Online Appendix for Dynamic Ex Post Equilibrium, Welfare, and Optimal Trading Frequency in Double Auctions

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1 Online Appendix for Dynamic Ex Post Equilibrium, Welfare, and Optimal Trading Frequency in Double Auctions Songzi Du Haoxiang Zhu September 2013 This document supplements Du and Zhu (2013. All results provided here are based on a static special case with =, namely trading occurs only once. Since no future information is relevant, we drop the time subscript and let (s 1, s 2,..., s n be the profile of initial signals and (z 1, z 2,..., z n be the profile of initial inventories. We denote bidder i s strategy by x i (p; s i, z i. Without loss of generality we normalize the discount rate r = 1, so that the perpetuity that pays $1 per unit of time is also priced today at 1/r = 1. Thus, the utility of bidder i from acquiring q i units at price p is: U(q i, p; v i, z i = (z i + q i v i λ 2 (z i + q i 2 q i p, (1 where v i = αs i + (1 α j i s j/(n 1. 1 Uniqueness of the Ex Post Equilibrium We show that under mild conditions, ex post optimality is a sufficiently strong equilibrium selection criterion such that it implies the uniqueness of the ex post equilibrium characterized in Corollary 1 of Du and Zhu (2013. Original paper URL: Du: Simon Fraser University, Department of Economics, 8888 University Drive, Burnaby, B.C. Canada, V5A 1S6. songzid@sfu.ca. Zhu: MIT Sloan School of Management, 100 Main Street E62-623, Cambridge, MA zhuh@mit.edu. 1

2 To simplify the exposition, in this section we make the following assumption on the set of signals: for every i {1, 2,..., n}, the set of bidder i s signals is (s, s R; this means that every (s 1, s 2,..., s n (s, s n is a possible signal profile. 1 On the other hand, we make no additional assumption on the set of inventory profiles Z, except that it is common knowledge that the total inventory is Z: for every (z 1, z 2,..., z n Z, n z i = Z. Proposition 1. In addition to nα > 2, suppose that either α < 1 and n 4, or α = 1 and n 3. Let (x 1,..., x n be an arbitrary ex post equilibrium in which every x i is continuously differentiable, x i p (p; s i, z i < 0, and x i s i (p; s i, z i > 0. Then, for any s (s, s n, z Z and i {1,..., n}, at the market-clearing price p = p (s, z, x i (p; s i, z i is equal to that given by Corollary 1 of Du and Zhu (2013 Proof. See Section A.1. For any fixed s i and z i, the uniqueness of x i (p; s i, z i in Proposition 1 applies only to market-clearing prices, i.e., p = p (s i, s i, z i, z i for some s i (s, s n 1 and (z i, z i Z, since the demands at non-market-clearing prices need not satisfy any optimality condition. Proposition 1 goes beyond the uniqueness results in Vives (2011 and Rostek and Weretka (2012, who establish uniqueness for linear and symmetric equilibrium. The proof of Proposition 1 is relatively involved, but its intuition is simple. For strategies to be ex post optimal, each bidder must be able to calculate an one-dimensional sufficient statistic of other bidders signals from variables that he observes the equilibrium allocation and price. Because the equilibrium allocations {x i (p ; s i, z i } satisfy the linear constraint n x i(p ; s i, z i = 0, and because valuations {v i } are linear in the signals {s i }, it is natural to conjecture that the ex post equilibrium condition holds only if each bidder s demand is linear in his signal and the price. The main theme of the proof of Proposition 1 is to establish this linearity. The uniqueness in Proposition 1 relies on the signals of different bidders being able to vary independently, and hence the ex post optimality of equilibrium holds for every s i (s, s n 1. When the support of signal profiles is restricted, the uniqueness of equilibrium may not apply. For example, if the signals are perfectly aligned, i.e., s 1 = s 2 = = s n, then the construction in Klemperer and Meyer (1989 implies 1 Proposition 1 still holds if different bidders have different supports of signals, so long as each support of signals is an open subset of the real line. 2

