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1 MIT Sloan School of Management Working Paper July 2002 BIDDING LOWER WITH HIGHER VALUES IN MULTI-OBJECT AUCTIONS David McAdams 2002 by David McAdams. All rights reserved. Short sections of text, not to exceed two paragraphs, may be quoted without explicit permission provided that full credit including notice is given to the source. This paper also can be downloaded without charge from the Social Science Research Network Electronic Paper Collection:

2 Bidding Lower with Higher Values in Multi-Object Auctions David McAdams MIT Sloan School of Management This version: July 17, 2002 First draft: June Abstract Multi-object auctions differ in an important way from single-object auctions. When bidders have multi-object demand, equilibria can exist in which bids decrease as values increase! Consider a model with n bidders who receive affiliated one-dimensional types t and whose marginal values are non-decreasing in t and strictly increasing in own type t i. In the first-price auction of a single object, all equilibria are monotone (over the range of types that win with positive probability) in that each bidder s equilibrium bid is non-decreasing in type. On the other hand, some or all equilibria may be non-monotone in many multi-object auctions. In particular, examples are provided for the asbid and uniform-price auctions of identical objects in which (i) some bidder reduces his bids on all units as his type increases in all equilibria and (ii) symmetric bidders all reduce their bids on some units in all equilibria, and for the as-bid auction of non-identical objects in which (iii) bidders have independent types and some bidder reduces his bids on some packages in all equilibria. Fundamentally, this difference in the structure of equilibria is due to the fact that payoffs fail to satisfy strategic complementarity and/or modularity in these multi-object auctions. I thank Susan Athey for starting me to thinking about monotone equilibria in auctions and Phil Reny for catching an error in a previous version. mcadams@mit.edu. Post: MIT Sloan School of Management, E52-448, 50 Memorial Drive, Cambridge, MA

3 1 Introduction To what extent does single-object auction theory extend to auctions of multiple objects? Some basic insights drawn from the single-object theory certainly continue to apply. For example, Ausubel and Cramton (1998) show that bidders with private values in the uniform-price auction of identical objects generally bid below their true value. Bid-shading extends to the multi-unit setting because its logic extends: bidders will shade their bids down whenever those bids may determine their own payment upon winning. The main point of this paper, however, is that other basic single-object insights do not extend to the multiple-object setting. In particular, I focus on the issue of whether bidders adopt monotone strategies and how this relates to strategic complementarity and modularity of payoffs. For illustration purposes, consider a simple model in which two riskneutral bidders compete in a first-price auction. Bidder i s valuation is given by v i (t), where t is a vector of affiliated, one-dimensional bidder types and v i is non-decreasing in t and strictly increasing in t i. Let π i (b, t) = (v i (t) b i )1{i wins} represent bidder i s ex post payoff. In this case, it is well-known that bidder i always has a non-decreasing best response strategy whenever j adopts a non-decreasing strategy. (See Athey (2001).) Less well appreciated is that, ultimately, this property derives from the fact that ex post payoff π i satisfies a form of strategic complementarity, single-crossing in b i, b j (shorthand SC(b i, b j )), as well as modularity in b i. (See page 5 of the introduction for definitions of these and other terms.) To see why π i (b; t) satisfies SC(b i, b j ) (for all states t), consider for example the incremental return to b i = 80 versus b i = 60 as a function of opponent s bid. When b j < 60, then either bid wins, so i gets negative incremental return 60 80; when b j (60, 80) i gets incremental return v i (t) 80; when b j > 80 i is obviously indifferent between the bids since they both lose; and when b j = 60 or 80, i s incremental return is an average of the returns in these regions. Thus, π i (b i, b j ; t) π i (b i, b j ; t) implies that π i (b i, b j; t) π i (b i, b j; t) whenever b i > b i, b j > b j. An implication of this fact is that π i (b i, b j (t j ); t) satisfies SC(b i, t j ) (for all t i and all non-decreasing opponent strategies b j ( )). Figure 1 graphs the incremental return to b i = 80 versus b i = 60 as a function of opponents type. 1 Finally, when combined with an assumption of affiliated types, this implies in the two-bidder case that in- 1 I thank Susan Athey for explaining this set of ideas to me using this sort of figure. 2

4 f(80, t 2 ) f(60, t 2 ) 20 j (60) b 1 j (80) b 1 t 2 Figure 1: SC(b 1, t 2 ) of f(b 1, t 2 ) = π 1 (b 1, b 2 (t 2 ), t 1, t 2 ) terim expected payoff E tj t i [π i (b i, b j (t j ), t) t i ] satisfies SC(b i, t i ) whenever the other bidder follows a non-decreasing strategy. (Athey (2001) Theorem 7.) Armed with this single-crossing condition, Athey (2001) proves in the case of n = 2 bidders that a monotone pure strategy equilibrium exists in the first-price auction, i.e. each bidder s bid is non-decreasing in his type. In related work, McAdams (2001) and Kazumori (2002) prove existence of a monotone pure strategy equilibrium in auctions of multiple identical objects with n bidders and multi-dimensional types but only in the case of independent types and risk-neutral bidders. Reny and Zamir (2002) prove existence of monotone pure strategy equilibrium in the first-price auction with affiliated types n > 2 bidders with affiliated types. 2 In this setting, Athey s single-crossing condition fails. When others follow non-decreasing strategies, a bidder may prefer b to b when he has type t i but prefer b to b when he has type t i for some bid levels b > b and types t i > t i. Reny and Zamir (2002) show, however, that this is not possible if type t i gets non-negative expected payoff from the higher bid b. This observation allows them to apply Athey (2001) s method of proof of existence since that method only requires 2 Reny and Zamir (2002) also provide an example with 3 bidders having multidimensional affiliated types in which the unique equilibrium is non-monotone. 3

