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MIT Sloan School of ManagementWorking Paper 4249-02
July 2002
BIDDING LOWER WITH HIGHERVALUES IN MULTI-OBJECT AUCTIONS
David McAdams
© 2002 by David McAdams. All rights reserved. Shortsections of text, not to exceed two paragraphs, may be quoted without explicit permission
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Bidding Lower with Higher Values inMulti-Object Auctions
David McAdams∗
MIT Sloan School of Management
This version: July 17, 2002First draft: June 2002.
AbstractMulti-object auctions differ in an important way from single-object
auctions. When bidders have multi-object demand, equilibria can ex-ist in which bids decrease as values increase! Consider a model withn bidders who receive affiliated one-dimensional types t and whosemarginal values are non-decreasing in t and strictly increasing in owntype ti. In the first-price auction of a single object, all equilibria aremonotone (over the range of types that win with positive probability)in that each bidder’s equilibrium bid is non-decreasing in type. Onthe other hand, some or all equilibria may be non-monotone in manymulti-object auctions. In particular, examples are provided for the as-bid and uniform-price auctions of identical objects in which (i) somebidder reduces his bids on all units as his type increases in all equi-libria and (ii) symmetric bidders all reduce their bids on some unitsin all equilibria, and for the as-bid auction of non-identical objects inwhich (iii) bidders have independent types and some bidder reduceshis bids on some packages in all equilibria. Fundamentally, this dif-ference in the structure of equilibria is due to the fact that payoffsfail to satisfy strategic complementarity and/or modularity in thesemulti-object auctions.
∗I thank Susan Athey for starting me to thinking about monotone equilibria in auctionsand Phil Reny for catching an error in a previous version. E-mail: [email protected]: MIT Sloan School of Management, E52-448, 50 Memorial Drive, Cambridge, MA02142
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1 Introduction
To what extent does single-object auction theory extend to auctions of mul-tiple objects? Some basic insights drawn from the single-object theory cer-tainly continue to apply. For example, Ausubel and Cramton (1998) showthat bidders with private values in the uniform-price auction of identicalobjects generally bid below their true value. Bid-shading extends to themulti-unit setting because its logic extends: bidders will shade their bidsdown whenever those bids may determine their own payment upon winning.The main point of this paper, however, is that other basic single-object in-sights do not extend to the multiple-object setting. In particular, I focus onthe issue of whether bidders adopt monotone strategies and how this relatesto strategic complementarity and modularity of payoffs.
For illustration purposes, consider a simple model in which two risk-neutral bidders compete in a first-price auction. Bidder i’s valuation isgiven by vi(t), where t is a vector of affiliated, one-dimensional bidder typesand vi is non-decreasing in t and strictly increasing in ti. Let πi(b, t) =(vi(t) − bi)1{i wins} represent bidder i’s ex post payoff. In this case, it iswell-known that bidder i always has a non-decreasing best response strategywhenever j adopts a non-decreasing strategy. (See Athey (2001).) Less wellappreciated is that, ultimately, this property derives from the fact that expost payoff πi satisfies a form of strategic complementarity, single-crossing inbi, bj (shorthand SC(bi, bj)), as well as modularity in bi. (See page 5 of theintroduction for definitions of these and other terms.)
To see why πi(b; t) satisfies SC(bi, bj) (for all states t), consider for ex-ample the incremental return to b′i = 80 versus bi = 60 as a function ofopponent’s bid. When bj < 60, then either bid wins, so i gets negativeincremental return 60 − 80; when bj ∈ (60, 80) i gets incremental returnvi(t) − 80; when bj > 80 i is obviously indifferent between the bids sincethey both lose; and when bj = 60 or 80, i’s incremental return is an averageof the returns in these regions. Thus, πi(b
′i, bj; t) ≥ πi(bi, bj; t) implies that
πi(b′i, b
′j; t) ≥ πi(bi, b
′j; t) whenever b′i > bi, b′j > bj. An implication of this
fact is that πi(bi, bj(tj); t) satisfies SC(bi, tj) (for all ti and all non-decreasingopponent strategies bj(·)). Figure 1 graphs the incremental return to b′i = 80versus bi = 60 as a function of opponents’ type.1 Finally, when combined withan assumption of affiliated types, this implies in the two-bidder case that in-
1I thank Susan Athey for explaining this set of ideas to me using this sort of figure.
2
t2
f(80, t2)
−f(60, t2)
−20
b−1j (60) b−1
j (80)
b
b
bb
r
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Figure 1: SC(b1, t2) of f(b1, t2) = π1(b1, b2(t2), t1, t2)
terim expected payoff Etj |ti [πi(bi, bj(tj), t)|ti] satisfies SC(bi, ti) whenever theother bidder follows a non-decreasing strategy. (Athey (2001) Theorem 7.)
