Characterization of equilibrium in pay-as-bid auctions for multiple units

Transcription

1 Economic Theory (2006) 29: DOI /s y RESEARCH ARTICLE Indranil Chakraborty Characterization of equilibrium in pay-as-bid auctions for multiple units Received: 26 April 2004 / Accepted: 21 June 2005 / Published online: 5 October 2005 Springer-Verlag 2005 Abstract Equilibrium bidding strategies under most multi-unit auction rules cannot be obtained as closed form expressions. Research in multi-unit auctions has, therefore, depended on implicit characterization of equilibrium strategies using the first-order conditions of the bidders expected payoff maximization problem. In this paper we consider the pay-as-bid auction with diminishing marginal values for two units and show that any symmetric equilibrium in continuous strategies has the necessary properties to allow such a characterization. Moreover, any increasing solution to the system of differential equations that is used to characterize the equilibrium strategies describes an equilibrium strategy. Keywords Multi-unit auctions Pay-as-bid rule JEL Classification Numbers D44 1 Introduction A large number of auctions involve multiple units of the same item. Some of them like the Treasury bill auctions and the procurement auctions together amount to trillions of dollars annually making studies into the working of such auctions a matters of practical interest. At the same time, it has been shown that results developed for single-unit auctions do not generally extend to the multi-unit set-up (Engelbrecht- Wiggans and Kahn, 1998a,b; Ausubel and Cramton, 1998). This has resulted in a recent interest in developing a separate theory for multi-unit auctions. Generous suggestions and comments from Richard Engelbrecht-Wiggans, Charles M. Kahn, and an anonymous referee are gratefully acknowledged. I. Chakraborty Department of Economics, National University of Singapore, 1 Arts Link, AS-2#05-36, Singapore, Singapore indro@nus.edu.sg

2 198 I. Chakraborty The exploration of multi-unit auctions has been impeded by the mathematical complexity involved in analyzing even the basic models. Most of the standard auctions do not have a closed form expression for the equilibrium bidding strategies making the analysis of the bidding behavior difficult. Many authors have, therefore, bypassed explicit calculations of the equilibrium bidding strategies, and resorted to less explicit characterizations of the equilibrium bidding behavior. For instance, Noussair (1995), Katzman (1995, 1999), Engelbrecht-Wiggans and Kahn (1998a,b) have considered auctions where two indivisible units of an object are on sale, and each bidder has diminishing marginal values for the successive units. Equilibrium bidding strategies in such auctions do not have closed form expressions under most auction rules. They have therefore analyzed the auctions with the equilibrium strategies implicitly characterized by (a system of) differential equations based on the first-order conditions of bidders expected payoff maximization problem. Engelbrecht-Wiggans and Kahn (1998a) even reduced the equations for a pay-as-bid auction 1 to a characterization on just the boundary of the bidders (two-dimensional) set of (marginal) values to make it numerically solvable. Under appropriate conditions such a reduced characterization is sufficient for extending the solution of the system to the interior of the set of values, thereby making the relatively simpler characterization an important tool for analyzing the auction. Chakraborty (2004) showed that this approach can be useful even for characterizing auctions where bidders have increasing marginal values. Such an approach to analyzing auctions through the reduced characterization, however, leaves a crucial gap. First, the assumptions under which the reduced characterization is valid, restricts the set of equilibria that can be considered making it less likely that such an equilibrium will in fact exist. Thus, the robustness of an analysis based on such a characterization will increase significantly if the characterization is shown to be valid in the absence of some of these assumptions. Second, a numerically computed solution (like Engelbrecht-Wiggans and Kahn 1998a) and analysis may not mean much if the first-order condition based characterization yields multiple solutions some of which do not describe the equilibrium. Imagine the worst scenario where the equilibrium of interest may not even exist. If a solution to the equations exists in that situation it describes anything but an equilibrium. An analysis based on a numerical description or an implicit characterization of this solution will be misleading. In short, the literature begs for results that examine the extent of the necessity and sufficiency of such characterizations as descriptions of some broad range of equilibria. The objective of this paper is to do that in the context of the pay-as-bid auction rule. The model of multi-unit auction studied in this paper is that of Engelbrecht- Wiggans and Kahn (1998a) (see also Katzman 1995) where two identical and indivisible units of an object are sold to a finite number of symmetric bidders, each with diminishing marginal values for the successive units. This approach to multi-unit auction differs from that taken by Ausubel and Cramton (1998) and Back and Zender (1993), among others. They consider auctions for a perfectly divisible object where bidders submit continuous demand/bid schedules for different shares of the object. While their work provide valuable insights into the bidding behavior and revenue ranking of alternative rules for multi-unit auctions, their approach avoids the multi-dimensionality problem of multi-unit auctions by considering 1 Also known as the pay-your-bid and discriminatory-price auctions.