3 that there is a continuum of equilibria. Intuitively, with perfectly aligned signals bidders no longer have private information; without private information the ex post optimality cannot rule out any equilibrium. However, as long as there is some small shocks to the perfect alignment of signals, the ex post optimality criterion selects a unique equilibrium. 2 Auctions of Multiple Assets In this section we extend the static special case of Du and Zhu (2013 to a simultaneous double auction of multiple assets. For example, NYSE s program trading allows simultaneous purchase or sale of more than 15 stocks; this trading method now accounts for about 25% of NYSE s trading volume. 2 In derivatives markets, the default of a member of a clearinghouse can often be resolved by auctioning the defaulting member s derivative portfolios to non-defaulting members. 3 In addition to bolstering the basic intuition of the ex post equilibrium, this section provides additional insight regarding how the complementarity and substitutability among multiple assets affect the bidding strategies. Suppose that there are m 2 distinct assets. Bidder i receives a vector of private signals s i (s i,1,..., s i,m (s, s m and values asset k (1 k m at v i,k = α k s i,k + (1 α k 1 n 1 s j,k, (2 where α k is a known constant. Moreover, bidder i has an inventory z i,k of asset k. The inventory vector z i = (z i,1,..., z i,m is bidder i s private information. As before, the total ex-ante vector of inventory, j i z i = Z, (3 is common knowledge, where Z (Z 1,..., Z m is a constant vector. Again, the joint 2 See for more details. The word program in program trading does not mean that trading is done by a computer program. 3 For example, see and management process.pdf for details of default management processes at SwapClear and Eurex Clearing. 3

4 probability distribution of ( s 1,..., s n and ( z 1,..., z n are inconsequential because we focus on ex post equilibrium. Let α (α 1,..., α m. With multiple assets, bidder i s utility after acquiring q i (q i,1,..., q i,m units of assets at the price vector p (p 1,..., p m is U( q i, p; v i, z i = m v i,k z i,k + k=1 k=1 m (v i,k p k q i,k 1 2 m m (z i,k + q i,k Λ k,l (z i,l + q i,l k=1 l=1 (4 v i z i + ( v i p q i 1 2 ( z i + q i Λ( z i + q i, where v i (v i,1,..., v i,m is the vector of bidder i s valuations and Λ {Λ k,l } is a symmetric, positive definite matrix. The matrix Λ captures the complementarity and substitutability among the assets. For example, a negative Λ k,l indicates that asset k and asset l are complements because holding one of them increases the marginal valuation of holding the other. In this double auction, each bidder i simultaneously bids on all assets by submitting a demand schedule vector x i ( p (x i,1 ( p,..., x i,m ( p. Bidder i s strategy is thus x i ( p; s i, z i. Due to the complementarity and substitutability among assets, bidder i s demand for any given asset can depend on the prices of all assets. The market-clearing price vector p (p 1,..., p m is determined such that, for each asset k {1,..., m}, x i,k ( p ; s i, z i = 0. (5 In an ex post equilibrium of this multi-asset auction, bidder i s demand schedule vector x i, which depends only on his own signal vector s i and inventory vector z i, is optimal even if he learns all other bidders signal and inventory vectors ex post. We now characterize such an ex post equilibrium in the following proposition, where we denote by Diag( a the diagonal matrix whose diagonal vector is a, and by vector whose k-th component is nα k 1 n(nα k 2. n α 1 the n(n α 2 Proposition 2. Suppose that nα k > 2 for every k {1,..., m}. In a double auction with multiple assets and interdependent values, there exists an ex post equilibrium in 4

5 which bidder i submits the demand schedule vector ( n α 2 x i ( p; s i, z i =Λ 1 Diag ( s i p Λ 1 Diag n 1 ( (n α 2(1 α + Λ 1 Diag ΛZ, (n α 1(n 1 and the equilibrium price vector is ( n α 2 Λ z i (6 n α 1 p = 1 n s i 1 n Λ Z. (7 Proof. See Section A.2. Proposition 2 reveals that a bidder s equilibrium demand for any asset can depend on his signals, prices and inventories on all other assets. This interdependence of strategies is a natural consequence of the complementarity and substitutability among multiple assets. And the equilibrium price vector (7 aggregates bidders dispersed information on all assets and is independent of any distributional assumption about the signals and inventories. 3 Heterogeneous λ s under Private Values In this section we explore the equilibrium bidding strategies if bidders have different declining rates of marginal valuations, i.e., different λ s. We focus on the case of pure private values (i.e. α = 1. Private values are common in the auction literature and are reasonable in many applications. For example, the value of a commodity can be specific to a firm s production function, just as the value of a treasury security or swap contract can be specific to an investor s hedging demand. We let λ i be the declining rate of bidder i s marginal valuation, where the profile {λ i } n is common knowledge. We can interpret the heterogenous λ i as heterogenous risk aversion or heterogeneous diminishing returns in production functions. We work with private valuations with α = 1 and v i = s i. Thus, bidder i s utility is U i (q i, p; v i, z i = v i z i + (v i pq i 1 2 λ i(z i + q i 2. (8 Each bidder submits a demand schedule x(p; v i, z i. Our objective is to find an ex 5