Armed with this “single-crossing condition”, Athey (2001) proves in thecase of n = 2 bidders that a monotone pure strategy equilibrium exists inthe 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 amonotone pure strategy equilibrium in auctions of multiple identical objectswith n bidders and multi-dimensional types but only in the case of indepen-dent types and risk-neutral bidders. Reny and Zamir (2002) prove existenceof monotone pure strategy equilibrium in the first-price auction with affil-iated types n > 2 bidders with affiliated types.2 In this setting, Athey’ssingle-crossing condition fails. When others follow non-decreasing strategies,a bidder may prefer b′ to b when he has type ti but prefer b to b′ when hehas type t′i for some bid levels b′ > b and types t′i > ti. Reny and Zamir(2002) show, however, that this is not possible if type ti gets non-negativeexpected payoff from the higher bid b′. This observation allows them to applyAthey (2001)’s method of proof of existence since that method only requires
2Reny and Zamir (2002) also provide an example with 3 bidders having multi-dimensional affiliated types in which the unique equilibrium is non-monotone.
3
that single-crossing holds for pairs or bids b′ > b and pairs of types t′i > tisuch that b′ is a best response for type ti, given non-decreasing strategies byothers. They call this single-crossing condition BR-SCC since it only applieswhen the higher bid is a best response for the lower type.
t2
b2
b′ wins, b loses@@I ���b wins6
b2
b′
b
��
��
���
��
���
���
���
���
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���
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Figure 2: Bids b′, b against non-monotone b2(·)
In Section 2, I prove that all equilibria in the first-price auction must bemonotone over the range of types who win with positive probability in thegeneral affiliated values model with n bidders and one-dimensional types.(A mixed-strategy equilibrium is monotone iff the highest bid played withpositive probability by a lower type is less than or equal to the highest bidplayed with positive probability by a higher type.) In particular, for eachbidder i there exists some type t̂i such that all types above (below) t̂i win withpositive (zero) probability and bidder i’s strategy is monotone over the rangeof all types greater than or equal to t̂i. At the heart of this positive result ismy observation that single-crossing holds for all pairs b′ > b and t′i > ti suchthat b′ is a best response for type ti 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 thiscondition NM-SCC since it applies even when others follow non-monotonestrategies. NM-SCC is a strengthening of Reny and Zamir (2002)’s BR-SCCcondition since b−i = ∞ whenever others follow non-decreasing strategies.
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Figure 2 illustrates the lowest trough b2 in bidder 2’s non-monotone bidfunction; see the Appendix for a formal definition. The essence of the proof isto exploit this limited single-crossing property to show that the lowest troughof any given bidder’s equilibrium bid function must be strictly higher thanthe lowest trough of some other bidder’s bid function. This of course leadsto a contradiction unless all bidders are following non-decreasing strategies.
On the other hand, in Section 3, I show why McAdams (2001) and Kazu-mori (2002)’s results for multiple identical-object auctions can not possiblyextend to cover the case of affiliated types (even with private values) by pro-viding examples in which all equilibria are non-monotone. In Examples 1and 3, some bidder reduces his bids on all units as his type increases in allequilibria. In Example 2, symmetric bidders all reduce their bids on someunits in all equilibria. And in Example 4 of Section 4, some bidder reduceshis bids on the individual objects in all equilibria given independent typesand non-identical objects. Fundamentally, as discussed next in Section 1.1,these negative examples flow from the fact that bidders’ ex post payoffs failto satisfy SC(bi, b−i) in identical object auctions and also fail to be modu-lar in own bid in non-identical object combinatorial auctions. Lastly, in thepaper’s conclusion, I share my thoughts on where I believe these negativeresults 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 inT . 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 ofelements x′, x are being compared and with respect to which pair of elementst′, t. The more standard “single-crossing in (x, t)” requires SC(x′, x; t′, t) forall comparable pairs x′, x and t′, t:
5
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
g(x′, t) ≥ g(x, t) ⇒ g(x′, t′) ≥ g(x, t′)
for all x′ > x ∈ X and all t′ > t ∈ T .
Definition (Modularity in x). Let X be a lattice. g : X → R is modularin x iff
g(x′) + g(x) = g(x′ ∨ x) + g(x′ ∧ x)
for all x′, x ∈ X.
Note that when X is a sublattice of Euclidean space, modularity in x =(x1, ..., xk) is equivalent to additive separability in x1, ..., xk. Modularity ofpayoffs in bi is trivially satisfied in single-object auctions since bids are one-dimensional.
Payoffs are not modular in own bid in combinatorial auctions becauseincreasing one’s bid on a bundle will typically make one more likely to winthe bundle the more that one simultaneously lowers bids on individual units.Thus, one can not apply McAdams (2002)’s monotone equilibrium existencetheorem. (Quasisupermodularity in own bid also fails.) As mentioned ear-lier, payoffs are modular in own bid in most commonly studied auctions ofidentical objects (given risk-neutral bidders). As a consequence, when biddertypes are independent a monotone equilibrium exists and if bidder values arealso strictly increasing in own type, then one can prove that all equilibria aremonotone over the range of all types who win with positive probability. Evenin this case, however, the conditions of the monotone equilibrium existencetheorem may fail when bidder types are positively or negatively correlated.The reason for this is that payoffs fail to satisfy single-crossing in own bidand others’ bids. The purpose of most of the examples in the paper is toillustrate why this property fails. Since a graphical intuition is helpful, Iinclude here an example drawn from McAdams (2001).
Figure 3 shows why ex post payoffs fail to satisfy single-crossing for(D′
i(·), Di(·); D′j(·), Dj(·)) 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/ordiscrete price grid, and price-elastic supply. Other multi-unit auctions suchas the pay-as-bid auction also fail to possess strategic complementarity.