3 Characterization of equilibrium in pay-as-bid auctions for multiple units 199 only single-dimensional bidder types. The resulting simplification of the strategic environment makes their approach complementary rather than a substitute. This paper is organized as follows: Section 2 provides the details of the model of multi-unit auction that we consider, and a representative bidder s problem. In Section 3 we prove the properties of equilibrium strategies that are vital for a reduced characterization of the equilibrium strategies of a pay-as-bid auction on the boundary of the set of bidder values. In Section 4 we show that any increasing solution of the system of differential equations is also an equilibrium of the multi-unit auction. Finally, we conclude in Section 5. The proofs of all results are gathered in Section 6. 2 The model Two identical units of an object are to be auctioned to n (symmetric) bidders through a sealed bid auction without a reserve. We follow Engelbrecht-Wiggans and Kahn (1998a) and assume that bidder i has privately known diminishing (marginal) values V 1i and V 2i for the successive units. From the perspective of the other bidders, V 1i and V 2i are distributed according to a continuous distribution F(v 1,v 2 ), where v 1 and v 2 are realizations of the random variables V 1i and V 2i, on the support V = [v :0 v 2 v 1 1]. The corresponding density function is assumed to exist and is denoted by f(v 1,v 2 ). The values are identically and independently distributed across the bidders, and the distributional information is common knowledge. Under the pay-as-bid rule each bidder submits sealed bids b 1 and b 2 for the two units. The highest two bids are awarded the units at prices equal to the respective bids. Thus, a bidder may win one or two units, and pay a price equal to her highest or the sum of her bids, depending on whether one or both her bids are among the highest two bids in the auction. Since a bidder is awarded the first unit based on the value of the higher of her two bids, the higher bid is effectively her bid for the first unit. Hence we can assume, without loss of generality, that b 1 b 2. For most parts we assume an equilibrium environment, and ask what properties must the bidding strategies have in that environment. We look at the auction from the perspective of a representative bidder i, and denote by b j (v 1,v 2 ) the bidding strategy of this bidder for the jth unit (which may or may not depend on the value of the other unit). From this bidder s point of view the rival bids are random variables that depend on her opponents bidding strategies and their value distributions. Let H j (b) denote the probability that the jth highest bid of the opponent bidders is less than or equal to b. Clearly, the probability that the bidder will win her first unit with a bid b 1 is H 2 (b 1 ) and that the bidder will win the second unit with a bid b 2 is H 1 (b 2 ). Thus, if bidder i submits bids {b 1,b 2 } her expected payoff is π(v 1,v 2,b 1,b 2 ) = (v 1 b 1 )H 2 (b 1 ) + (v 2 b 2 )H 1 (b 2 ). Given the bid distributions of the other bidders, bidder i maximizes her expected payoff π(v 1,v 2,b 1,b 2 ) subject to the bidding constraint b 1 b 2. If each bidder s strategy is consistent with such a maximization, the strategies constitute a Bayes Nash equilibrium that we simply refer to as the equilibrium. In this paper we consider only symmetric equilibria (i.e., equilibria where all bidders use the

4 200 I. Chakraborty same strategy) in continuous bidding strategies and refer to only these equilibria throughout this paper. 2.1 Types of possible bids Given the bidding strategies of the other bidders, bidder i solves the following maximization problem: max b 1,b 2 b 1 b 2 π(v 1,v 2,b 1,b 2 ) (1) The solution b 1 (v 1,v 2 ) and b 2 (v 1,v 2 ) of this constrained maximization may or may not coincide with the solution of the corresponding unconstrained maximization. If the solution to the unconstrained maximization max π(v 1,v 2,b 1,b 2 ) (2) b 1,b 2 gives rise to solution {b1,b 2 }, where b 1 b 2, then this is also a solution to the maximization equation (1). However, if the solution satisfies b1 <b 2, then, the bidding constraint in equation (1) is binding, and the solution to equation (1) may either satisfy b1 >b 2 or b 1 = b 2.2 If the solution satisfies b1 = b 2 then it is also a solution to the problem max π(v 1,v 2,b,b). (3) b Since we have no reason to assume concavity or single-peakedness of the objective function in the bids, there are possibly three types of equilibrium bids that could result from the maximization problem: Separated bids. Bids in equilibrium will be called separated if they are also a solution to the unconstrained problem equation (2). Pooled bids. Equilibrium bids will be called pooled if the constraint in problem euation (1) is binding and can be obtained by solving problem equation (3). Pseudo-separated bids. Bids in equilibrium will be called pseudo-separated if they are neither separated nor pooled. In other words, pseudo-separated bids are equilibrium bids for which the constraint in problem equation (1) binds, and that satisfy the inequality b 1 >b 2. Obviously, the separated equilibrium bids can be obtained as solutions to the maximization problems max b 1 (v 1 b 1 )H 2 (b 1 ) and max b 2 (v 2 b 2 )H 1 (b 2 ). (4) Thus, separated bid for unit j is a function of v j alone. 2 b 1 >b 2 may result from the functions (v i b i )H 3 i having multiple local peaks that we do not rule out yet. Single-peakedness of the objective function is assumed under the standard approach.