6 post equilibrium. Proposition 3. Suppose that n > 2. In a double auction with private values and private inventories, there exists an ex post equilibrium in which bidder i submits the demand schedule where x i (p; v i, z i = b i (v i p λ i z i, (9 b i = 2 + λ ib λ 2 i B λ i, (10 and where B n b i is the unique positive solution to the equation B = 2 + λ i B λ 2 i B λ i. (11 The equilibrium price of the double auction is p = n b i(v i λ i z i n b. (12 i Proof. See Section A.3. Note that the equilibrium demand schedules x i in (9 is independent of the total inventory Z. Therefore, Proposition 3 remains an equilibrium even if bidders face uncertainties regarding Z. This feature is reminiscent to Klemperer and Meyer (1989, who characterize supply function equilibria that are ex post optimal with respect to demand shocks. In their model, however, bidders s marginal values are common knowledge. Similarly, in a setting with a commonly known asset value, Ausubel, Cramton, Pycia, Rostek, and Weretka (2011 characterize an ex post equilibrium with uncertain supply. The final price p with heterogenous {λ i } is the weighted average of the marginal values v i λ i z i. The smaller is λ i, the larger is b i, and the more influence bidder i has on the final price. 6

7 A Appendix: Proofs A.1 Proof of Proposition 1 We fix an ex post equilibrium strategy (x 1,..., x n such that for every i, x i is continuously differentiable, x i p (p; s i, z i < 0 and x i s i (p; s i, z i > 0 for every p, (s 1,..., s n (s, s n and (z 1,..., z n Z. Fix an arbitrary s = (s 1,..., s n (s, s n and z = (z 1,..., z n Z. There exists a unique market-clearing price p (s, z 4. We will prove that there exist a δ > 0 sufficiently small and constants A, B, D and E such that x i (p; s i, z i = A s i B p + D z i + E (13 holds for every p (p (s, z δ, p (s, z+δ, s i (s i δ, s i +δ, and i {1,..., n}, Once (13 is established, the values of A, B, D and E are pinned down by the construction of Corollary 1 in Du and Zhu (2013; in particular, the values of A, B, D and E are independent of (s 1,..., s n, (z 1,..., z n, and δ. Since s = (s 1,..., s n and z = (z 1,..., z n are arbitrary, the same constants A, B, D and E in (13 apply to any s = (s 1,..., s n (s, s n, z = (z 1,..., z n Z, and p = p (s, z. This proves Proposition 1. To prove (13, we work with the inverse function of x i (p;, z i, to which we refer as s i (p;, z i. That is, for any realized allocation y i R, we have x i (p; s i (p; y i, z i, z i = y i. Because x i (p; s i, z i is strictly increasing in s i, s i (p; y i, z i is strictly increasing in y i. Throughout the proof, we will denote bidder i s realized allocation by y i and his demand schedule by x i ( ;,. With an abuse of notation, we denote x i p (p; y i, z i x i p (p; s i(p; y i, z i, z i. Fix s = (s 1,..., s n (s, s n and z = (z 1,..., z n Z. Let p = p (s, z and ȳ i = x i (p (s; s i, z i. By continuity, there exists some δ > 0 such that, for any i and 4 Because x i (p; s i, z i is monotone in p, we know that if a market-clearing price exists, it is unique. To show that a market-clearing price exists, for the sake of contradiction suppose that it does not exist at some realization of signals and inventories. Then, by the intermediate value theorem, either n x i(p; s i, z i < 0 for all p, or n x i(p; s i, z i > 0 for all p. Without loss of generality, we suppose that n x i(p; s i, z i < 0 for all p. By the rule of the auction, each bidder gets q i = 0 (no trade. There exists a bidder i such that j i x j(p; s j, z j < 0 for all p (otherwise, we would have n j i x j(p; s j, z j 0 when p is sufficiently negative. But for sufficiently negative p, bidder i is strictly better off by taking the positive quantity j i x j(p; s j, z j at p than taking zero quantity. This is a contradiction to the ex post optimality of (x 1,..., x n. 7