6
quantity
price
v
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��
��
��
��
��
��
��
��
�� D1(·)
D′1(·)
S −D′2(·) S −D2(·)
Figure 3: Single-crossing in own bid and others’ bids fails in uniform-priceauction
Example. For some type profile t, bidder 1 has a constant marginal valuev for shares, i.e. v1 (q; t) = qiv. S − D′
2(·), S − D2(·) are the residualsupply curves that would result if bidder 2 submits the bid D′
2(·) or D2(·).The unlabelled curves in Figure 3, finally, are isoprofit curves of bidder 1.Thus, D1(·) is strictly preferred to D′
1(·) when bidder 2 submits D′2(·) whereas
D′1(·) is strictly preferred to D1(·) when bidder 2 submits D2(·). This violates
SC(D′i(·), Di(·); D′
j(·), Dj(·)) of bidder 1’s ex post payoff.
2 First-Price Auction
First-price auction model: There are n bidders and one object. Each bid-der has type ti ∈ [0, 1] and a randomization variable τi ∈ [0, 1], where tis affiliated, τ is independent of t, and the joint density of (t, τ) has fullsupport on [0, 1]n. (τi need not be independent of τj.) In particular, thisimplies that for any T ⊂ [0, 1]2n−2, Pr(t−i, τ−i ∈ T ) > 0 implies thatPr((t−i, τ−i) ∈ T |ti, τi) > 0 for all ti, τi. Each bidder’s utility upon losingis normalized to zero and upon winning has the form ui(vi(t)−bi), where thevaluation vi is non-decreasing in t and strictly increasing in ti and utility ui
is strictly increasing in surplus. The set of permissible bids is {∅}∪ [pmin,∞)where bidding {∅} ensures that one will never win the object. Ties are bro-
7
ken randomly when the high bid is at least pmin and the auction is cancelledif the high bid is {∅}.
Theorem 1. A monotone pure strategy equilibrium exists in the first-priceauction and every equilibrium is monotone over the range of types who winwith positive probability. Specifically, if b̂ is played with positive probabilityby type t̂i, Pr(̂b wins) > 0, t′i > ti > t̂i, and b′, b are played with positive
probability by types t′i, ti (respectively), then b′ ≥ b ≥ b̂.
Proof. The proof is in the Appendix.
Reny and Zamir (2002) prove existence of a monotone pure strategy equi-librium, 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 equi-libria with 3 bidders and consider only a special case in which several simplify-ing conditions are satisfied. Suppose that b∗(·) is a pure-strategy equilibriumwith (a) no atoms in i’s bid, i.e. Pr(b∗i (ti) = b) = 0 for all b, (b) all but thelowest bids win, i.e. b > infti b∗i (ti) implies that Pr(b > maxj 6=i b
∗j(tj)) > 0,
and (c) Pr(bi > b∗i (ti)) > 0, where
bi ≡ inf{b : b∗i (t′) < b, b∗i (t) > b for some t′i > ti}
≡ ∞ if this set is empty
bi 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 toassuming (c1) b∗i (t) < bi 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 thatbi > minj 6=i bj ≡ b−i whenever b−i < ∞. Thus, b1 = b2 = b3 = ∞ and anyequilibrium must be monotone! Why must the lowest trough in (say) bidder1’s equilibrium bid function be higher than the lowest trough of some oneelse’s equilibrium bid function? Recall the usual trade-off between bidding b′
versus b when others are following monotone strategies : (a) bidding b′ leadsbidder 1 to pay more in the event “b wins” and this event is a rectangle inthe lower corner of T−1 = [0, 1]n−1; (b) bidding b′ leads him to win sometimeswhen 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-offis very similar. The event “b wins” is still a rectangle in the lower corner ofT−1 and the event “b′ wins” still has a product structure, although it is not
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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 followa strategy with a single trough as in Figure 2 such that (i) b∗j(tj) ∈ [0, b) forall tj ∈ [0, t1j), (ii) b∗j(tj) ∈ (b, b′) for tj ∈ (t1j , t
2j) ∪ (t3j , t
4j), and (iii) b∗j(tj) > b′
for tj ∈ (t2j , t3j) ∪ (t4j , 1].
t2t12 t22 t32 t42
t3
t13
t23
t33
t43
X0,0
X0,1
X1,0
X1,1
X0,2 X1,2
X2,0
X2,2
X2,1
Figure 4: X0,0 = “b wins”. ∪0≤m2,m3≤2Xm2,m3 = “b′ wins”
Standard arguments prove that, when all others follow monotone strate-gies and bidder 1’s valuation v1(t) is strictly increasing in t1, bidder 1’s payoffhas strict single-crossing for (b′, b; t′1, t1) whenever b′, b both win with positiveprobability and t′1, t1 find b′, b to be best responses, respectively. Such a prop-erty implies that all of bidder 1’s best response strategies are monotone overthe range of types that win with positive probability. The key steps in theproof are applications of standard facts about functions of affiliated randomvariables, in particular Theorem 5 of Milgrom and Weber (1982) (hereafter“MW”). The crucial elements of the problem that allow one to apply MWare 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
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in which others follow non-monotone strategies, as long as the lower bid b isless than or equal to the lowest trough in others’ bid functions. Thus, it isimpossible that b′ and b ≤ b−1 both win with positive probability and t′1, t1find b′, b to be best responses, respectively. By the definition of b1, then,b1 > b−1.