5 Characterization of equilibrium in pay-as-bid auctions for multiple units Bid distributions Let G j (b) be the bidder distribution of bids for the jth unit that results from the bidding strategies of each of the other bidders. Then from the point of view of a bidder the probability that the highest and the second-highest bids her opponents make is less than b is given by the following: H 1 (b) = [G 1 (b)] n 1 H 2 (b) = [G 1 (b)] n 1 + N[G 1 (b)] n 2 [G 2 (b) G 1 (b)]. 3 Structure and characterization of equilibrium strategy Single unit auctions often have a relatively simple structure in the equilibrium bidding strategies (Vickrey 1961; Milgrom and Weber 1982). A bidder s equilibrium strategy is obtained through a maximization of her expected payoff over a single dimension which greatly contributes to the simplicity. In multi-unit auctions, on the other hand, not only is the maximization problem two dimensional, there is also the presence of a bidding constraint which greatly increases the complexity of the problem. However, some basic properties of equilibrium strategies hold in the multi-unit setting making it possible to describe the general structure of the equilibrium strategies. It can be shown that an equilibrium bidding strategy is always nondecreasing, 3 and that a bidder with a positive value for a unit always bids a positive amount (but less than the value) thereby guaranteeing an interior solution to the bidder s maximization problem. Moreover, the distribution of a bidder s equilibrium bids is always non-atomic, i.e., no particular bid is made with a positive probability. 4 Observe that if the pseudo-separated bids are not made in equilibrium then a bidder needs to solve only the unconstrained maximizations equations (3) and (4) for the best response bids. We start by ruling out the possibility of having pseudo-separated bids in equilibrium thereby restricting our attention to the simpler maximization problems. Proposition 1 Pseudo-separated bids are not made in equilibrium. That only pooled and separated bids can be made in equilibrium also makes the geometric structure of the bids more manageable. Note that the domain V of bidder values is the triangular area defined by the three points {0, 0}, {1, 0} and {1, 1} (see Fig. 1). Clearly, the bidder bids zero on both units in equilibrium at value {0, 0}. Now fix the value for the second unit at 0 and increase v 1. At a value {v 1, 0} with v 1 > 0, the bidder bids zero on the second unit but bids a positive amount on the first unit. Hence the bids are separated. Now as v 1 increases with v 2 fixed at zero the bid for the second unit will still be zero but the bid for the first unit will increase until the point {1, 0}. Thus the bids are always separated along this line. Starting with a separated equilibrium bid at {v 1, 0} (v 1 > 0), and increasing v 2, the bid for the first unit remains unchanged 5 while that for the second unit increases. 3 In fact, one can show that separating bids are strictly increasing. 4 The details can be obtained from the author. 5 This follows from the continuity of equilibrium strategies.

6 202 I. Chakraborty Fig. 1 Regions for pooled and separating bids This happens as long as the constraint in the bidder s maximization problem is not binding. If the constraint becomes binding at some point, then it is straightforward to see that for all value pairs right above this point the bidding constraint binds, and the bids become pooled (since pseudo-separated bids cannot happen). Finally, consider the bid for the second unit when the value of the first unit is fixed at 1, and the value for the second unit is increased from 0 to 1. The second bid increases from zero while the first bid remains fixed. Now one of two possible situations may arise: The bids can remain separated throughout this line. Alternatively, the bidding constraint can start to bind beyond some point C on this line (see Fig. 1). This structure is the first step in obtaining a reduced characterization of the strategies on the boundary of V. The characterization is ultimately made possible by a valid first-order condition representation of the equilibrium bids, and piecewise linearity of the iso-bid lines both of which are guaranteed by the differentiability of the bidder s objective function if bidding strategies are separated all along the vertical boundary of V. If the differentiability of the objective function and separated bids along the vertical edge are assumed (Engelbrecht-Wiggans and Kahn, 1998a), the reduced description of the bidding strategy along the boundary of the domain becomes sufficient for describing the bidding strategies in the interior. We show in the following Proposition that in fact these two features hold as equilibrium properties (thus making the assumptions unnecessary). Proposition 2 In equilibrium a bidder s expected payoff function is differentiable at all relevant bids. Moreover, all bids can be made as separated bids. Proposition 2 guarantees the first-order condition representations of the solutions to the maximization problems (equations (2) and (3)). The proof of Proposition 2 also implies that the bids at {1, 1} are equal. Let us denote this highest bid by b. The first-order conditions for problems (equations (2) and (3)) are given by the equations:

7 Characterization of equilibrium in pay-as-bid auctions for multiple units 203 (v 1 b 1 )h 2 (b 1 ) = H 2 (b 1 ) (5) when the bids are separated, and by (v 2 b 2 )h 1 (b 2 ) = H 1 (b 2 ) (6) (v 1 b)h 2 (b) + (v 2 b)h 1 (b) = H 1 (b) + H 2 (b) (7) when the bids are pooled, where h i is the derivative of H i. (Observe that Proposition 2 implies that C ={1, 1} in Fig. 1.) In order to get the reduced characterization from here let S denote the set of values for which the bids are separated and P denote the set of values for which the bids are pooled. Let S and P denote the closures of the two regions. Then for values in S P, equilibrium bids can be computed either from equations (5) and (6), or from equation (7). Again, define I = [{v 1,v 2 } : v 1 = v 2 ]. Note that if a bidder bids b 1 (v1,v 2 ) = b 2 (v1,v 2 ) for a certain pair {v 1,v 2 }, then b 1(v 1,v 2 ) = b 2 (v 1,v 2 ) for all v 1 and v 2 satisfying v 1 v1 and v 2 v2.6 Also, due to the continuity of the bidding strategies, if at {v1,v 2 } the bids are separated and the bid for the two units are b 1 and b 2, then for all values {v1,v 2} with v 2 <v2 the first bid is b 1. Similarly, at {v 1,v2 } with v 1 >v1 the second bid is b 2. An examination of the equation (7) shows that if {v 1,v 2 } S P then there exists a { v, v} P I such that the bids made on {v 1,v 2 } and { v, v} are identical. Bids at both pairs are pooled and are equal to β, say, and for 0 <α<1, the bids at {αv 1 + (1 α) v, αv 2 + (1 α) v} are pooled and are equal to β. This means that a description of separated bids for values {v 1, 0} where v 1 [0, 1] and {1,v 2 } where v 2 [0, 1] and of pooled bids on I is sufficient to describe the equilibrium bids at other values. Moreover, the bids b i (v, v) must be strictly increasing in v, else the bidders will be making a particular bid with a positive probability, which is impossible in equilibrium. 7 Now we can follow Engelbrecht-Wiggans and Kahn (1998a) to define the following inverse functions (strict monotonicity guarantees the existence of the inverses) v 1 (b 1 ) = the inverse of the function b 1 (v 1, 0), v 2 (b 2 ) = the inverse of the function b 2 (1,v 2 ) and v(b) = the inverse of the function b 1 (v, v) defined for v I. The above functions map the possible bid combinations to values on the boundary of V, and because of the above discussion, They are sufficient to describe the equilibrium bidding strategies for all values. Thus, we have a complete description 8 of the system on the boundary of V : 6 A straightforward application of monotonicity of bidding strategies. 7 This construction is a slight generalization of Engelbrecht-Wiggans and Kahn (1998a,b) in that we do not assume that equilibrium bids are necessarily pooled everywhere on I. Note that for Figure 1 to show this possibility the line describing S P must touch I in at least one point that is not an endpoint. The reduced characterization is valid regardless of the nature of bids on I. 8 Similar characterization is possible for asymmetric equilibrium bidding strategies over regions where the first-order conditions are valid. In that case we will have a system of equations for each bidder and the distributions H 1 and H 2 will take a more general form.

8 204 I. Chakraborty (v 1 (b) b)h 2 (b) = H 2 (b) (v 2 (b) b)h 1 (b) = H 1 (b) (8) ( v(b) b)(h 2 (b) + h 1 (b)) = H 1 (b) + H 2 (b) b [0, b], where v 1 (0) = v 2 (0) = v(0) = 0 and v 1 ( b) = v 2 ( b) = v( b) = 1, and H 1 (b) = [G 1 (b)] n 1 H 2 (b) = [G 1 (b)] n 1 + N[G 1 (b)] n 2 [G 2 (b) G 1 (b)] G 1 (b) = f(x,y) dy dx = 0 G 2 (b) = A v(b) x B 0 = G 1 (b) + f(x,y)dy dx + f(x,y) dy dx 1 v 1 (b) v2 (b) 0 v1 (b) v(b) f(x,y)dy dx v(b)v 1 (b) v(b)v 2 (b) v 1 (b) v(b) v(b) v 2 (b) v 1 (b) v(b) x 0 f(x,y)dy dx The set A consists of values for which the bidders bid less than or equal to b on the first unit, and B consists of the values for which a bidder bids less than or equal to b on the second unit. This discussion along with the above propositions can be summarized by the following proposition: Proposition 3 Every symmetric and continuous equilibrium bidding strategy can be characterized by a system of differential equations on the boundary of the set of bidder values. 4 Sufficiency of the characterization In the case of a single-object auction a closed form solution from a system such as equation (8) is then verified to be the description of an equilibrium. However, as mentioned earlier, the differential equations (equation (8)) do not have a closed form solution. Thus, an analysis depends either on a numerical description of a solution, or the properties of the above system of equations without any guarantee that the actual solution, if it exists, does indeed represents an equilibrium. 9 The next result shows that in fact a verification is not necessary as long as the solution in question is increasing. Proposition 4 Every increasing solution to the system of differential equations (equation (8)) characterizes an equilibrium bidding strategy. This result implies that a solution such as that calculated numerically in the example by Engelbrecht-Wiggans and Kahn (1998a) can indeed be used to analyze the bidding behavior in the auction. 9 Chakraborty (2004), for instance, gives an example of multi-unit auction where a solution of a similar first order condition does not describe equilibrium strategies.