8 any (p, y i ( p δ, p + δ (ȳ i δ, ȳ i + δ, there exists some s i (s, s such that x i (p; s i, z i = y i. In other words, every price and allocation pair in ( p δ, p + δ (ȳ i δ, ȳ i + δ is realizable given some signal. We will prove that there exist constants A 0, B, D and E such that s i (p; y i, z i = Ay i + Bp + Dz i + E (14 for every (p, y i ( p δ/n, p + δ/n (ȳ i δ/n, ȳ i + δ/n, i {1,..., n}. Clearly, this implies (13. We now proceed to prove (14. There are two cases. In Case 1, α < 1 and n 4. In Case 2, α = 1 and n 3. Since z = (z 1,..., z n Z is fixed in the rest of the proof, we omit the dependence on z i in s i (p; y i, z i and x i p (p; y i, z i to simplify notations. A.1.1 Case 1: α < 1 and n 4 The proof for Case 1 consists of two steps. Step 1 of Case 1: Lemma 1 and Lemma 2 below imply equation (14. Lemma 1. There exist functions A(p, {B i (p} such that s i (p; y i = A(py i + B i (p, (15 holds for every p ( p δ, p + δ and every y i (ȳ i δ/n, ȳ i + δ/n, 1 i n. Proof. This lemma is proved in Step 2 of Case 1. For this lemma we need the condition that n 4; in the rest of the proof n 3 suffices. Lemma 2. Suppose that l 2 and for every i {1,..., l}, Y i is an open subset of R, P is an arbitrary set, and f i (p; y i is a differentiable function of y i Y i for every p P. Moreover, suppose that l f i (p; y i = f l+1 (p; l y i, (16 for every p P and (y 1,..., y l l Y i. Then there exist functions G(p and {H i (p} such that f i (p; y i = G(py i + H i (p 8

9 holds for every i {1,..., l}, p P and y i Y i. Proof. We differentiate (16 with respect to y i and to y j, where i, j {1, 2,..., l}, and obtain f i y i (p; y i = f l+1 y i ( p; l y j = f j y j (p; y j for any y i Y i and y j Y j. Because (y 1,..., y l are arbitrary, the partial derivatives above cannot depend on any particular y i. Thus, there exists some function G(p such that f i y i (p; y i = G(p for all y i. Lemma 2 then follows. In Step 1 of the proof of Case 1 of Proposition 1, we show that Lemma 1 and Lemma 2 imply equation (14. Recall from Section of Du and Zhu (2013 that β = (1 α n 1, (17 and rewrite bidder i s ex post first-order condition as: ( y i + α s i (p; y i + β ( s j (p; y j p λ(z i + y i j i j i x j p (p; y j = 0, (18 where y n = n 1 y j, p ( p δ, p + δ and y j (ȳ j δ/n, ȳ j + δ/n. 5 Our strategy is to repeatedly apply Lemma 1 and Lemma 2 to (18 in order to arrive at (14. First, we plug the functional form of Lemma 1 into (18. Without loss of generality, we let i = n and rewrite (18 as n 1 x j p (p; y j }{{} left-hand side of (16 = y n α(a(py n + B n (p + β n 1 (A(py. j + B j (p p λ(z n + y n }{{} right-hand side of (16 Applying Lemma 2 to the above equation, we see that there exist functions G(p and {H j (p} such that x j p (p; y j = G(py j + H j (p, (19 5 We restrict y j to (ȳ j δ/n, ȳ j + δ/n so that y n = n 1 y j (ȳ n δ, ȳ n + δ, and as a result s(p; y n and xn y n (p; y n are well-defined. 9

