ i ) ] [ θi Eθ i ui (g(s i, s i(θ i )), θ i ) ] θi, si S i. (15) Definition 4.2. A mechanism Γ = (S 1,, S I, g( )) implements f( ) in BNE if there exists a BNE, s ( ), such that g(s (θ)) = f(θ), θ Θ. Definition 4.3. An SCF f( ) is truthfully implementable in BNE if the direct revelation mechanism Γ = (Θ 1,, Θ I, f( )) has a BNE where everyone tells the truth, i.e. s i (θ i ) = θ i, θ i, i. Proposition 4.4 (Revelation Principle for Bayesian Implementation). Suppose that there exists a mechanism Γthat implements f( ) in BNE. Then, f( ) is truthfully implementable in BNE. Proof. Let s ( ) denote the BNE for the mechanism Γ. Then, the inequality (15) must hold for all s i S i, which means that it also holds with s i = s i (θ i) for any θ i Θ i. By substituting s i = s i (θ i) into (15) and using the fact that g(s (θ)) = f(θ), θ, we obtain E θ i [u i (f(θ i, θ i ), θ i ) θ i ] E θ i [u i (f(θ, θ i ), θ i ) θ i ], θ i S i, which means that given the direct mechanism (Θ 1,, Θ I, f), it is optimal for agent i to tell the truth if others are doing so. 21

Remark 4.5. Proposition 4.4 says that given any (possibly non-direct) mechanism Γ and its BNE s, one can find a direct mechanism that implements the same outcome via a truthful BNE. This means that whatever SCF we can implement, we must be able to implement it using a direct mechanism. In other words, if we cannot find any direct mechanism that implements a certain SCF, there is no, direct or indirect, mechanism that implements it. 4.1 General Results in Quasilinear Environment Suppose that each agent i s preference is given as in (5). We also assume that agents types are independently distributed so for any function τ : Θ R, E θ i [τ(θ i, θ i ) θ i ] = E θ i [τ(θ i, θ i )], θ i. AGV (d Aspremont and Gerard-Varet) Mechanism Here we go back to the problem of whether a mechanism can be both efficient and budget balanced. It is possible to show that such mechanism does exist if the equilibrium concept is relaxed from the dominant strategy to the Bayesian Nash equilibrium. For doing so, define [ ] ξ i (θ i ) E θ i v j (k (θ i, θ i ), θ j ), j i that is the expected utility of other agents than i when the latter s type is θ i. AGV (direct) mechanism with SCF f( ) = (k ( ), t( )) is given as t i (θ) = ξ i (θ i ) + h i (θ i ), i, θ (16) where ( ) 1 h i (θ i ) = ξ j (θ j ). I 1 j i Proposition 4.6. AGV mechanism has the truth-telling as a BNE and is budget balanced. Proof. To see the budget-balancedness, note t i (θ) = ξ i (θ i ) + i I i I i I h i (θ i ) 22

For the truth-telling part, note = ( ) 1 ξ i (θ i ) ξ j (θ j ) I 1 i I i I j i = ( ) 1 ξ i (θ i ) (I 1)ξ i (θ i ) = 0. I 1 i I i I E θ i [v i (k (θ i, θ i ), θ i ) + t i (θ i, θ i )] ] =E θ i [ j I E θ i [ j I v j (k (θ i, θ i ), θ j ) v j (k (θ i, θ i ), θ j ) ] + E θ i [h i (θ i )] + E θ i [h i (θ i )] =E θ i [v i (k (θ i, θ i ), θ i )) + t i (θ i, θ i )], where the two equalities follow from (16) and the definition of ξ i ( ) while the inequality from the efficiency of k (θ) when the type profile is θ. Bayesian Implementation with Linear Utility Suppose now that the utility is linear in that given a social alternative x = (k, t), u i (x, θ i ) = θ i v i (k) + t i. We also assume that each θ i is distributed on Θ i = [ θ i, θ i ] following a cdf F i. Given a direct mechanism with SCF f( ) = (k( ), t( )), let us define v i (θ i ) E θ i [v i (k(θ i, θ i ))] t i (θ i ) E θ i [t i (k(θ i, θ i ))]. If agent i of type θ i announces θ i, then his payoff is E θ i [θ i v i (k i (θ i, θ i )) + t i (θ i, θ i )] =θ i v i (θ i) + t i (θ i). Let U i (θ i ) θ i v i (θ i ) + t i (θ i ), that is agent i s payoff in the truthful equilibrium of the direct mechanism with f( ) = (k( ), t 1 ( ),, t I ( )). 23