3 Auctions of Identical Objects
Identical objects model: The examples in this section vary but they all fit intothe following framework. There are n risk-neutral bidders and two identicalobjects being auctioned. Each bidder has type ti ∈ [0, 1] and t is affiliated.Each bidder’s marginal values vi(1, t), vi(2, t) are non-decreasing in t.3 Insome examples, the set of permissible bids is a finite grid while in others bidsare allowed on a continuum and the tie-breaking rule varies.
Note also that I maintain the assumption of one-dimensional, affiliatedtypes. These assumptions are less natural in the multi-unit context4 but use-ful from an expository point of view. When types are negatively correlated ormulti-dimensional, the first-price auction sometimes possesses non-monotoneequilibria for very natural reasons. (See Jackson and Swinkels (2001) for anexample with negative correlation, Reny and Zamir (2002) for an exam-ple with multi-dimensional affiliated types.) Maintaining these assumptionshelps me to isolate important differences between single- and multi-objectauctions, namely the non-monotonicity of equilibria that is caused by thefailure of payoffs to satisfy strategic complementarity and/or modularity.
Finally, one may easily concoct examples of non-monotone equilibria inwhich the non-monotonicity occurs because some bidder is indifferent to sub-
3Marginal 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 corre-sponding to a range of types in the unit interval (while maintaining affiliation). Over thisrange, values are constant in own type.
4Bidder types may be multi-dimensional if different information is relevant to the valueof different objects or (in the identical object case) to initial versus later units, amongother reasons. For example, suppose that a buyer may consume one unit himself at utilityt1i and/or resell any units that he wins at price t2i . The affiliation assumption may alsobecome less natural in that t1i , t
2i may be negatively correlated (even if ti, tj are positively
correlated). In an auction of rare coins, for instance, it seems plausible that dealers whoexpect to be able to resell the coins at a high price may have a relatively low personalwillingness to pay for the coins. Such dealers may have a relatively large inventory andhence already possess a personal collection that satisfies them.
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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! Thesimplest example that I have found applies to the so-called uniform N +1-stprice auction in which price equals the highest losing bid. (Examples 1 and2 each apply to the uniform N + 1-st price auction which does not extendthe first-price rule. Similar logic applies in the as-bid and uniform N -thprice auctions, however, which each do extend the first-price rule. See alsoExample 3.)
Example 1 (Uniform N + 1-st price auction). There are two identicalobjects for sale and two bidders. Bidder 1 has unit demand and privatevalue v1 ∈ [0, 1]. Bidder 2 has (constant) marginal value v2 = 1 for bothunits and receives a signal s2 ∈ {H, L} that is informative about 1’s value.In particular, the conditional density of v1 given s2 is f(v1|s2 = L) = 2−2v1,f(v1|s2 = H) = 2v1.
Claim. An equilibrium in weakly undominated strategies exists and all suchequilibria are non-monotone in this example.
Proof. Bidder 1 bids v1 on the first unit and zero on the second unit andbidder 2 bids his true value v2 = 1 on a first unit in any equilibrium inweakly undominated strategies. (Note however that bidder 2’s first unit bidnever sets the price.) Bidder 2 has two basic options for his bid on thesecond unit: (a) concede one unit to bidder 1 and bid 0, thereby winningone unit at price 0, or (b) attempt to win two units and bid b > 0, therebysometimes winning one unit at price b (if v1 > b) and other times winningtwo units at per unit price v1 (if v1 < b). Under option (a), bidder 2’s profitequals 1. Under option (b), bidder 2’s profit and marginal return to a highersecond-unit bid are
πi(b, s2) =
∫ b
0
(2− 2v1)f(v1|s2)dv1 + (1− b) Pr(v1 > b|s2)
∂πi/∂b(b, s2) = (2− 2b)f(b|s2)− (1− b)f(b|s2)− Pr(v1 > b|s2)
When s2 = L, f(v1|L) = 2−2v1, Pr(v1 > b|L) = (1−b)2, and ∂πi/∂b(b, L) =(1 − b)2 > 0 for all b < 1. Thus, b∗2(L) = (1, 1). On the other hand,
11
when s2 = H, f(v1|H) = 2v1, Pr(v1 > b|H) = 1 − b2, and πi(b, H) =−b3/3 + b2 − b + 1 < 1 = πi(0, H) for all b ∈ (0, 1]. Thus, b∗2(H) = (1, 0).Note that this non-monotone equilibrium is the unique equilibrium in weaklyundominated strategies.
The key feature driving Example 1 is that the multi-unit demand bidderfaces a decision about whether to bid with an eye toward winning both unitsor just one unit (i.e. “conceding”). Given the particular structure of theexample – two goods, one opponent who only wants a single unit – it isclear that conceding becomes relatively more attractive the higher that theopponent bids. Since types are affiliated and the opponent’s bid is increasingin its type, then, it is natural that the two-unit demander’s bid may fall withhis own type, even if higher own type were also to somewhat increase hisown value for the objects.