9 Characterization of equilibrium in pay-as-bid auctions for multiple units Concluding remarks The biggest difficulty in studying multi-unit auctions is the impossibility of describing the equilibrium bidding strategies as closed form expressions. Characterizations via first-order conditions of a bidder s maximization problem, therefore, plays an important role in the analysis of such auctions. We considered the pay-as-bid auction rule for two units, and showed that properties like nonatomicity of bid distribution, differentiability of the bidder s expected payoff functions, obtaining all bids as separated bids, that are vital for the reduced characterization, in fact follow as equilibrium properties in any symmetric equilibrium in continuous strategies. These are properties that are generally assumed about the equilibrium for the validity of the reduced characterization by differential equations on the boundary of the set of values. While an equilibrium may be described by a system of differential equations, there is generally no guarantee that a solution to such a system gives rise to an equilibrium strategy. We showed that in the pay-as-bid auction every increasing solution to the differential equations gives rise to an equilibrium bidding strategy. Since the closed form solutions to such equations cannot be obtained, this result allows an analysis of equilibrium strategies based on the first-order characterization. Moreover, our results reduce the study of uniqueness and existence of the equilibrium in the auction to a study of the same for the solution of the differential equations a particularly well studied area. Research in multi-unit auctions may increasingly depend on characterization of equilibrium strategies via the first-order conditions as in Noussair (1995), Katzman (1995, 1999) and Engelbrecht-Wiggans and Kahn (1998a,b). From that perspective, results such as ours are necessary to validate the approach taken in the different contexts. 6 Appendix Define the functions q i (H i ) = inf{b H i (b) H i } K i (v, H ) = (v q i (H ))H The notation i stands for 3 i. Note that the q i s are left continuous, and since H i is continuous, q i is a strictly increasing function. Also, given the continuity of bidding strategies, the distribution functions, H 1 and H 2 are strictly increasing. Hence, the functions K 1 and K 2 are continuous in both arguments. Lemma A Let v 1 >v 1 > 0 and H 2 >H 2 > 0. Suppose for some H 1, H 1 Then K 1 (v 1,H 2 ) + K 2(v 2,H 1 ) K 1(v 1,H 2) + K 2 (v 2,H 1 ). K 1 (v 1,H 2 ) + K 2(v 2,H 1 )<K 1(v 1,H 2 ) + K 2 (v 2,H 1 ). Similar inequality holds if we change the value of the second unit, and replace v 1, v 1, H 2, H 2, H 1 and H 1 by appropriate values.