Example 4.7. In the auction environment of Example 2.8, v i (k) = y i and thus v i (θ i ) = E θ i [y i (θ i, θ i )] is the expected winning probability for bidder i of type θ i. Theorem 4.8. A direct mechanism (k( ), t( )) is truthfully implementable in BNE if and only if, for all i I and θ i Θ i, ` (i) v i ( ) is nondecreasing (ii) For any given θi 0 Θ i, U i (θ i ) = U i (θi 0 ) + θi θ 0 i v i (s)ds, θ i. Proof. To prove the only if part, choose any θ i and θ i > θ i. Since θ i and θ i (weakly) prefer telling the truth to reporting θ i and θ i, respectively, we have U i (θ i ) θ i v i (θ i) + t i (θ i) = U i (θ i) + (θ i θ i) v i (θ i) (17) U i (θ i) θ i v i (θ i ) + t i (θ i ) = U i (θ i ) + (θ i θ i ) v i (θ i ), which can be rearranged to yield v i (θ i) U i(θ i) U i (θ i ) θ i θ i v i (θ i ). (18) So, v i ( ) is nondecreasing. Also, as θ i converges to θ i, (18) yields U i(θ i ) = v i (θ i ),which establishes (ii). To prove the if part, let us consider any θ i and show that θ i (weakly) prefers telling the truth to reporting any other type θ i < θ i, 2 which will hold if (17) is satisfied. To see it, use (ii) to derive U i (θ i ) U i (θ i) = θi θ i θi θ i v i (s)ds v i (θ i)ds = (θ i θ i) v i (θ i), where the inequality holds due to (i). 2 The argument for the case θ i > θ i is analogous. 24

Proposition 4.9. Consider any two mechanisms, Γ = (k, t) and Γ = (k, t ) that are truthfully implementable in BNE, and their equilibrium payoffs U i ( ) and U i( ). there exists some constant c i for each i such that U i (θ i ) = U i(θ i ) + c i, θ i. Proof. From the Proposition 4.8, we must have U i (θ i ) = U i ( θ i ) + U i(θ i ) = U i( θ i ) + θi θi θi θi v i (s)ds v i (s)ds. Letting c i U i ( θ i ) U i( θ i ), we obtain the desired result. Then, We now introduce the reservation utility, or the utility that an agent can obtain outside the mechanism. Letting U i (θ i ) denote the reservation utility for θ i, we call a mechanism (k, t) individual rational if U i (θ i ) = θ i v i (θ i ) + t i (θ i ) U i (θ i ), θ i, i, (19) that is each type θ i is weakly better off participating in the mechanism. So the set of inequalities (19) is also referred to as the participation constraint. Fixing an efficient allocation rule at the efficient rule k, one can ask if there exists a mechanism (k, t) that is truthfully implementable while being individual rational and satisfying no ex ante budget deficit. To answer this question, we define a critical type for agent i as θ C i ( θi ) arg min U i (θ i ) U i (θ i ) = arg min v i (s)ds U i (θ i ), (20) θ i Θ i θ i Θ i θi 0 which does not depend on the specific form of the transfer function t i ( ). Construct a VCG mechanism (k, t ) as follows: t i (θ) W i (θ) W (θ C i, θ i ) + U i (θ C i ), (21) 25

where W (θ) j I θ j v j (k (θ), θ j ) and W i (θ) j i θ j v j (k (θ), θ j ). Clearly, this mechanism is truthfully implementable in BNE ( a VCG mechanism is truthfully implmentable in DS). Note that the critical type obtains just its reservation payoff Ui (θi C ) = E θ i [θi C v i (k (θi C, θ i ), θ i ) + t i (θi C, θ i )] = E θ i [θi C v i (k (θi C, θ i ), θ i ) + W i (θi C, θ i ) W (θi C, θ i ) + U i (θi C )] = U i (θi C ). This mechanism is individual rational since applying (20) to the mechanism (k, t ) implies U i (θ i ) U i (θ i ) U i (θ C i ) U i (θ C i ) = 0. Moreover, this mechanism minimizes the expected budget among all mechanisms that truthfully implement k and are individual rational: For any individually rational mechanism (k, t), E[t i (θ) t i (θ)] = E[ t i (θ i ) t i (θ i )] = E[U i (θ i ) θ i v i (θ i ) (U i (θ i ) θ i v i (θ i ))] = E[U i (θ C i ) U i (θ C i )] = E[U i (θ C i ) U i (θ C i )] 0, where the third inequality follows from Proposition 4.9 and the inequality from the fact that the mechanism (k, t) is individual rational. From the discussion so far, the following result is immediate: Theorem 4.10. There exists a mechanism that truthfully implements the efficient allocation k ( ) while satisfying the individual rationality and no budget deficit if and only if [ E i I ] t i (θ) = E [ i I ] W i (θ) W (θi C, θ i ) + U i (θi C ) 0. (22) 26