5 that single-crossing holds for pairs or bids b > b and pairs of types t i > t i such that b is a best response for type t i, given non-decreasing strategies by others. They call this single-crossing condition BR-SCC since it only applies when the higher bid is a best response for the lower type. b 2 b 2 b b b b wins wins, b loses Figure 2: Bids b, b against non-monotone b 2 ( ) t 2 In Section 2, I prove that all equilibria in the first-price auction must be monotone over the range of types who win with positive probability in the general affiliated values model with n bidders and one-dimensional types. (A mixed-strategy equilibrium is monotone iff the highest bid played with positive probability by a lower type is less than or equal to the highest bid played with positive probability by a higher type.) In particular, for each bidder i there exists some type t i such that all types above (below) t i win with positive (zero) probability and bidder i s strategy is monotone over the range of all types greater than or equal to t i. At the heart of this positive result is my observation that single-crossing holds for all pairs b > b and t i > t i such that b is a best response for type t i and b is less than or equal to b i, the lowest trough in the bid functions of any bidder other than i. I call this condition NM-SCC since it applies even when others follow non-monotone strategies. NM-SCC is a strengthening of Reny and Zamir (2002) s BR-SCC condition since b i = whenever others follow non-decreasing strategies. 4

6 Figure 2 illustrates the lowest trough b 2 in bidder 2 s non-monotone bid function; see the Appendix for a formal definition. The essence of the proof is to exploit this limited single-crossing property to show that the lowest trough of any given bidder s equilibrium bid function must be strictly higher than the lowest trough of some other bidder s bid function. This of course leads to a contradiction unless all bidders are following non-decreasing strategies. On the other hand, in Section 3, I show why McAdams (2001) and Kazumori (2002) s results for multiple identical-object auctions can not possibly extend to cover the case of affiliated types (even with private values) by providing examples in which all equilibria are non-monotone. In Examples 1 and 3, some bidder reduces his bids on all units as his type increases in all equilibria. In Example 2, symmetric bidders all reduce their bids on some units in all equilibria. And in Example 4 of Section 4, some bidder reduces his bids on the individual objects in all equilibria given independent types and non-identical objects. Fundamentally, as discussed next in Section 1.1, these negative examples flow from the fact that bidders ex post payoffs fail to satisfy SC(b i, b i ) in identical object auctions and also fail to be modular in own bid in non-identical object combinatorial auctions. Lastly, in the paper s conclusion, I share my thoughts on where I believe these negative results should lead future multi-unit auction research. 1.1 Single-crossing and modularity Definition (Single-crossing for (x, x; t, t), single-crossing for (x, x)). Let X, T be partially ordered sets with elements x > x in X and t > t in T. g : X T R satisfies (strict) single-crossing for (x, x; t, t) iff g(x, t) g(x, t) g(x, t ) (>)g(x, t ) Similarly, g : X R satisfies (strict) single-crossing for x, x iff g(x) 0 g(x ) (>)0 Note that single-crossing for (x, x; t, t) explicitly specifies which pair of elements x, x are being compared and with respect to which pair of elements t, t. The more standard single-crossing in (x, t) requires SC(x, x; t, t) for all comparable pairs x, x and t, t: 5

7 Definition (Single-crossing in (x, t)). Let X, T be partially ordered sets. g : X T R satisfies single-crossing in (x, t) (shorthand SC(x, t)) iff for all x > x X and all t > t T. g(x, t) g(x, t) g(x, t ) g(x, t ) Definition (Modularity in x). Let X be a lattice. g : X R is modular in x iff for all x, x X. g(x ) + g(x) = g(x x) + g(x x) Note that when X is a sublattice of Euclidean space, modularity in x = (x 1,..., x k ) is equivalent to additive separability in x 1,..., x k. Modularity of payoffs in b i is trivially satisfied in single-object auctions since bids are onedimensional. Payoffs are not modular in own bid in combinatorial auctions because increasing one s bid on a bundle will typically make one more likely to win the bundle the more that one simultaneously lowers bids on individual units. Thus, one can not apply McAdams (2002) s monotone equilibrium existence theorem. (Quasisupermodularity in own bid also fails.) As mentioned earlier, payoffs are modular in own bid in most commonly studied auctions of identical objects (given risk-neutral bidders). As a consequence, when bidder types are independent a monotone equilibrium exists and if bidder values are also strictly increasing in own type, then one can prove that all equilibria are monotone over the range of all types who win with positive probability. Even in this case, however, the conditions of the monotone equilibrium existence theorem may fail when bidder types are positively or negatively correlated. The reason for this is that payoffs fail to satisfy single-crossing in own bid and others bids. The purpose of most of the examples in the paper is to illustrate why this property fails. Since a graphical intuition is helpful, I include here an example drawn from McAdams (2001). Figure 3 shows why ex post payoffs fail to satisfy single-crossing for (D i( ), D i ( ); D j( ), D j ( )) in the uniform-price auction in a simple example with two bidders and S perfectly divisible units. The observation extends, however, to settings with any number of bidders, indivisible units and/or discrete price grid, and price-elastic supply. Other multi-unit auctions such as the pay-as-bid auction also fail to possess strategic complementarity. 6