10 for j {1,..., n 1}. Note that we have used the condition n 3 when applying Lemma 1. By the same argument, we apply Lemma 2 to (18 for i = 1, and conclude that (19 holds for j = n as well. Using (15 and (19, we rewrite bidder i s ex post first-order condition as: ( ( ( (α β s i (p; y i + β B j (p p λ(z i + y i G(p( y i H j (p y i = 0. j i (20 Solving for s i (p; y i in terms of p and y i from equation (20, we see that for the solution to be consistent with (15, we must have G(p = ( 0. Otherwise, i.e. if G(p 0, then (20 implies that s i (p; y i contains the term y i / G(p( y i j i H j(p, contradicting the linear form of Lemma 1. Inverting (15, we see that x i (p; s i = (s i B i (p/a(p. Therefore, for x i p (p; s i to be independent of s i (i.e., G(p = 0, A(p must be a constant function, i.e. A(p = A for some constant A R. This implies that by the definition of H i (p in (19. Given G(p = 0 and A(p = A, (20 can be rewritten as H i (p = B i(p A, (21 ( y i (α β s i (p; y i + β B j (p p λ(z i + y i j i H j(p = 0. (22 For (22 to be consistent with s i (p; y i = Ay i + B i (p, we must have that H j (p = H j for some constants H j, j {1,..., n}, and that 1 j i H = j 1 j i H j, for all i i, which implies that for all i, H i H for some constant H. By (21, this means that B i (p = Bp + F i, where B = HA, and {F i } are some constants. Finally, (22 implies that F i = Dz i + E for some constants D and E. Hence, we have shown that Lemma 1 implies (14. This completes Step 1 of the proof of Case 1 of Proposition 1. In Step 2 below, we prove Lemma 1. 10

11 Step 2 of Case 1: Proof of Lemma 1. Bidder n s ex post first order condition can be written as: n 1 x j p (p; y j = y n α s n (p; y n + β n 1 s j(p; y j p λ(z n + y n, (23 where y n = n 1 y j. Differentiating (23 with respect to y i, i {1,..., n 1}, gives: y i where ( xi p (p; y i = Γ(y 1,..., y n 1 + y n ( α sn y n (p; y n + β s 1 y 1 (p; y 1 + λ, (24 Γ(y 1,..., y n 1 2 n 1 Γ(y 1,..., y n 1 = α s n (p; y n + β s j (p; y j p λ(z n + y n. (25 Solving for Γ(y 1,..., y n 1 in (24, we get for some function ρ i, i {1, 2,..., n 1}. n 1 Γ(y 1,..., y n 1 = ρ i (y i, y j (26 We let ρ i,1 be the partial derivative of ρ i with respect to its first argument, and let ρ i,2 be the partial derivative of ρ i with respect to its second argument. For each pair of distinct i, k {1,..., n 1}, differentiating (26 with respect to y i and y k, we have dγ(y 1,..., y n 1 dy i = ρ i,1 + ρ i,2 = ρ k,2, dγ(y 1,..., y n 1 dy k = ρ k,1 + ρ k,2 = ρ i,2, which imply that for all i k {1,..., n 1}, ρ i,1 + ρ k,1 = 0. (27 Clearly, (27 together with n 4 imply that ρ i,1 = ρ i,1, i.e., ρ i,1 = 0 for all 11

12 i {1,..., n 1}. That is, each ρ i is only a function of its second argument: ( n 1 n 1 ρ i (y i, y j = ρ i y j. (28 Then, using (25, (26 and (28 for i = 1, we have ( n 1 n 1 β s j (p; y j = ρ 1 y j + p + λy n α s n (p; y n. (29 Applying Lemma 2 to (29 (recall that y n = n 1 y j, we conclude that, for all j {1,..., n 1}, s j (p; y j = A(py j + B j (p. (30 Finally, we repeat this argument to bidder 1 s ex post first-order condition and conclude that (30 holds for j = n as well. This concludes the proof of Lemma 1. A.1.2 Case 2: α = 1 and n 3 We now prove Case 2 of Proposition 1. Bidder n s ex post first order condition in this case is: n 1 x j p (p; y j = y n s n (p ; y n p λ(z n + y n, (31 for every p ( p δ, p + δ and (y 1,..., y n 1 n 1 (ȳ j δ/n, ȳ j + δ/n, and where y n = n 1 y j. Applying Lemma 2 to (31 gives: x j p (p; y j = G(py j + H j (p, (32 for j {1,..., n 1}. Applying Lemma 2 to the ex post first-order condition of bidder 1 shows that (32 holds for j = n as well. Substituting (32 back into the first-order condition (31, we obtain: ( s i (p; y i p λ(z i + y i ( G(p( y i j i H j (p y i = 0, 12