4.2 Single-Unit Auctions We adopt the auction setup introduced in Example 2.8. Each bidder is assumed to have the reservation payoff equal to zero, i.e. U i (θ i ) = 0, θ i, i. 4.2.1 Second-Price Auctions and English Auctions. The auction rules are as follows: Second-Price Auction: Each bidders submits a bid. Then, the bidder who submits the highest bidder wins the object and pays the second highest bid. If there is a tie, then the winner is randomly determined among the highest bidders. English Auction(Japaneses Version): There is a price clock that continuously rises starting from zero. The bidders gradually drop out of the auction and the clock stops as soon as only one bidder remains. The remaining bidders is awarded the object and pays the current price of the clock. It is a (weakly) dominant strategy for each bidder i to submit a bid equal to his value θ i. Proof. Let b i denote the bid by bidder i. Consider bidder 1, say, and suppose that p 1 := max j 1 b j is the highest competing bid. By bidding b 1, bidder 1 will win if b 1 > p 1 and not win if b 1 < p 1. First, bidding θ 1 is always (weakly) better than bidding some b 1 < θ 1 : If p 1 < b 1 < θ 1 or b < θ 1 p 1, then bidder 1 is indifferent between θ 1 and b 1. If b 1 = p 1 < θ 1, then θ 1 is strictly better than b 1. A similar argument shows that bidding θ 1 is always better than bidding some b 1 > θ 1. Thus, it is weakly dominant to bid θ 1. By the same logic, it is weakly dominant for a bidder in English auction to drop out when the price reaches his value. As a result, the object is allocated to the highest value bidder, who pays the second highest value. So, the equilibrium allocation is efficient. It doesn t matter whether bidders know the others values or not since they play the weakly dominant strategy. 27

There are other equilibria which are less reasonable: Bidder 1, say, always bids while others bid 0 These equilibria might be exploited by the collusive bidders. Assuming that bidders are symmetric, that is F i ( ) = F ( ), the seller s revenue is E[θ (2) ] = θ θ si(i 1)(1 F (s))f(s)f (s) I 2 ds, where θ (2) is the second highest order statistic. Under the symmetry assumption, the bidder i of type θ i pays in expectation θi θ s ( F (s) I 1) ds, (23) which will prove useful for comparing the revenues between SPA and FPA. 4.2.2 First-Price Auctions The rule is the same as in the second-price auction except that the winner pays his own bid. It is very difficult to fully characterize the equilibrium bidding strategy if bidders are asymmetric. So, we assume that bidders are symmetric that is F i ( ) = F ( ), i I. We focus on BNE in which bidders use the symmetric and increasing bidding strategy, β : [ θ, θ] R +. Given that other bidders employ the bidding strategy β( ), each bidder i with value θ i must solve max(θ i b)f (β 1 (b)) I 1, b R + which, by substitution θ β 1 (b) or b = β(θ), becomes equivalent to max (θ i β(θ))f (θ) I 1. (24) θ [ θ, θ] Let U i (θ i ) denote the equilibrium payoff that results from the maximization problem above. By the envelope theorem, we obtain U i(θ i ) = [ (θi β(θ))f (θ) I 1] θ i = F (θ i ) I 1 θ=θi 28