8 price S D 2( ) S D 2 ( ) v D 1( ) D 1 ( ) quantity Figure 3: Single-crossing in own bid and others bids fails in uniform-price auction Example. For some type profile t, bidder 1 has a constant marginal value v for shares, i.e. v 1 (q; t) = q i v. S D 2( ), S D 2 ( ) are the residual supply curves that would result if bidder 2 submits the bid D 2( ) or D 2 ( ). The unlabelled curves in Figure 3, finally, are isoprofit curves of bidder 1. Thus, D 1 ( ) is strictly preferred to D 1( ) when bidder 2 submits D 2( ) whereas D 1( ) is strictly preferred to D 1 ( ) when bidder 2 submits D 2 ( ). This violates SC(D i( ), D i ( ); D j( ), D j ( )) of bidder 1 s ex post payoff. 2 First-Price Auction First-price auction model: There are n bidders and one object. Each bidder has type t i [0, 1] and a randomization variable τ i [0, 1], where t is affiliated, τ is independent of t, and the joint density of (t, τ) has full support on [0, 1] n. (τ i need not be independent of τ j.) In particular, this implies that for any T [0, 1] 2n 2, Pr(t i, τ i T ) > 0 implies that Pr((t i, τ i ) T t i, τ i ) > 0 for all t i, τ i. Each bidder s utility upon losing is normalized to zero and upon winning has the form u i (v i (t) b i ), where the valuation v i is non-decreasing in t and strictly increasing in t i and utility u i is strictly increasing in surplus. The set of permissible bids is { } [p min, ) where bidding { } ensures that one will never win the object. Ties are bro- 7

9 ken randomly when the high bid is at least p min and the auction is cancelled if the high bid is { }. Theorem 1. A monotone pure strategy equilibrium exists in the first-price auction and every equilibrium is monotone over the range of types who win with positive probability. Specifically, if b is played with positive probability by type t i, Pr( b wins) > 0, t i > t i > t i, and b, b are played with positive probability by types t i, t i (respectively), then b b b. Proof. The proof is in the Appendix. Reny and Zamir (2002) prove existence of a monotone pure strategy equilibrium, so my contribution is to prove that all equilibria must be monotone. To illustrate the essential idea of the proof, I will focus on pure strategy equilibria with 3 bidders and consider only a special case in which several simplifying conditions are satisfied. Suppose that b ( ) is a pure-strategy equilibrium with (a) no atoms in i s bid, i.e. Pr(b i (t i ) = b) = 0 for all b, (b) all but the lowest bids win, i.e. b > inf ti b i (t i ) implies that Pr(b > max j i b j(t j )) > 0, and (c) Pr(b i > b i (t i )) > 0, where b i inf{b : b i (t ) < b, b i (t) > b for some t i > t i } if this set is empty b i is the level of the lowest trough of bidder i s bid function and equals exactly when b i ( ) is monotone non-decreasing. Thus, (c) amounts to assuming (c1) b i (t) < b i over some interval of types including zero and (c2) b i ( ) is monotone non-decreasing over this range. (See Figure 2 on page 4.) The proof proceeds by showing that b ( ) an equilibrium implies that b i > min j i b j b i whenever b i <. Thus, b 1 = b 2 = b 3 = and any equilibrium must be monotone! Why must the lowest trough in (say) bidder 1 s equilibrium bid function be higher than the lowest trough of some one else s equilibrium bid function? Recall the usual trade-off between bidding b versus b when others are following monotone strategies : (a) bidding b leads bidder 1 to pay more in the event b wins and this event is a rectangle in the lower corner of T 1 = [0, 1] n 1 ; (b) bidding b leads him to win sometimes when b would have lost and the event b wins is also a product of intervals. When others are following non-monotone strategies but b b 1, the trade-off is very similar. The event b wins is still a rectangle in the lower corner of T 1 and the event b wins still has a product structure, although it is not 8

10 the product of intervals. Figure 4 provides an example of the structure of b wins and b wins when there are three bidders and bidders 2,3 follow a strategy with a single trough as in Figure 2 such that (i) b j(t j ) [0, b) for all t j [0, t 1 j), (ii) b j(t j ) (b, b ) for t j (t 1 j, t 2 j) (t 3 j, t 4 j), and (iii) b j(t j ) > b for t j (t 2 j, t 3 j) (t 4 j, 1]. t 3 t 4 3 t 3 3 X 0,2 X 1,2 X 2,2 t 2 3 t 1 3 X 0,1 X 1,1 X 2,0 X 0,0 X 1,0 X 2,1 t 1 2 t 2 2 t 3 2 t 4 2 t 2 Figure 4: X 0,0 = b wins. 0 m2,m 3 2X m 2,m 3 = b wins Standard arguments prove that, when all others follow monotone strategies and bidder 1 s valuation v 1 (t) is strictly increasing in t 1, bidder 1 s payoff has strict single-crossing for (b, b; t 1, t 1 ) whenever b, b both win with positive probability and t 1, t 1 find b, b to be best responses, respectively. Such a property implies that all of bidder 1 s best response strategies are monotone over the range of types that win with positive probability. The key steps in the proof are applications of standard facts about functions of affiliated random variables, in particular Theorem 5 of Milgrom and Weber (1982) (hereafter MW ). The crucial elements of the problem that allow one to apply MW are that b wins is a product of intervals including zero and that b wins also has a product structure. Both of these elements carry over to the case 9