10 206 I. Chakraborty Proof of Lemma A We have K 1 (v 1,H 2 ) + K 2(v 2,H 1 ) K 1(v 1,H 2) + K 2 (v 2,H 1 ) or, K 1 (v 1,H 2 ) + K 2(v 2,H 1 ) + (v 1 v 1)H 2 K 1 (v 1,H 2 ) + K 2 (v 2,H 1 ) + (v 1 v 1)H 2 or, K 1 (v 1,H 2 ) + K 2(v 2,H 1 ) K 1(v 1,H 2 ) + K 2 (v 2,H 1 ) (v 1 v 1)(H 2 H 2) and the result follows. Proof of Proposition 1 First observe that a bidder s maximization problem can be restated alternatively as one where the bidder chooses the probability of winning each unit rather than the bid in the following manner. max K 1 (v 1,H 2 ) + K 2 (v 2,H 1 ) H 1,H 2 s.t. q 2 (H 2 ) q 1 (H 1 ). (9) Suppose the bids are pseudo-separated at some value {v1,v 2 }. Define v 2 to be the supremum of second values v 2 such that bids are not separated at {v 1,v 2} for v 2 >v 2 and are not pseudo-separated at {v 1,v 2} for v 2 v 2. Due to the continuity of the bidding strategies it must be the case that the bids are equal at {v1,v 2 }. Clearly, v1 >v 2. Claim. In equation (9) let the bidder choose {H2,H 1 } at {v 1,v 2 }. There is a H 2 > H2 such that K 1 (v1,h 2 ) = K 1(v1,H 2 ). Proof of Claim. Consider a sequence v2 n v 2 such that the bids on {v 1,vn 2 } are pseudo-separated. Let {H2 n,hn 1 } be the corresponding selections of probabilities. Because of weak monotonicity of the bidding strategy for the second unit and the pseudo-separated bids, we have H2 n >H 2 for all n. If the claim is not true then the only possibility is that K 1 (v1,h 2 )>K 1(v1,Hn 2 ) for all n. Since K 1 is continuous in the second argument this means that in a small right neighborhood of H2,K 1(v1,H)is strictly decreasing in H. Also, since the bidding strategy for the second unit is continuous in the value of the second unit, for n large enough, H 2 (q 1 (H1 n)) must be in that right neighborhood. But then H 2 n satisfying q 2 (H2 n)>q 1(H1 n) cannot be optimal since K 1 is strictly decreasing in the right neighborhood a contradiction. Hence the claim must be true. Now let v1 m v 1, the choice {H 2 m,hm 1 } at {vm 1,v 2 } satisfy H 2 m H 2 for all m, by a direct application of Lemma A. lim H2 m exists due to weak monotonicity and lim H2 m H 2. Also let v1 n v 1. The choice {H 2 n,hn 1 } at {vn 1,v 2 } satisfies H 2 n H 2 by weak monotonicity of strategies and thus we have lim H2 n H 2 <H 2 lim H 2 m. This is a contradiction to the continuity of b 1 (v 1,v 2 ) in v 1. This completes the proof.

11 Characterization of equilibrium in pay-as-bid auctions for multiple units 207 Proof of Proposition 2. Define b i to be the supremum of the bids for the ith unit by the bidder, and ˆb i to be the supremum of the separated bids for the ith unit by the bidder. Step 1: First we show that the function H i (x) is differentiable at all bids for the (3 i)th unit that can be made as a separated bid. Consider a first bid b1 (0, ˆb 1 ), by continuity there exists a v1 (0, 1) such that the bidder bids {b1, 0} at {v 1, 0}. Let vn 1 v 1 and the corresponding bids b1 n b 1. Then at {vn 1, 0}, bidding {bn 1, 0} is at least as good as bidding {b 1, 0}. Hence we have, or, or, or, or, (v n 1 bn 1 )H 2(b n 1 ) (vn 1 b 1 )H 2(b 1 ) (v n 1 bn 1 )(H 2(b n 1 ) H 2(b 1 )) (bn 1 b 1 )H 2(b 1 ) H 2 (b 1 ) H 2(b n 1 ) b 1 bn 1 H 2(b1 ) v1 n. bn 1 Again at {v1, 0}, bidding {b 1, 0} is at least as good as bidding {bn 1, 0}. Hence, (v1 b 1 )H 2(b1 ) (v 1 bn 1 )H 2(b1 n ) (v 1 b 1 )(H 2(b 1 ) H 2(b n 1 )) (b 1 bn 1 )H 2(b n 1 ) H 2 (b 1 ) H 2(b n 1 ) b 1 bn 1 H 2(b1 n) v1. b 1 Now letting n we see that the left derivative of H 2 (b) at b1 exists and is equal to H 2 (b1 )/(v 1 b 1 ). Similarly, we can show that the right derivative of H 2(b 1 ) at b1 exists and is equal to H 2(b1 )/(v 1 b 1 ). Thus H 2(b) is differentiable at b1. Similar steps also show that H 1 (b) has a derivative at b where b1 is in the interior of the set of possible separated second bids that the bidders can make in equilibrium. Also, this derivative is equal to H 1 (b )/(v2 b ) where v2 is the second value such that the bidder under consideration has a second bid equal to b at {1,v2 }. Step 2: Next we show that the function H 1 (x) + H 2 (x) is differentiable at all bids that can be made as pooled bids at values like {v, v}. Follows similarly as in Step 1 that the derivative is equal to H 1(b)+H 2 (b) v b. Step 3: Finally, we show that b 1 = b 2 = ˆb 1 = ˆb 2. Suppose b 1 > b 2. First consider the case where there are only two bidders. This means that the highest separated first bid that a bidder makes is b 1, i.e., b 1 = ˆb 1.