and thus U i (θ i ) = U i ( θ) + θi θ F (s) I 1 ds = θi since the equilibrium payoff for the lowest type, U i ( θ), is zero. By definition of U i (θ i ) and (25), it must be true that (θ i β(θ i ))F (θ i ) I 1 = U i (θ i ) = which can be rearranged to yield β(θ i ) = θ i θi θ θ θi θ F (s) I 1 ds (25) F (s) I 1 ds, F (s) I 1 ds F (θ i ) I 1. (26) The equilibrium bidding function in (26) has resulted from only considering the first-order condition. To check that β(θ) is indeed a global maximum (or second-order condition) for each type θ i, substitute (26) into (24) to obtain (θ i θ)f I 1 (θ) + Differentiate this with θ to verify θ θ F (s) I 1 ds. (θ i θ) ( F (θ) I 1) 0 if θ θi, implying that θ = θ i achieves the global maximum. For example, assuming the uniform distribution F (θ i ) = θ i on [0, 1] yields θi 0 β(θ i ) = θ i si 1 ds = I 1 θ θ I 1 i. i I Note that the equilibrium allocation is efficient as in the second-price auction. The seller s revenue is equal to θ θ β(s) ( F (s) I) ds. The seller s revenue is the same across SPA and FPA since in FPA, the amount each bidder i of type θ i pays in expectation is given as F (θ i ) I 1 β(θ i ) = θ i F (θ i ) I 1 which is equal to (23). 29 θi θ F (s) I 1 ds = θi θ s ( F (s) I 1) ds,

4.2.3 Revenue Equivalence and Optimal Auctions We can use the revelation principle to focus on the direct mechanism (y, t) : Θ [0, 1] I R I such that i I y i(θ) 1, θ Θ. Applying Theorem 4.8 to the auction setup, the direct mechanism (y, t) is truthfully implementable in BNE if and only if, for all i I, (i) ȳ i ( ) is non-decreasing and (ii) U i (θ i ) = U i ( θ i ) + θi Using the fact that U i (θ i ) = θ i ȳ i (θ i ) + t i (θ i ), (27) can be rewritten as t i (θ i ) = θ i ȳ i (θ i ) θi θi θi ȳ i (s)ds, θ i. (27) ȳ i (s)ds U i ( θ i ). (28) Note that t i (θ i ) is the amount paid to the seller by the buyer i of type θ i. So given a truthfully implementable mechanism (y, t), the seller s revenue can be expressed as E [ t i (θ i )] i I = [ E θ i ȳ i (θ i ) i I θi where the equality follows from substituting (28). θi ] ȳ i (s)ds i I U i ( θ i ), (29) Theorem 4.11 (Revenue Equivalence Theorem). The seller s revenue must be the same across any two auction mechanisms (y, t) and (y, t ) satisfying: (a) y i ( ) = y i( ), i, i.e. the good is allocated to the same bidder across two auctions; (b) U i ( θ i ) = U i( θ i ), i, i.e. the lowest type obtains the same equilibrium payoff across two auctions. Proof. The proof is immediate by noting that the seller s revenue expressed in (29) only depends on (y i, U i ( θ i )) I i=1. The design of optimal auction boils down to choosing (y i, U i ( θ i )) I i=1 that maximizes the expression in (29) subject to the constraint (i). Let us temporarily ignore the constraint (i). 30

First of all, it is optimal to set U i ( θ i ) = 0 for all i I. To see how (y 1,, y I ) should be determined, rewrite (29) as follows: [ ] θi E θ i ȳ i (θ i ) ȳ i (s)ds where i I θi = i I = i I θi θi θi ( θ i ȳ i (θ i ) θi θi ( θ i 1 F i(θ i ) f i (θ i ) θi ȳ i (s)ds ) f i (θ i )dθ i ) ȳ i (θ i )f i (θ i )dθ i = E [J i (θ i )ȳ i (θ i )] i I [ ] =E J i (θ i )y i (θ), (30) i I J i (θ i ) θ i 1 F i(θ i ). f i (θ i ) The second equality follows from the integration by parts that yields ( ) θi θi θi θi ȳ i (s)ds f i (θ i )dθ i ( θi = (1 F i (θ i )) ȳ i (s)ds) = θi θi θi (1 F i (θ i ))ȳ i (θ i )dθ i θ i θi + θi θi ( θi (1 F i (θ i )) ȳ i (s)ds) dθ i The last equality follows since the function J i ( ) does not depend on θ i. From (30), the optimal allocation rule (y1,, yi ) has to be θi y i (θ) = { 1 if Ji (θ i ) > max{max j i J j (θ j ), 0} 0 otherwise. (31) Note that ȳ i (θ i ) is non decreasing as desired if J i ( ) is non-decreasing, which is the case with many cdf s. So the constraint (i) is verified. 31