11 in which others follow non-monotone strategies, as long as the lower bid b is less than or equal to the lowest trough in others bid functions. Thus, it is impossible that b and b b 1 both win with positive probability and t 1, t 1 find b, b to be best responses, respectively. By the definition of b 1, then, b 1 > b 1. 3 Auctions of Identical Objects Identical objects model: The examples in this section vary but they all fit into the following framework. There are n risk-neutral bidders and two identical objects being auctioned. Each bidder has type t i [0, 1] and t is affiliated. Each bidder s marginal values v i (1, t), v i (2, t) are non-decreasing in t. 3 In some examples, the set of permissible bids is a finite grid while in others bids are allowed on a continuum and the tie-breaking rule varies. Note also that I maintain the assumption of one-dimensional, affiliated types. These assumptions are less natural in the multi-unit context 4 but useful from an expository point of view. When types are negatively correlated or multi-dimensional, the first-price auction sometimes possesses non-monotone equilibria for very natural reasons. (See Jackson and Swinkels (2001) for an example with negative correlation, Reny and Zamir (2002) for an example with multi-dimensional affiliated types.) Maintaining these assumptions helps me to isolate important differences between single- and multi-object auctions, namely the non-monotonicity of equilibria that is caused by the failure of payoffs to satisfy strategic complementarity and/or modularity. Finally, one may easily concoct examples of non-monotone equilibria in which the non-monotonicity occurs because some bidder is indifferent to sub- 3 Marginal values are not necessarily strictly increasing in own type. In some examples, each bidder has only finitely many types. One may reinterpret each such type as corresponding to a range of types in the unit interval (while maintaining affiliation). Over this range, values are constant in own type. 4 Bidder types may be multi-dimensional if different information is relevant to the value of different objects or (in the identical object case) to initial versus later units, among other reasons. For example, suppose that a buyer may consume one unit himself at utility t 1 i and/or resell any units that he wins at price t2 i. The affiliation assumption may also become less natural in that t 1 i, t2 i may be negatively correlated (even if t i, t j are positively correlated). In an auction of rare coins, for instance, it seems plausible that dealers who expect to be able to resell the coins at a high price may have a relatively low personal willingness to pay for the coins. Such dealers may have a relatively large inventory and hence already possess a personal collection that satisfies them. 10

12 mitting a range of bids. I want to avoid such examples. As a consequence, in my examples I focus on equilibria in weakly undominated strategies. 3.1 Non-Monotone Equilibria: Examples Monotonicity of equilibria does not extend to multi-object settings! The simplest example that I have found applies to the so-called uniform N + 1-st price auction in which price equals the highest losing bid. (Examples 1 and 2 each apply to the uniform N + 1-st price auction which does not extend the first-price rule. Similar logic applies in the as-bid and uniform N-th price auctions, however, which each do extend the first-price rule. See also Example 3.) Example 1 (Uniform N + 1-st price auction). There are two identical objects for sale and two bidders. Bidder 1 has unit demand and private value v 1 [0, 1]. Bidder 2 has (constant) marginal value v 2 = 1 for both units and receives a signal s 2 {H, L} that is informative about 1 s value. In particular, the conditional density of v 1 given s 2 is f(v 1 s 2 = L) = 2 2v 1, f(v 1 s 2 = H) = 2v 1. Claim. An equilibrium in weakly undominated strategies exists and all such equilibria are non-monotone in this example. Proof. Bidder 1 bids v 1 on the first unit and zero on the second unit and bidder 2 bids his true value v 2 = 1 on a first unit in any equilibrium in weakly undominated strategies. (Note however that bidder 2 s first unit bid never sets the price.) Bidder 2 has two basic options for his bid on the second unit: (a) concede one unit to bidder 1 and bid 0, thereby winning one unit at price 0, or (b) attempt to win two units and bid b > 0, thereby sometimes winning one unit at price b (if v 1 > b) and other times winning two units at per unit price v 1 (if v 1 < b). Under option (a), bidder 2 s profit equals 1. Under option (b), bidder 2 s profit and marginal return to a higher second-unit bid are π i (b, s 2 ) = b 0 (2 2v 1 )f(v 1 s 2 )dv 1 + (1 b) Pr(v 1 > b s 2 ) π i / b(b, s 2 ) = (2 2b)f(b s 2 ) (1 b)f(b s 2 ) Pr(v 1 > b s 2 ) When s 2 = L, f(v 1 L) = 2 2v 1, Pr(v 1 > b L) = (1 b) 2, and π i / b(b, L) = (1 b) 2 > 0 for all b < 1. Thus, b 2(L) = (1, 1). On the other hand, 11