One feature of Example 1 is that bidder 2 can “lock in” a unit by makingthe minimal possible bid, but this is not crucial. Suppose that we add anadditional unit-demand bidder (call it bidder 0) who has private value v0.Let fv(x|s2) be the conditional density of v ≡ max{v0, v1} and fv(x|s2) theconditional density of v ≡ min{v0, v1}. Then 2’s profit as a function of hisbid b on the second unit is
πi(b, s2) =
∫ b
0
(2− 2x)fv(x|s2)dx + (1− b) Pr(b ∈ (v, v)|s2)
+
∫ 1
b
(1− x)fv(x|s2)dx
∂π/∂b = (2− 2b)fv(b|s2) + (1− b)(fv(b|s2)− fv(b|s2))
− Pr(b ∈ (v, v)|s2)− (1− b)fv(b|s2)
= (2− 2b)fv(b|s2)− (1− b)fv(b|s2)− Pr(b ∈ (v, v)|s2)
(As always in the uniform-price auction in which price equals highest losingbid, bidder 2’s weakly dominant strategy is to bid his true value on thefirst unit.) Note that the term representing bidder 2’s marginal return toincreasing b differs from that in the previous example only in that the densityfor v1 is replaced by the density for v and that Pr(v1 > b|s2) is replaced byPr(b ∈ (v, v)|s2). As long as the distribution of v0 is heavily weighted enoughby values near zero, then, the logic of the previous example carries throughwithout modification.
Another feature of Example 1 is that the bidders are asymmetric, butthis also is not crucial. More important is that, as the unit demander’s
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type increases, the difference between his equilibrium bid on the first unitand the second unit increases. As this difference in one’s opponents’ bidsincreases, it increases the relative attractiveness of conceding a unit. In theN +1-st uniform-price auction, it also tends to increase the relative likelihoodthat one’s second-unit bid will set the price, thereby further increasing one’sincentive to concede a unit. These points are illustrated with the followingsymmetric example in the N + 1-st uniform-price auction.
Example 2 (Symmetric bidders). Two bidders receive types t1, t2 ∈ [0, 1].vi(1, ti) = v1(2, ti) = 200ti for all ti ∈ [0, 1/2). vi(1, ti) = 200ti and vi(2, ti) =100 for all ti ∈ (1/2, 1]. The joint density of t is given by f(t) = 2 whent1, t2 > 1/2 or t1, t2 < 1/2 and f(t) = 0 otherwise. In other words, t1 >(<)1/2 implies that t2 > (<)1/2 and t are independent conditional on bothbeing greater than or less than 1/2.
Claim. An equilibrium in weakly undominated strategies exists and all suchequilibria are non-monotone in this example.
Proof. Conditional on t1, t2 < 1/2 or > 1/2, t1, t2 are independent. Thus,Kazumori (2002) proves that an equilibrium exists in “sub-auctions” in whichthe types are less than or greater than 1/2. And any combination of equi-libria in these sub-auctions is an equilibrium in the auction. Furthermore,it is straightforward to prove that an equilibrium in undominated strategiesexists.5
In any equilibrium, b∗1(1, t1), b∗2(1, t2) ≥ 100 and b∗1(2, t1) = b∗2(2, t2) ≤ 100
for all t1, t2 > 1/2. (If the second-unit bids are not tied with probability oneconditional on types greater than 1/2, then some bidder could sometimeslower the uniform-price – without otherwise altering the auction outcome –by lowering his bid.) Furthermore, b∗1(2, t1) = b∗2(2, t2) = 0 in any equilibriumin weakly undominated strategies. On the other hand, it is straightforwardto show that b∗i (2, ti) > 0 in any equilibrium when ti < 1/2.
5Consider variations in which bidders submit random bids with positive probability.Kazumori (2002) proves existence of isotone equilibria in each sub-auction given suchtrembling. A limit of such equilibria, as the probability of trembling goes to zero, is anequilibrium in undominated strategies without trembling. (A limit exists because the setof isotone strategies is compact in the weak topology and this limit is an equilibrium sinceexpected payoffs are continuous with respect to this topology. See Athey (2001) for atemplate of this argument in the context of the first-price auction.)
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What about the pay-as-bid auction and the version of the uniform-priceauction in which price equals the lowest winning bid? As one probablyexpects by now, similar examples can be constructed in which some or allequilibria are non-monotone in these auctions as well. Calculating equilibriabecomes more cumbersome in these auctions since it is no longer true thata multi-unit bidder will bid truthfully on the first unit. I provide here anexample 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 orin the uniform N -th price auction. Each bidder i wins at least one unitif bi(1) > bj(2) and two units if bi(2) > bj(1); ties are broken randomly.The set of permissible bids is {0, 10, 20} and each bidder receives a perfectlycorrelated signal in the set {L, H}. Bidder i’s valuation for a kth unit givensignal s will be denoted vi(k, s).
v1(1, L) = v1(2, L) = 0, v1(1, H) = 24, v1(2, H) = 0
v2(1, s) = v2(2, s) = 24 for each s
Claim. An equilibrium in weakly undominated strategies exists in both thepay-as-bid and the uniform N-th price auction and all such equilibria arenon-monotone.