13 which can be rewritten as: x i p (p; y i = G(py i + H i (p = y i s i (p; y i p λ(z i + y i + H j (p. (33 We claim that G(p = 0. Suppose for contradiction that G(p 0. Then matching the coefficient of y i in (33, we must have s i (p; y i = λy i + B i (p for some function B i (p. But this implies that x i p (p; y i = B i(p/λ, which is independent of y i. This implies G(p = 0, a contradiction. Thus, G(p = 0. Then, (33 implies that s i (p; y i p λz i = A i (py i for some function A i (p. And since x i (p; y p i is independent of y i, A i (p must be a constant function, i.e., s i (p; y i p λz i = A i y i for some A i R. Substitute this back to (33 gives: which implies x i p (p; y i = 1 A i = 1 A i λ 1 A j, 1 A i λ 1 A j λ = 1 1, for all i j, A j A i which is only possible if A i = A j A R for all i j. Thus, s i (p; y i p λz i = Ay i, which concludes the proof of this case. A.2 Proof of Proposition 2 We define β (β 1,..., β m where, for each k {1,..., m}, β k = 1 α k n 1. We conjecture an ex post equilibrium in which bidder i uses the demand schedule: x i ( p; s i, z i = B( s i p + D z i + E Z, (34 where B, D and E are m-by-m matrices. Furthermore, we assume that B is symmetric and invertible. Fix a profile of signals ( s 1,..., s n and inventories ( z 1,..., z n. Bidder i s ex post first order condition at the market-clearing prices p is: 13

14 x i ( p ; s i, z i +(n 1B ( Diag( α s i + Diag( β j i s j p Λ( z i + x i ( p ; s i, z i = 0. The demand schedules in (34 yield the market-clearing price vector of p = 1 ( 1 s i + B 1 n n D + E Z, (35 where I is the identity matrix. Substituting this expressions of p into (35 and rearranging, we have: (I + (n 1BΛ x i ( p ; s i, z i ( =(n 1B (Diag( α β( s i p Diag(nβB 1n 1 D + E Z Λ z i. Matching coefficients with the conjecture in (34, we obtain (where n α 2 n α 1 denotes the vector whose k-th component is nα k 2 nα k 1, etc.: ( n α 2 B = Λ 1 Diag, n 1 ( n α 2 D = Λ 1 Diag n α 1 E = Λ 1 Diag A.3 Proof of Proposition 3 Λ, ( (n α 2(1 α (n α 1(n 1 Λ. We conjecture that bidder i submits the demand schedule x i (p; v i, z i = b i (v i p λ i z i, where b i > 0. Then, bidder i s ex post first order condition is: ( x i (p ; v i, z i + (v i p λ i (z i + x i (p ; v i, z i b j = 0. (36 j i 14

15 Solving for x i (p ; v i, z i in (36 and matching coefficients with x i (p ; v i, z i = b i (v i p λ i z i, we obtain b i = j i b j 1 + λ i j i b j b i + (λ i b i 1(B b i = 0, (37 where we use the fact that j i b j = B b i. Solving for b i in (37, we get (10. (The quadratic equation has two solutions, but only the smaller one is the correct solution. 6 Thus, B must solve the equation (11. To show that (11 has a unique positive solution B, we rationalize the numerators of (11 and rewrite it as 0 = B ( 1 + Under the conjecture that B > 0, we have Bλ i + λ 2 i B = f(b Bλ i + λ 2 i B It is straightforward to see that f (B < 0, f(0 = n 2 B. Thus, (11 has a unique positive solution B. 1 > 0, and f(b 1 as References Ausubel, L. M., P. Cramton, M. Pycia, M. Rostek, and M. Weretka (2011: Demand Reduction, Inefficiency and Revenues in Multi-Unit Auctions, Working paper. Du, S. and H. Zhu (2013: Dynamic Ex Post Equilibrium, Welfare, and Optimal Trading Frequency in Double Auctions, Working paper. Klemperer, P. D. and M. A. Meyer (1989: Supply Function Equilibria in Oligopoly under Uncertainty, Econometrica, 57, Rostek, M. and M. Weretka (2012: Price Inference in Small Markets, Econometrica, 80, Vives, X. (2011: Strategic Supply Function Competition with Private Information, Econometrica, 79, If b i = 2+λiB+ λ 2 i B2 +4 2λ i, then we would have b i > B, which contradicts the definition of B. 15