13 when s 2 = H, f(v 1 H) = 2v 1, Pr(v 1 > b H) = 1 b 2, and π i (b, H) = b 3 /3 + b 2 b + 1 < 1 = π i (0, H) for all b (0, 1]. Thus, b 2(H) = (1, 0). Note that this non-monotone equilibrium is the unique equilibrium in weakly undominated strategies. The key feature driving Example 1 is that the multi-unit demand bidder faces a decision about whether to bid with an eye toward winning both units or just one unit (i.e. conceding ). Given the particular structure of the example two goods, one opponent who only wants a single unit it is clear that conceding becomes relatively more attractive the higher that the opponent bids. Since types are affiliated and the opponent s bid is increasing in its type, then, it is natural that the two-unit demander s bid may fall with his own type, even if higher own type were also to somewhat increase his own value for the objects. One feature of Example 1 is that bidder 2 can lock in a unit by making the minimal possible bid, but this is not crucial. Suppose that we add an additional unit-demand bidder (call it bidder 0) who has private value v 0. Let f v (x s 2 ) be the conditional density of v max{v 0, v 1 } and f v (x s 2 ) the conditional density of v min{v 0, v 1 }. Then 2 s profit as a function of his bid b on the second unit is π i (b, s 2 ) = b 0 (2 2x)f v (x s 2 )dx + (1 b) Pr(b (v, v) s 2 ) + 1 b (1 x)f v (x s 2 )dx π/ b = (2 2b)f v (b s 2 ) + (1 b)(f v (b s 2 ) f v (b s 2 )) Pr(b (v, v) s 2 ) (1 b)f v (b s 2 ) = (2 2b)f v (b s 2 ) (1 b)f v (b s 2 ) Pr(b (v, v) s 2 ) (As always in the uniform-price auction in which price equals highest losing bid, bidder 2 s weakly dominant strategy is to bid his true value on the first unit.) Note that the term representing bidder 2 s marginal return to increasing b differs from that in the previous example only in that the density for v 1 is replaced by the density for v and that Pr(v 1 > b s 2 ) is replaced by Pr(b (v, v) s 2 ). As long as the distribution of v 0 is heavily weighted enough by values near zero, then, the logic of the previous example carries through without modification. Another feature of Example 1 is that the bidders are asymmetric, but this also is not crucial. More important is that, as the unit demander s 12

14 type increases, the difference between his equilibrium bid on the first unit and the second unit increases. As this difference in one s opponents bids increases, it increases the relative attractiveness of conceding a unit. In the N +1-st uniform-price auction, it also tends to increase the relative likelihood that one s second-unit bid will set the price, thereby further increasing one s incentive to concede a unit. These points are illustrated with the following symmetric example in the N + 1-st uniform-price auction. Example 2 (Symmetric bidders). Two bidders receive types t 1, t 2 [0, 1]. v i (1, t i ) = v 1 (2, t i ) = 200t i for all t i [0, 1/2). v i (1, t i ) = 200t i and v i (2, t i ) = 100 for all t i (1/2, 1]. The joint density of t is given by f(t) = 2 when t 1, t 2 > 1/2 or t 1, t 2 < 1/2 and f(t) = 0 otherwise. In other words, t 1 > (<)1/2 implies that t 2 > (<)1/2 and t are independent conditional on both being greater than or less than 1/2. Claim. An equilibrium in weakly undominated strategies exists and all such equilibria are non-monotone in this example. Proof. Conditional on t 1, t 2 < 1/2 or > 1/2, t 1, t 2 are independent. Thus, Kazumori (2002) proves that an equilibrium exists in sub-auctions in which the types are less than or greater than 1/2. And any combination of equilibria in these sub-auctions is an equilibrium in the auction. Furthermore, it is straightforward to prove that an equilibrium in undominated strategies exists. 5 In any equilibrium, b 1(1, t 1 ), b 2(1, t 2 ) 100 and b 1(2, t 1 ) = b 2(2, t 2 ) 100 for all t 1, t 2 > 1/2. (If the second-unit bids are not tied with probability one conditional on types greater than 1/2, then some bidder could sometimes lower the uniform-price without otherwise altering the auction outcome by lowering his bid.) Furthermore, b 1(2, t 1 ) = b 2(2, t 2 ) = 0 in any equilibrium in weakly undominated strategies. On the other hand, it is straightforward to show that b i (2, t i ) > 0 in any equilibrium when t i < 1/2. 5 Consider variations in which bidders submit random bids with positive probability. Kazumori (2002) proves existence of isotone equilibria in each sub-auction given such trembling. A limit of such equilibria, as the probability of trembling goes to zero, is an equilibrium in undominated strategies without trembling. (A limit exists because the set of isotone strategies is compact in the weak topology and this limit is an equilibrium since expected payoffs are continuous with respect to this topology. See Athey (2001) for a template of this argument in the context of the first-price auction.) 13