If bidder are symmetric, i.e. F i ( ) = F ( ), i I and the function J( ) is increasing, then (31) becomes { 1 if θi > max{max j i θ j, J 1 (0)} y i (θ) = 0 otherwise, that is, bidder with the highest value is awarded the good if his value is at least J 1 (0) or otherwise the good is not allocated at all. A practical mechanism that raises the same revenue as the optimal mechanism above is, for instance, the first-price or second-price auction with reserve price r = J 1 (0). Consider the following example with two asymmetric bidders: F 1 (θ) = θ and F 2 (θ) = θ 2, so bidder 2 is stronger in terms of first-order stochastic dominance. Note that J 1 (θ) = θ (1 θ) > θ 1 θ2 = J 2 (θ). 2θ Thus, the optimal auction favors the weaker bidder, or handicaps the stronger bidder. 4.2.4 Optimal Auction with Correlated Values We now relax the assumption that values are independently distributed. Relaxing this assumption leads to a striking result in terms of the design of optimal auction: The seller can achieve the first-best allocation and extract all the surplus from the bidders. Here, this result is shown in a simple 2 2 model, though it is much more general. Assume that there are 2 bidders and 2 possible valuations, θ H and θ L < θ H, for each bidder. Let p ij denote the probability that bidder 1 has θ i and bidder 2 has θ j. Suppose that valuations are correlated: p LL p HH p LH p HL 0: Letting ψ := p HH p LL p HL p LH if ψ > 1 (< 1), then values are positively (negatively) correlated. 32

Consider the general auction rule: x ij = the probability that bidder 1 gets the object and t ij = bidder 1 s payment, when his value is i and bidder 2 s value is j. The full extraction means that (i) the allocation must be efficient x HH = x LL = 1 2, x HL = 1, and x LH = 0. (ii) the individual rationality for both types must be biding: ( ) θl p LL 2 t LL p LH t LH = 0 ( ) θh p HH 2 t HH + p HL (θ H t HL ) = 0, from which we have t LH = p ( LL θl p LH t HL = θ H + p HH p HL ) (32) ( ) θh 2 t HH. (33) 2 t LL Also, we need to satisfy the incentive compatibility condition: ) ) 0 = p LL ( θl 2 t LL p LH t LH p LL (θ L t HL ) + p LH ( θl 2 t HH 0 = p HH ( θh 2 t HH Substituting (32) and (33) into this yields ) (34) + p HL (θ H t HL ) p HH t LH + p HL ( θ H 2 t LL). (35) 0 p LH 2 (θ L ψθ H ) p LL (θ H θ L ) + p LH (ψ 1)t HH (36) 0 p LH 2 (θ H ψθ L ) + p HL (ψ 1)t LL. (37) Then, we are done by choosing t HH and t LL that satisfy(36) and (37). Some remarks are in order. If ψ = 1 or values are independent, then it is impossible to satisfy the above inequalities, which is what we already know from the analysis of independent types. 33

As ψ gets larger beyond 1 or values become more positively correlated, (36) and (37) can be satisfied by setting t HH and t LL lower while setting t HL and t LH relatively higher: This is effective in giving bidders an incentive to tell truthfully whey types are highly positively correlated. If ψ > 1 but ψ 1, thent HL and t LH have to be very large, which may cause some problem for a budget-constrained bidder. 4.3 Bilateral Trade Suppose that there are a seller, denoted as S, with one indivisible object and a buyer, denoted as B. The seller s value from keeping the object is denoted as θ S [ θ S, θ S ] while the buyer s value from obtaining the object is denoted as θ B [ θ B, θ B ]. Let us assume ( θ S, θ S ) ( θ B, θ B ). (38) The efficient allocation in this setup is given by (0, 1) k (θ) = (ys(θ), yb(θ)) = (1, 0) if θ B > θ S. if θ S θ B (39) Due to (38), both allocations k = (0, 1) and (1, 0) take place with positive probability in the efficient allocation. If (38) is violated, for instance, B θ θ S, then the efficient allocation can be easily achieved by letting S sell the object to B at some fixed price p [ θ S, θ B ]. 4.3.1 Impossibility Result We want a mechanism (y, t) to satisfy the following properties: 1. Individual Rationality: U S (θ S ) θ S, θ S U B (θ B ) 0, θ B. 34