12 208 I. Chakraborty But H 2 (b 2 ) = 1 b 2 b 2. Hence the bidder should not bid more than b 2 on the first unit. Thus b 1 = b 2. Next consider the case where there are three or more bidders. Since b 1 > b 2 all bids can be made as separated bids, hence the first-order conditions must be satisfied and Note that and (1 b 2 )h 1 ( b 2 ) = H 1 ( b 2 ). H 1 (b) = (G 1 (b)) n 1 H 2 (b) = (G 1 (b)) n 1 + (n 1)(G 1 (b)) n 2 [G 2 (b) G 1 (b)]. Therefore, we have, 10 (1 b 2 )(n 1)g 1 ( b 2 ) = G 1 ( b 2 ). (10) Also, using the continuity of b 1 (.,.) and the first-order condition for selecting b 1 there is a v<1 s.t. Therefore, (v b 2 )h 2 ( b 2 ) = H 2 ( b 2 ). (1 b 2 )h 2 ( b 2 )>H 2 ( b 2 ). Now, since for c b 2, G 2 (c) = 1, we have from above, (1 b 2 )(n 1)(n 2)(G 1 ( b 2 )) n 3 (G 1 ( b 2 )) n 2 )g 1 ( b 2 ) >(G 1 ( b 2 )) n 1 + (n 1)(G 1 ( b 2 )) n 2 [1 G 1 ( b 2 )]. Substituting equation (10) into the inequality we have (n 2)((G 1 ( b 2 )) n 2 (G 1 ( b 2 )) n 1 ) >(G 1 ( b 2 )) n 1 + (n 1)(G 1 ( b 2 )) n 2 [1 G 1 ( b 2 )] = (n 1)(G 1 ( b 2 )) n 2 (n 2)(G 1 ( b 2 )) n 1 or, (G 1 ( b 2 )) n 2 < 0 a contradiction. Now suppose that b 1 = b 2 but ˆb 1 < b 1. Clearly, (1 ˆb 1 )H 2 ( ˆb 1 ) (1 b)h 2 (b 1 ) b ˆb 1 since ˆb 1 is optimal at (1,0). Bids that are greater than ˆb 1 occur only as pooled bids. Hence H 2 (b) = H 1 (b) b ˆb 1 10 Differentiability of G 1 ( ) follows from that of H 1 ( ).

13 Characterization of equilibrium in pay-as-bid auctions for multiple units 209 which implies (1 ˆb 1 )H 1 ( ˆb 1 ) (1 b 2 )H 1 (b 2 ) whenever b 2 > ˆb 1. By Lemma A if b 2 > ˆb 1 and v 2 < 1 Put differently, for any v 2 < 1, if (v 2 ˆb 1 )H 1 ( ˆb 1 )>(v 2 b 2 )H 1 (b 2 ) b 2 arg max b 2 (v 2 b 2 )H 1 (b 2 ), then b 2 ˆb 1. Therefore, for values (1,v 2 ) where v 2 < 1 the bidder should not bid more than ˆb 1 on the second unit. This is a contradiction to the hypothesis b 1 = b 2 > ˆb 1. Thus we have b 1 = b 2 = ˆb 1 which in turn implies that b 1 = b 2 = ˆb 1 = ˆb 2. Proof of Proposition 4 First observe that if bid b1 maximizes (v 1 b 1 )H 2 (b 1 ) and b2 maximizes (v 2 b 2 )H 1 (b 2 ) then if at (v 1,v 2 ) the equilibrium bids are separated they are given by {b1,b 2 }. Thus to check that a given bidding strategy is optimal for all values over which bidding strategies are separated it is enough to check that the bidding strategies are optimal for all values like {v 1, 0} and {1,v 2 }.Now suppose the same bidding strategy maximizes the expected payoff with pooled bids {b,b } for some value {v,v }. Also, suppose there is a pair {v 1,v 2 } at which the bidder maximizes her expected payoff by bidding {b,b }, then for every value {αv 1 + (1 α)v,αv 2 + (1 α)v },α [0, 1], the bids {b,b } maximize the expected payoff as mentioned before. Thus it is enough to check that the solutions characterize bidding strategies that maximize the bidder s expected payoff at values like {v 1, 0}, {1,v 2 } and {v, v}. 11 Suppose v 1 (b), v 2 (b) and v(b) are solutions of the system of differential equations. Then we have (v 1 (b) b)h 2 (b) H 2 (b) = 0 (v 2 (b) b)h 1 (b) H 1 (b) = 0 ( v(b) b)(h 1 (b) + h 2 (b)) (H 1 (b) + H 2 (b)) = 0. Let the solution prescribe bid b 1 for the first unit at value {v 1, 0} with v 1 = v 1 (b 1 ). Consider any bid 0 b 1 <b 1. The derivative12 of the expected payoff function with values {v 1, 0} and with respect to the bid b 1 is (v 1 (b 1 ) b 1)h 2 (b 1 ) H 2 (b 1 ). That v 1 ( ) is increasing implies that v 1 (b 1 ) v 1(b 1 ). This, combined with the equation (v 1 (b 1 ) b)h 2 (b 1 ) H 2 (b 1 ) = 0, gives: 13 (v 1 (b 1 ) b 1)h 2 (b 1 ) H 2 (b 1 ) It is necessary to check at {v, v} only when the bids are pooled at {v, v}. 12 Observe that when the rival bidders are using the bidding strategies as prescribed by the equation system, the distributions of their bids are differentiable. 13 Since H i (b) > 0 when v 1, v 2, and v are increasing.