15 What about the pay-as-bid auction and the version of the uniform-price auction in which price equals the lowest winning bid? As one probably expects by now, similar examples can be constructed in which some or all equilibria are non-monotone in these auctions as well. Calculating equilibria becomes more cumbersome in these auctions since it is no longer true that a multi-unit bidder will bid truthfully on the first unit. I provide here an example that applies to both of these auctions at the same time. Example 3 (Pay-as-bid auction or Uniform N-th price auction). There are two bidders and two units for sale in the pay-as-bid auction or in the uniform N-th price auction. Each bidder i wins at least one unit if b i (1) > b j (2) and two units if b i (2) > b j (1); ties are broken randomly. The set of permissible bids is {0, 10, 20} and each bidder receives a perfectly correlated signal in the set {L, H}. Bidder i s valuation for a kth unit given signal s will be denoted v i (k, s). v 1 (1, L) = v 1 (2, L) = 0, v 1 (1, H) = 24, v 1 (2, H) = 0 v 2 (1, s) = v 2 (2, s) = 24 for each s Claim. An equilibrium in weakly undominated strategies exists in both the pay-as-bid and the uniform N-th price auction and all such equilibria are non-monotone. Proof. Pay-as-bid: Clearly, b 1(1, L) = b 1(2, L) = b 1(2, H) = 0. After either signal, bidder 2 s unique best response is (0, 0) if bidder 1 bids (10, 0) or (20, 0) since this ensures that he wins an expected 1 1/2 units one unit for profit 3/2(v 2 (s) 0) > 2(v 2 (s) 10). Similarly, after either signal bidder 2 s unique best response is (10, 10) if bidder 1 bids (0, 0). After (0, 0) he wins an expected 1 units for profit 24, after (10, 0) he wins an expected 1 1/2 units for profit 36 10, after (10, 10) he always wins two units for profit 2(24 10). After signal H, bidder 1 s best response is to bid (10, 0) as long as b 2(2, H) < 20. Bidding (0, 0) rather than (10, 0) leads bidder 1 to lose entirely if b 2(2, H) = 10 and to decrease expected winnings from 1 to 1/2 if b 2(2, H) = 0, but > 1/2(24 0). Thus, in the unique equilibrium in weakly undominated strategies, b 1(, L) = (0, 0), b 1(, H) = (10, 0), b 2(, L) = (10, 10), b 2(, H) = (0, 0) Uniform N-th price: After either signal, bidder 2 s unique best response is still (0, 0) if bidder 1 bids (10, 0) or (20, 0). One difference, however, is that 14

16 now bidder 2 s best response is to bid (10, 0) if bidder 1 bids (0, 0) since 3/2(24 0) > 2(24 10). After signal H, bidder 1 s best response to both (0, 0) and (10, 0) remains to bid (10, 0) for the same reason as before. Thus, in the unique equilibrium here in weakly undominated strategies, b 1(, L) = (0, 0), b 1(, H) = (10, 0), b 2(, L) = (10, 0), b 2(, H) = (0, 0) 4 Non-Monotone Equilibria in Auctions of Non-Identical Objects Non-identical objects model: Same as identical objects model except that each bidder has valuations for objects A, B separately and together, v i (A; t i ), v i (B; t i ), v i (A, B; t i ). Existence of non-monotone equilibria is perhaps not so surprising in combinatorial auctions of non-identical objects. After all, a bidder with a low type may only bid seriously on an individual object whereas with a higher type he may bid with an eye to winning a bundle. In this case, it may make sense to withdraw one s bid on individual objects to increase the chances of winning the bundle. The logic of non-monotonicity here ultimately relies on the fact that a bidder s payoff is not additively separable in his bids on various bundles: increasing one s bid on the bundle will typically make one more likely to win the bundle the more that one simultaneously lowers bids on individual units. Consequently, while some sort of correlation of types is necessary for all equilibria to be non-monotone in the identical objects case, it is quite easy to construct examples with non-identical objects in which types are independent and all equilibria are non-monotone. Example 4. There are two bidders in the pay-as-bid combinatorial auction of two objects A, B. Each bidder i submits bids on all combinations: b i (A), b i (B), b i (A, B) {0, 10, 20,...}. If max i b i (A) + max i b i (B) > (<) max b i (A, B), i then the goods are allocated separately (bundled) to the relevant high bidders with ties broken randomly; if these two allocations bring in the same 15

17 revenue, then the method of allocation is chosen randomly, again with subsequent ties broken randomly. All of the bidders have additive valuation functions. Bidder 1 is a single-object demander and doesn t receive any private information (his values are known): v 1 (A) = 25, v 1 (B) = 0. Bidder 2 may want both objects and receives signal t 2 {L, H}, each with probability 1/2. v 2 (A, L) = 0, v 2 (B, L) = 25 whereas v 3 (A, H) = v 3 (B, H) = 100. Claim. An equilibrium in weakly undominated strategies exists and all such equilibria are non-monotone. Proof. Obviously b 1(B) = b 2(A; L) = 0 in any equilibrium. Also, v 2 (, H) is chosen to be so high that bidder 2 certainly wins both objects when his type is high. Thus, b 1(A) = 10 in any equilibrium: 1 only wins if t 2 = L and the objects are sold separately and, in this case, bidding 0 leads to a tie for expected payoff is 12 1/2 while bidding 10 leads to certain victory for payoff 15. For the same reasons, b 2(B; L) = 10 in any equilibrium and the objects are sold separately when t 2 = L. If t 2 = H, finally, bidder 2 s unique best response is to bid b 2(A; H) = b 2(B; H) = 0 and b 2(A, B; H) = 20 thereby winning both units for certain. This equilibrium is non-monotone since b 2(B; L) > b 2(B; H). 5 Conclusion This paper has shown why all equilibria in the first-price auction are monotone when bidders have one-dimensional affiliated types but also why some or all equilibria may be non-monotone in auctions of multiple identical objects given these same assumptions. In addition, I showed that all equilibria may be non-monotone in non-identical object auctions even given independent types. In my mind, these negative results should spur and focus future study into the important question of when and why bidders follow monotone equilibrium strategies, especially in auctions of identical objects. Example 2 shows why assuming symmetric bidders and private values is not enough to guarantee existence of a monotone equilibrium, but several other natural conjectures remain to be tested. For instance, suppose that each of two bidders has constant marginal values, v i (1; t) = v i (2; t) = v i (t), for two objects on sale. Will all (or some) equilibria in this setting be monotone and does this remain true with n > 2 bidders and k > 2 objects? If not, what if there 16