Proof. Pay-as-bid: Clearly, b∗1(1, L) = b∗1(2, L) = b∗1(2, H) = 0. After eithersignal, 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 forprofit 3/2(v2(s) − 0) > 2(v2(s) − 10). Similarly, after either signal bidder2’s unique best response is (10, 10) if bidder 1 bids (0, 0). After (0, 0) hewins an expected 1 units for profit 24, after (10, 0) he wins an expected 11/2 units for profit 36− 10, after (10, 10) he always wins two units for profit2(24− 10). After signal H, bidder 1’s best response is to bid (10, 0) as longas b∗2(2, H) < 20. Bidding (0, 0) rather than (10, 0) leads bidder 1 to loseentirely if b∗2(2, H) = 10 and to decrease expected winnings from 1 to 1/2 ifb∗2(2, H) = 0, but 24− 10 > 1/2(24− 0). Thus, in the unique equilibrium inweakly 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 isstill (0, 0) if bidder 1 bids (10, 0) or (20, 0). One difference, however, is that
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now bidder 2’s best response is to bid (10, 0) if bidder 1 bids (0, 0) since3/2(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 thateach bidder has valuations for objects A, B separately and together, vi(A; ti),vi(B; ti), vi(A, B; ti).
Existence of non-monotone equilibria is perhaps not so surprising in com-binatorial auctions of non-identical objects. After all, a bidder with a lowtype may only bid seriously on an individual object whereas with a highertype he may bid with an eye to winning a bundle. In this case, it may makesense to withdraw one’s bid on individual objects to increase the chancesof winning the bundle. The logic of non-monotonicity here ultimately relieson the fact that a bidder’s payoff is not additively separable in his bids onvarious bundles: increasing one’s bid on the bundle will typically make onemore likely to win the bundle the more that one simultaneously lowers bidson individual units. Consequently, while some sort of correlation of types isnecessary for all equilibria to be non-monotone in the identical objects case,it is quite easy to construct examples with non-identical objects in whichtypes are independent and all equilibria are non-monotone.
Example 4. There are two bidders in the pay-as-bid combinatorial auc-tion of two objects A, B. Each bidder i submits bids on all combinations:bi(A), bi(B), bi(A, B) ∈ {0, 10, 20, ...}. If
maxi
bi(A) + maxi
bi(B) > (<) maxi
bi(A, B),
then the goods are allocated separately (bundled) to the relevant high bid-ders with ties broken randomly; if these two allocations bring in the same
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revenue, then the method of allocation is chosen randomly, again with sub-sequent ties broken randomly. All of the bidders have additive valuationfunctions. Bidder 1 is a single-object demander and doesn’t receive any pri-vate information (his values are known): v1(A) = 25, v1(B) = 0. Bidder 2may want both objects and receives signal t2 ∈ {L, H}, each with probability1/2. v2(A, L) = 0, v2(B, L) = 25 whereas v3(A, H) = v3(B, H) = 100.
Claim. An equilibrium in weakly undominated strategies exists and all suchequilibria are non-monotone.
Proof. Obviously b∗1(B) = b∗2(A; L) = 0 in any equilibrium. Also, v2(·, H)is chosen to be so high that bidder 2 certainly wins both objects when histype is high. Thus, b∗1(A) = 10 in any equilibrium: 1 only wins if t2 = Land the objects are sold separately and, in this case, bidding 0 leads to atie for expected payoff is 12 1/2 while bidding 10 leads to certain victoryfor payoff 15. For the same reasons, b∗2(B; L) = 10 in any equilibrium andthe objects are sold separately when t2 = L. If t2 = H, finally, bidder 2’sunique best response is to bid b∗2(A; H) = b∗2(B; H) = 0 and b∗2(A, B; H) = 20thereby winning both units for certain. This equilibrium is non-monotonesince b∗2(B; L) > b∗2(B; H).
5 Conclusion
This paper has shown why all equilibria in the first-price auction are mono-tone when bidders have one-dimensional affiliated types but also why someor all equilibria may be non-monotone in auctions of multiple identical ob-jects given these same assumptions. In addition, I showed that all equilibriamay be non-monotone in non-identical object auctions even given indepen-dent types. In my mind, these negative results should spur and focus futurestudy into the important question of when and why bidders follow monotoneequilibrium strategies, especially in auctions of identical objects. Example2 shows why assuming symmetric bidders and private values is not enoughto guarantee existence of a monotone equilibrium, but several other naturalconjectures remain to be tested. For instance, suppose that each of two bid-ders has constant marginal values, vi(1; t) = vi(2; t) = vi(t), for two objectson sale. Will all (or some) equilibria in this setting be monotone and doesthis remain true with n > 2 bidders and k > 2 objects? If not, what if there
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are common values as well, v1(t) = v2(t)? In any event, there is a reassur-ing sense in which non-monotone equilibria are a phenomenon restricted tosettings with small numbers of bidders. Pesendorfer and Swinkels (2000)’sresults on asymptotic efficiency of auctions imply that any equilibrium withmany6 bidders must be monotone. Many important real-world auctions haverelatively few bidders, however, and it remains an important task to build amore complete theory of multi-object auctions in such settings. This paperserves 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 (ti) is the set of bids madeby type ti with positive probability; I do not specify these probabilities sincethey are not needed. When necessary, b∗i (ti, τi) will refer to i’s actual bid forsome realization of his randomization variable τi. The only important fact isthat any bid in b∗i (ti) is a best response.