14 210 I. Chakraborty Similarly, for b 1 >b1 we have (v 1 (b1 ) b 1)h 2 (b 1 ) H 2 (b 1 ) 0. The two inequalities together imply that the bidder s expected payoff is maximized at {v 1, 0} by the bidding strategy derived from the solution. The same argument shows that at value {1,v 2 } the bidding strategy described by the solution indeed maximizes the expected payoff. Now consider values like { v(b ), v(b )} where the bidder bids b for both units. The bids and the values must satisfy the differential equation ( v(b) b)(h 1 (b) + h 2 (b)) (H 1 (b) + H 2 (b)) = 0. (11) Observe that if the differential equations yield v 1 (b) v 2 (b) then strictly speaking the bid b is not made as a pooled (in the sense that the constraint is binding) bid. This implies that v(b) is irrelevant for the characterization of the equilibrium bidding strategy for values like {v, v} at which such bids are made. Since we are looking at pooled bids we can assume that v 1 (b)>v 2 (b) for the relevant bids. Now suppose v(b) v 1 (b)>v 2 (b). Using this in the other two differential equations we have and ( v(b) b)h 2 (b) H 2 (b) 0 ( v(b) b)h 1 (b) H 1 (b) > 0. This is inconsistent with equation (11) hence we cannot have v(b) v 1 (b)>v 2 (b) from the differential equations. Again if v(b) v 2 (b)<v 1 (b) then we have and ( v(b) b)h 2 (b) H 2 (b) < 0 ( v(b) b)h 1 (b) H 1 (b) < 0 which is again inconsistent with equation (11). Hence we must have v 1 (b) > v(b)>v 2 (b) whenever v(b) is relevant for our calculation of equilibrium strategy. Now we are ready to prove that the bid {b,b } maximizes a bidder s expected payoff at { v(b ), v(b )}. To show that consider the derivative of the expected payoff if a bidder bids {b 1,b 2 } with b 1 >b 2 at { v(b ), v(b )}. The derivative with respect to the second bid is ( v(b ) b 2 )h 1 (b 2 ) H 1 (b 2 )>0 whenever b 2 <b by a previous argument. However, for b 1 close enough to b but smaller than b we have ( v(b ) b 1 )h 2 (b 1 ) H 2 (b 1 )<0. In other words, a deviation like b b 1 >b 2 cannot maximize the expected payoff at this value. Similarly, deviations like b b 2 <b 1 or b 2 b b 1 with one of

15 Characterization of equilibrium in pay-as-bid auctions for multiple units 211 the inequalities being strict are suboptimal. Hence the only meaningful deviations that we need to consider are the ones for which the bids are equal. Now by arguments similar to that used before we see that for b<b and ( v(b ) b)(h 1 (b) + h 2 (b)) (H 1 (b) + H 2 (b)) > 0 ( v(b ) b)(h 1 (b) + h 2 (b)) (H 1 (b) + H 2 (b)) < 0. Therefore, the expected payoff is maximized by bidding (b,b ). The proof is completed upon extending the solution on the boundary to the interior of the set V along the lines of the construction in Section 3. References Ausubel, L.M., Cramton, P.: Demand reduction and inefficiency in multi-unit auctions. Working Paper, University of Maryland (1998) Back, K., Zender, J.: Auctions of divisible goods: on the rationale for the treasury experiments. Rev Financial Stud 6, (1993) Chakraborty, I.: Multi-unit auctions with synergies. Eco Bullet 4 (2004). Engelbrecht-Wiggans, R., Kahn, C.M.: Multi-unit pay-as-bid auctions with variable awards. Games Econ Behav 23, (1998a) Engelbrecht-Wiggans, R., Kahn, C.M.: Multi-unit auctions with uniform prices. Econ Theory 12, (1998b) Katzman, B.: Asymptotic properties of equilibria in discriminatory and uniform price auctions. University of Miami 1999 Katzman, B.: Multi-unit auctions with incomplete information. University of Miami (1995) Milgrom, P., Weber, R.J.: A theory of auctions and competitive bidding. Econometrica 50, (1982) Noussair, C.: Equilibria in a multi-object uniform sealed bid auction with multi-unit demands. Econ Theory 5, (1995) Vickrey, W.: Counterspeculation, auctions and competitive sealed tenders. J Finance 16, 8 37 (1961)