18 are common values as well, v 1 (t) = v 2 (t)? In any event, there is a reassuring sense in which non-monotone equilibria are a phenomenon restricted to settings with small numbers of bidders. Pesendorfer and Swinkels (2000) s results on asymptotic efficiency of auctions imply that any equilibrium with many 6 bidders must be monotone. Many important real-world auctions have relatively few bidders, however, and it remains an important task to build a more complete theory of multi-object auctions in such settings. This paper serves to highlight one of the more prominent holes in the current theory. Appendix Proof of Theorem 1 Let b ( ) be an equilibrium. In my notation b i (t i ) is the set of bids made by type t i with positive probability; I do not specify these probabilities since they are not needed. When necessary, b i (t i, τ i ) will refer to i s actual bid for some realization of his randomization variable τ i. The only important fact is that any bid in b i (t i ) is a best response. Let b low j be the level above which j needs to bid to win with positive probability and let b j be the level of the lowest trough in j s bid function over the range of types who win with positive probability: b low j sup{b : Pr(b max i j b i (t i, τ i )) = 0} b j inf{b b low j : max τ i if Z is empty b i (t i, τ i ) > b > min b i (t τ i i, τ i ) for t i > t i e j)} Z empty when e j = 1, i.e. j always loses or, more generally, when b j( ) is non-decreasing over the range of types who submit a bid that wins with positive probability. Consider the following conditions on bidder j s equilibrium bid (not assumptions!): 6 In their model, the number of objects and the number of bidders scales at the same rate in a sequence of auctions. The limit of equilibrium outcomes is efficient so, at least in the limit, all bidders strategies must be monotone in any model in which bidder values are strictly increasing in own type. I suspect that their results could also be leveraged to show that all equilibria must be monotone when there are n > N bidders, i.e. that monotonicity is obtained along the sequence as well as in the limit. 17

19 1. Pr(b b j(t j )) = 0 for all b > b low j. 2. e j such that Pr(b wins) = (>) 0 for all t j < (>) e j and all b b j(t j ). 3. e j < 1 implies b(e 1 ) < b(e 2 ) b j for some 1 e 2 > e 1 e j, all b(e 1 ) b j(e 2 ), and all b(e 2 ) b j(e 1 ). Conditions 1, 2, 3 are satisfied whenever b j( ) is non-decreasing and nonconstant. Condition 2 says that the set of types who win with positive probability is an interval containing the highest possible type. Condition 3 has several important implications: b low j < b j. b j( ) is non-decreasing and less than b j over the range [e j, e 1 ]. Proof: b j(e 1 ) b j(t j ) for all t j < e 1, else b j b j(e 1 ) by definition of b j. Similarly, b j(t j) < b j(t j ) for some e j t < t e 1 implies that b j b j(t ) b j(e 1 ). Pr(b j(t j ) < b) > 0 for all b (b j(e 1 ), b j ). The set {t j : b j(t j ) b} is a non-empty lower-comprehensive set for all b (b low j, b j ). The proof proceeds as follows. (Part 1) Any equilibrium in which each bidder s equilibrium bid satisfying conditions 1, 2, 3 must be monotone over the range of all types that win with positive probability. (Part 2) Any equilibrium must satisfy these conditions. Part 1: Let b > b, t 1 > t 1 be such that both bids win with positive probability, b b 1, type t 1 finds b to be a best response, and type t 1 finds b to be a best response. I need to show that this leads to a contradiction, since then b 1 > b 1 or b 1 = b 1 by the definition of b 1. This will complete part 1 of the proof, since the same argument applies to each bidder and so b i = for all i. There are two cases to consider in which both b, b win with positive probability: (a) Pr(b wins) > 0 and Pr(b wins, b loses) = 0, ruled out since all types strictly prefer b to b. (b) Pr(b wins) > 0, Pr(b wins, b loses) > 0. The rest of the proof addresses this case. For each bidder j, let t + j (b) be the set of types at which j s bid crosses b from below and t j (b) the set of types at which j s bid crosses b from 18

20 above. More precisely, t + j (b) consists of types t j such that b j(t j ε) < b and b j(t j +ε) > b for all small enough ε > 0. (By condition 1, b j( ) is not constant over any interval, so all crossings occur at specific types.) Elements of t j (b) are defined similarly. For any x, the elements of t + j (x), t j (x) clearly alternate: there can not be two crossings from below without a crossing from above in between them. Furthermore, the minimal element of t + j (x) is less than the minimal element of t j (x) (when the latter is non-empty) for all x b. This is because each bidder j must bid less than b (and hence less than x) with positive probability but all bids less than b j must be made on the lowest set of types, over which j s bid function is monotone non-decreasing. Finally, by definition of b 1, for all x < b 1 it must be that t + j (x) = 1 (and there are no crossings from above for such x) for all j 1. For simplicity I will treat t + j (b) as having finitely many elements. (Extending the proof to the countable case is straightforward.) By convention, for b > max tj b j(t j ) define t + j (b) = 1. If b j( ) s last crossing of x is from above, then there are equally many crossings from below and from above; by convention in this case, let t + j (x). This guarantees that t+ j (x) = t j (x) + 1. Let t + j (x) = {t1+ j,..., t (k+1)+ j } and t j (x) = {t1 j,..., t k j } be ordered labellings of these sets. Observe that (up to a zero measure set) b < b j(t j ) < b iff t j k m=1xj m b j(t j ) < b iff t j Xj 0 wherexj 0 (0, t + j (b)), Xj 1 (t + j (b), t1+ j (b )), Xj m (t m j (b ), t (m+1)+ j (b )) for 1 < m k As shorthand, let X m Π j 1 k m=0x m j j for all m = (m 2,..., m n ) {0,..., k} n 1. Bidder 1 wins with bid b iff t 1 X 0 and wins with bid b iff t 1 m X m. The expected incremental return to b versus b, as a function ( ) of 1 s type, equals (x) Pr(X 0 x)(b b ) + m 0 Pr(X m x)e [v 1 (t) b x, X m ]. Since type t 1 finds b to be a best response, (t 1 ) 0 and the expected payoff to b is non-negative. I will show that (t 1) > 0 for all t 1 > t 1. Note first that Milgrom and Weber (1982) s Theorem 5 ( MW ) implies that E [v 1 (t) b t 1, X m ] E [v 1 (t) b t 1, X m ] 19