Let blowj be the level above which j needs to bid to win with positive
probability and let bj be the level of the lowest “trough” in j’s bid functionover the range of types who win with positive probability:
blowj ≡ sup{b : Pr(b ≥ max
i6=jb∗i (ti, τi)) = 0}
bj ≡ inf{b ≥ blowj : max
τi
b∗i (ti, τi) > b > minτi
b∗i (t′i, τi) for t′i > ti ≥ e∗j)}
≡ ∞ if Z is empty
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 withpositive probability.
Consider the following conditions on bidder j’s equilibrium bid (not as-sumptions!):
6In their model, the number of objects and the number of bidders scales at the samerate in a sequence of auctions. The limit of equilibrium outcomes is efficient so, at leastin the limit, all bidders’ strategies must be monotone in any model in which bidder valuesare strictly increasing in own type. I suspect that their results could also be leveragedto show that all equilibria must be monotone when there are n > N bidders, i.e. thatmonotonicity is obtained along the sequence as well as in the limit.
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1. Pr(b ∈ b∗j(tj)) = 0 for all b > blowj .
2. ∃ e∗j such that Pr(b wins) = (>) 0 for all tj < (>) e∗j and all b ∈ b∗j(tj).
3. e∗j < 1 implies b(e1) < b(e2) ≤ bj for some 1 ≥ e2 > e1 ≥ e∗j , allb(e1) ∈ b∗j(e2), and all b(e2) ∈ b∗j(e1).
Conditions 1, 2, 3 are satisfied whenever b∗j(·) is non-decreasing and non-constant. Condition 2 says that the set of types who win with positiveprobability is an interval containing the highest possible type. Condition 3has several important implications:
• blowj < bj.
• b∗j(·) is non-decreasing and less than bj over the range [e∗j , e1]. Proof:b∗j(e1) ≥ b∗j(tj) for all tj < e1, else bj ≤ b∗j(e1) by definition of bj.Similarly, b∗j(t
′j) < b∗j(tj) for some e∗j ≤ t < t′ ≤ e1 implies that bj ≤
b∗j(t′) ≤ b∗j(e1).
• Pr(b∗j(tj) < b) > 0 for all b ∈ (b∗j(e1), bj).
• The set {tj : b∗j(tj) ≤ b} is a non-empty lower-comprehensive set for allb ∈ (blow
j , bj).
The proof proceeds as follows. (Part 1) Any equilibrium in which each bid-der’s equilibrium bid satisfying conditions 1, 2, 3 must be monotone over therange of all types that win with positive probability. (Part 2) Any equilibriummust satisfy these conditions.
Part 1: Let b′ > b, t′1 > t1 be such that both bids win with positiveprobability, b ≤ b−1, type t1 finds b′ to be a best response, and type t′1 findsb′ to be a best response. I need to show that this leads to a contradiction,since then b1 > b−1 or b1 = b−1 by the definition of b1. This will completepart 1 of the proof, since the same argument applies to each bidder and sobi = ∞ for all i.
There are two cases to consider in which both b′, b win with positiveprobability: (a) Pr(b wins) > 0 and Pr(b′ wins, b loses) = 0, ruled out sinceall 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 crossesb from below and t−j (b) the set of types at which j’s bid crosses b from
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above. More precisely, t+j (b) consists of types tj such that b∗j(tj − ε) < b andb∗j(tj +ε) > b for all small enough ε > 0. (By condition 1, b∗j(·) is not constantover 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 inbetween them. Furthermore, the minimal element of t+j (x) is less than theminimal element of t−j (x) (when the latter is non-empty) for all x ≥ b. Thisis because each bidder j must bid less than b (and hence less than x) withpositive probability but all bids less than bj must be made on the lowest setof types, over which j’s bid function is monotone non-decreasing. Finally, bydefinition of b−1, for all x < b−1 it must be that |t+j (x)| = 1 (and there areno crossings from above for such x) for all j 6= 1.
For simplicity I will treat t+j (b) as having finitely many elements. (Ex-tending the proof to the countable case is straightforward.) By convention,for b > maxtj b∗j(tj) 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 conven-tion 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 , ..., tk−j } be ordered labellings
of these sets. Observe that (up to a zero measure set)
b < b∗j(tj) < b′ iff tj ∈ ∪km=1X
mj
b∗j(tj) < b iff tj ∈ X0j whereX0
j ≡ (0, t+j (b)),
X1j ≡ (t+j (b), t1+
j (b′)), Xmj ≡ (tm−
j (b′), t(m+1)+j (b′)) for 1 < m ≤ k
As shorthand, let Xm ≡ Πj 6=1∪km=0X
mj
j for all m = (m2, ...,mn) ∈ {0, ..., k}n−1.Bidder 1 wins with bid b iff t−1 ∈ X0 and wins with bid b′ iff t−1 ∈ ∪mXm.