2. No budget deficit: E[t S (θ) + t B (θ)] 0. Proposition 4.12. Suppose that θ B < θ S. Then, there exists no mechanism that truthfully implements the efficient allocation k ( ) in BNE while satisfying the individual rationality and no budget deficit. Proof. Let us make use of Theorem 4.10. Note first that ( ) d (U S (θ S ) U S (θ S )) = d θs ȳ dθ S dθ S(s)ds θ S = ȳs(θ S ) 1 0 S θs ( 0 ) and d (U B (θ B ) U B (θ B )) = d θb ȳ dθ B dθ B(s)ds = ȳb(θ B ) 0, B which implies θs C = θ S and θb C = Using this and (21), we obtain θb. t i (θ) = max{θ S, θ B } max{ θ S, θ B } max{θ S, θ B } + θ S i=s,b θ 0 B 0 if θ S θ B = θ B max{θ S, θ B } 0 if θ S < θ B θ S. θ S max{θ S, θ B } 0 if θ S θ S < θ B Thus, (22) is violated so the desired conclusion is reached. 4.3.2 Double Auction Consider the following trading mechanism: Let the seller submit an ask price, p S, and the buyer submit a bid price, p B. If p B p S, then then the buyer takes the good from the seller and pays p S+p B to him. Otherwise, no trade occurs. So 2 p S +p B θ 2 S if p B p S seller s net payoff = 0 otherwise and θ B p S+p B if p 2 B p S buyer s net payoff =. 0 otherwise For simplicity, we assume that both θ S and θ B are drawn from the uniform distribution. 35

Letting p i : [0, 1] R + denote i s equilibrium strategy, we conjecture that p i (θ i ) = a i + b i θ i with b i > 0. The seller s problem is given as { } ps + E [p B (θ B ) p B (θ B ) p S ] max θ S Pr[p B (θ B ) p S ] p S 2 { } ps + E [a B + b B θ B a B + b B θ B p S ] = θ S Pr[a B + b B θ B p S ] 2 { ( 1 = p S + p ) } ( S + a B + b B θ S 1 p ) S a B, 2 2 b B which, differentiated by p S, yields the first-order condition So we must have p S = a B + b B 3 + 2 3 θ S. a S = a B + b B and b S = 2 3 3. (40) The buyer s problem is given as { max θ B E [p } S(θ S ) p B p S (θ S )] + p B Pr[p B p S (θ S )] p B 2 { = θ B E [a } S + b S θ S p B a S + b S θ S ] + p B Pr[p B a S + b S θ S ] 2 { = θ B 1 ( )} ( ) pb + a S pb a S + p B, 2 2 which, differentiated by p B, yields the first-order condition So we must have a B = a S 3 Combining (40) and (41), we obtain b S p B = a S 3 + 2 3 θ B. and b B = 2 3. (41) a S = 1 4, a B = 1 12, and b S = b B = 2 3, which implies that the trade occurs if and only if 1 12 + 2 3 θ B = p B (θ B ) p S (θ S ) = 1 4 + 2 3 θ S or θ B θ S 1 4 36

Thus, the inefficiency arises, as should be expected from Proposition 4.12, but its magnitude is tolerable since the inefficient outcome (that is no trade) arises only when the value difference is less than 1/4. In fact, one can show that if the distribution is uniform, then the double auction achieves the highest social surplus among all mechanisms that satisfy the individual rationality and no budget deficit.. 4.4 Dissolving Partnerships Efficiently Consider the same setup as in the previous subsection only except that two agents now have an equal ownership of the object at the beginning. That is, one half of the object belongs to the agent 1 and the other half to the agent 2. To allocate the (entire) object to one of the agents is so called the partnership dissolution problem. The efficient way of dissolving the partnership is to allocate the object to whoever values it the highest. We show that it is possible to truthfully implement the efficient allocation in a way to satisfy the individual rationality and no budget deficit, which in contrast with the impossibility result in the bilateral setup. Theorem 4.13. Suppose that the object is initially equally owned by two agents. Suppose also that two agents have the same value distribution, F : [ θ, θ] [0, 1]. Then, there exists a mechanism that truthfully implements the efficient outcome while satisfying the individual rationality and no budget-deficit. Proof. We again make use of Theorem 4.10. To first determine the critical type, note that the efficient allocation requires yi (θ i ) = F (θ i ), θ, i = 1, 2 and thus ( ) d (U i (θ i ) U i (θ i )) = d θi F (s)ds 1 dθ i dθ i 2 θ i = F (θ i ) 1 2 0 if and only if θ i F 1 (1/2), θ 0 i which implies that the critical type for each agent is θ m F 1 (1/2). Therefore, from (21), we obtain t i (θ) = max{θ 1, θ 2 } max{θ m, θ 2 } max{θ 1, θ m } + θ m i=1,2 37