21 for all m and in fact that this inequality is strict since v 1 (t) is strictly increasing in t 1. Thus, using the fact that Pr(b wins, b loses) > 0, (t 1 ) < Pr(X 0 t 1 )(b b ) + Pr(X m t 1 )E [v 1 (t) b t 1, X m ] = E [φ(t 1 ) t 1 ] where φ(t 1 ) (b b ) if X 0 (k 2,k 3 ) (0,0) E [v 1 (t) b t 1, X m ] if X m (t 1) = E [φ(t 1 ) t 1], so it suffices to show that φ(t 1 ) is non-decreasing since then we may again apply MW to conclude that E [φ(t 1 ) t 1] E [φ(t 1 ) t 1 ]. For any m 0 and t 1 X m, t 1 X 0, φ(t 1) = E [v 1 (t) b X m, t 1] E [ ] v 1 (t) b X 0, t 1 = b b + E [ ] v 1 (t) b X 0, t 1 b b = φ(t 1 ) The first inequality follows from MW and the second from the working assumption that type t 1 finds b to be a best response (and hence has nonnegative expected payoff). Finally, for all m > m 0 in the product order and t 1 X m, t 1 X m, φ(t 1) φ(t 1 ) follows from yet another application of MW. Part 2: Conditions 1, 2, 3 must be satisfied in any equilibrium. Condition 11: by standard arguments Pr(b b i (t i )) = 0 for all b > b low j. Condition 2: Also standard. Suppose that t i wins with positive probability and t i > t i. t i must get positive payoff from its bid implying that its expected utility from winning is positive. Since types are affiliated, MW implies that t i gets weakly higher expected utility conditional from winning with this same bid, so it must get positive payoff in equilibrium as well and hence win with positive probability. Condition 3: Define I 0 {i : b i ( ) violates condition 2} I 1 {i : b i ( ) satisfies condition 2} Let i 1, i 2 I 0. When e i = 1, condition 3 is satisfied trivially. Else, by 20

22 the definition of b i, there exists some type t i > e i such that b i (t i ) b i. 7 By condition 2, it must be that Pr(b i wins) > 0 for all i. On the other hand, note that Pr(min{b i1, b i2 } wins) = 0 since Pr(b i 1 (t i1 ) > b i1 ) = Pr(b i 2 (t i2 ) > b i2 ) = 1. This is a contradiction unless #(I 0 ) 1. Suppose finally that {i} = I 0. Since all other bidders are following strategies satisfying conditions 1, 2, 3, part 1 of the proof implies that bidder i s incremental expected payoff from bidding b i versus any b > b i has strict single-crossing in t i. This provides the contradiction since b i (t i) b i, b i (t i ) b > b i. References Athey, S. (2001): Single Crossing Properties and the Existence of Pure- Strategy Equilibria in Games of Incomplete Information, Econometrica, 69(4), Ausubel, L., and P. Cramton (1998): Demand Reduction and Inefficiency in Multi-Unit Auctions, U. Maryland Working Paper, Available at Jackson, M., and J. Swinkels (2001): Existence of Equilibrium in Single and Double Private Value Auctions, manuscript, Available at Kazumori, E. (2002): Toward a Theory of Strategic Markets with Incomplete Information: Existence of Isotone Equilibrium, manuscript. McAdams, D. (2001): Monotone Equilibrium in Multi-Unit Auctions, Manuscript, MIT Sloan, Available at mcadams. (2002): Isotone Equilibrium in Games of Incomplete Information, MIT Sloan Working Paper # , Available at mcadams. Milgrom, P., and R. Weber (1982): A Theory of Auctions and Competitive Bidding, Econometrica, 50(5), It is possible instead that there is a sequence of types t i (ε) converging to t i > e j with b i (t i(ε)) converging to a set containing b. But one can show that the strict single-crossing property holds when comparing any bid in the neighborhood of b i with b. To shorten the proof, I leave out these details. 21

23 Pesendorfer, W., and J. Swinkels (2000): Efficiency and Information Aggregation in Auctions, American Economic Review, 90(3), Reny, P., and S. Zamir (2002): On the Existence of Pure Strategy Monotone Equilibria in Asymmetric First-Price Auctions, Available at 22