The expected incremental return to b′ versus b, as a function 4(·) of 1’stype, equals
4(x) ≡ Pr(X0|x)(b− b′) +∑m6=0
Pr(Xm|x)E [v1(t)− b′|x, Xm] .
Since type t1 finds b′ to be a best response, 4(t1) ≥ 0 and the expectedpayoff to b′ is non-negative. I will show that 4(t′1) > 0 for all t′1 > t1. Notefirst that Milgrom and Weber (1982)’s Theorem 5 (“MW”) implies that
E [v1(t)− b′|t′1, Xm] ≥ E [v1(t)− b′|t1, Xm]
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for all m and in fact that this inequality is strict since v1(t) is strictly in-creasing in t1. Thus, using the fact that Pr(b′wins, b loses) > 0,
4(t1) < Pr(X0|t1)(b− b′) +∑
(k2,k3) 6=(0,0)
Pr(Xm|t1)E [v1(t)− b′|t′1, Xm]
= E [φ(t−1)|t1] where
φ(t−1) ≡ (b− b′) if X0
≡ E [v1(t)− b′|t′1, Xm] if Xm
4(t′1) = E [φ(t−1)|t′1], so it suffices to show that φ(t−1) is non-decreasing sincethen we may again apply MW to conclude that E [φ(t−1)|t′1] ≥ E [φ(t−1)|t1].
For any m 6= 0 and t′−1 ∈ Xm, t−1 ∈ X0,
φ(t′−1) = E [v1(t)− b′|Xm, t′1]
≥ E[v1(t)− b′|X0, t′1
]= b− b′ + E
[v1(t)− b|X0, t′1
]≥ b− b′ = φ(t−1)
The first inequality follows from MW and the second from the working as-sumption that type t′1 finds b to be a best response (and hence has non-negative expected payoff). Finally, for all m′ > m 6= 0 in the productorder and t′−1 ∈ Xm′
, t−1 ∈ Xm, φ(t′−1) ≥ φ(t−1) follows from yet anotherapplication of MW.
Part 2: Conditions 1, 2, 3 must be satisfied in any equilibrium. Condition11: by standard arguments Pr(b ∈ b∗i (ti)) = 0 for all b > blow
j . Condition 2:Also standard. Suppose that ti wins with positive probability and t′i > ti.ti must get positive payoff from its bid implying that its expected utilityfrom winning is positive. Since types are affiliated, MW implies that t′i getsweakly higher expected utility conditional from winning with this same bid,so it must get positive payoff in equilibrium as well and hence win withpositive probability.
Condition 3: Define
I0 ≡ {i : b∗i (·) violates condition 2}I1 ≡ {i : b∗i (·) satisfies condition 2}
Let i1, i2 ∈ I0. When e∗i = 1, condition 3 is satisfied trivially. Else, by
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the definition of bi, there exists some type ti > e∗i such that b∗i (ti) 3 bi.7 By
condition 2, it must be that Pr(bi wins) > 0 for all i. On the other hand, notethat Pr(min{bi1
, bi2} wins) = 0 since Pr(b∗i1(ti1) > bi1
) = Pr(b∗i2(ti2) > bi2) =
1. This is a contradiction unless #(I0) ≤ 1. Suppose finally that {i} = I0.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 frombidding bi versus any b′ > bi has strict single-crossing in ti. This providesthe contradiction since b∗i (t
′i) 3 bi, b∗i (ti) 3 b′ > bi.
References
Athey, S. (2001): “Single Crossing Properties and the Existence of Pure-Strategy Equilibria in Games of Incomplete Information,” Econometrica,69(4), 861–889.
Ausubel, L., and P. Cramton (1998): “Demand Reduction and Ineffi-ciency in Multi-Unit Auctions,” U. Maryland Working Paper, Available atwww.cramton.umd.edu.
Jackson, M., and J. Swinkels (2001): “Existence of Equilibrium inSingle and Double Private Value Auctions,” manuscript, Available athttp://masada.hss.caltech.edu/jacksonm.
Kazumori, E. (2002): “Toward a Theory of Strategic Markets with Incom-plete Information: Existence of Isotone Equilibrium,” manuscript.
McAdams, D. (2001): “Monotone Equilibrium in Multi-Unit Auctions,”Manuscript, MIT Sloan, Available at http://www.mit.edu/˜mcadams.
(2002): “Isotone Equilibrium in Games of Incomplete In-formation,” MIT Sloan Working Paper #4248-02, Available athttp://www.mit.edu/˜mcadams.
Milgrom, P., and R. Weber (1982): “A Theory of Auctions and Com-petitive Bidding,” Econometrica, 50(5), 1089–1122.
7It is possible instead that there is a sequence of types ti(ε) converging to ti > e∗j withb∗i (ti(ε)) converging to a set containing b. But one can show that the strict single-crossingproperty holds when comparing any bid in the neighborhood of bi with b′. To shorten theproof, I leave out these details.
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Pesendorfer, W., and J. Swinkels (2000): “Efficiency and InformationAggregation in Auctions,” American Economic Review, 90(3), 499–525.
Reny, P., and S. Zamir (2002): “On the Existence of Pure StrategyMonotone Equilibria in Asymmetric First-Price Auctions,” Available athttp://www.src.uchicago.edi/users/preny.
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