Albano2006-Japanese Eng Ascending Auction B JOURNAL

7/29/2019 Albano2006-Japanese Eng Ascending Auction B JOURNAL

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Rev. Econ. Design (2006) 10:18DOI 10.1007/s10058-006-0006-z

O R I G I N A L PA P E R

Retaliatory equilibria in a Japanese ascending auction

for multiple objects

Gian Luigi Albano Fabrizio Germano Stefano Lovo

Received: 8 February 2005 / Accepted: 23 January 2006 /

Published online: 23 June 2006 Springer-Verlag 2006

Abstract We construct a family of retaliatory equilibria for the Japaneseascending auction for multiple objects and show that, while it is immune tomany of the tacitly collusive equilibria studied in the literature, it is not entirelyimmune when some bidders are commonly known to be interested in a specificobject.

Keywords Ascending auctions for multiple objects Clock auctions FCC auctions Collusion Retaliation

JEL Classification C72 D44

1 Introduction

Simultaneous ascending auctions for multiple objects have been widely used for

over a decade, most notably in the sale of spectrum licenses by the Federal Com-munications Commission (FCC) in the US, but also by corresponding agencies

G. L. AlbanoDepartment of Economics and ELSE, University College London,WC1E 6BT London, UK

F. Germano (B)

Departament dEconomia i Empresa, Universitat Pompeu Fabra,08005 Barcelona, Spaine-mail: [email protected]

B. To look at review & biblio


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2 G. L. Albano et al.

throughout Europe and in several other countries. In spite of the different vari-ations adopted, these auction formats typically ensure a transparent biddingprocess leading to extensive information revelation of bidders valuations andreasonably efficient aggregations of the objects among the bidders. However,

the more transparent the bidding process the easier it is for bidders to exploitthe auction rules and to (tacitly) communicate and coordinate on collusiveoutcomes.1

When few bidders are competing for few objects, the incentives to reducedemands and hence prices may be large. Engelbrecht-Wiggans and Kahn (2005)and Brusco and Lopomo (2002) show that, for auction formats close to theFCCs, there are equilibria where bidders can coordinate a division of the avail-able objects at very low prices. On the other hand, Albano et al. (2001, 2006)study a variant of the simultaneous ascending auction, namely the Japanese

auction for multiple objects (JAMO), which is basically a clock version of theFCC auction mechanisms and differs from them in that (a) prices are exog-enously raised by the auctioneer, and (b) closing is object-by-object. Amongother things, they show that because of these two features all the collusive low-revenue equilibria constructed by Engelbrecht-Wiggans and Kahn (2005) andBrusco and Lopomo (2002) are not possible in the JAMO.

This note shows that collusive equilibria of a retaliatory type, (related to onesreported in Cramton and Schwartz 2000, 2002), nonetheless exist in the JAMO.Their logic is as follows. Suppose two objects are put for sale to two bidders, a

bundle bidder interested in both objects, and a unit bidder interested only inobject 1. Assume this to be common knowledge. The two bidders have overlap-ping interests on object 1; in particular, the unit bidder wants the bundle bidderto exit early from object 1. In order to achieve this, the unit bidder actively bidson object 2, though he has zero value for it. Such a strategy is potentially costlyto both the unit and the bundle bidder; we refer to the unit bidders behavior asretaliatory strategy. The extent to which the unit bidder is successful in inducingthe bundle bidder to drop early from object 1 depends on whether he succeedsin making his threat credible. We show that the JAMO admits equilibria with

such strategies.

2 A Japanese ascending auction

2.1 Framework

Two objects are auctioned to a set of participants of two types: M 1 bundlebidders who are interested in both objects and one unit bidder interested inobject 1 only. Bundle and unit bidders draw their values independently from

some smooth distribution Fwith positive densityf, both defined over [0, 1]. Let


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Retaliatory equilibria in a Japanese ascending auction for multiple objects 3

vk and uk denote the value of object k = 1, 2 to a bundle and to a unit bidderrespectively, with u2 = 0. The value of the bundle vB to a bundle bidder isvB = v1 + v2. The nature of bidders, bundle and unit, is also commonly known.

2.2 The auction mechanism

We consider a Japanese (or English clock) auction for multiple objects. Pricesstart from zero on all objects and are simultaneously and continuously increaseduntilonlyoneagentisleftonagivenobject,inwhichcasethepriceonthatobjectstops and continues to rise on the remaining ones. Once an agent has droppedfrom a given object, the exit is irrevocable. Whenever an agent exits one object,the clock (price) temporarily stops on both objects giving the opportunity toother bidders to exit at the same price. The last agent receives the object at theprice at which bidding on that object stopped. The number and the identity ofagents active on any object is publicly known at any given time. The overallauction ends when all agents but one have dropped out from all objects. Werefer to this mechanism as the Japanese auction for multiple objects (JAMO);this auction is also studied in Albano et al. (2001, 2006).

3 Retaliatory equilibria

Our results are summarized as follows.

Proposition 1 There exist Perfect Bayesian Equilibria of the JAMO where bid-

ders use retaliatory strategies effectively.

We show this by means of the following two examples, presented in order ofgenerality.

Example 1 Consider the JAMO with two objects and two bidders; one unitbidder interested in object 1 and one bundle bidder interested in both objects

1 and 2, with the same value for the two objects, v1 = v2 = x; assume alsoall values are drawn according to a cumulative distribution function F on [0, 1]

satisfying1

0 udF(u) 1/2. The following constitutes a PBE of the JAMO:

All types of the unit bidder bid on both objects and stay on object 1 until u1and on object 2 until min(u1, t

21), where t

21 is the bundle bidders exiting time

from object 1; All types of the bundle bidder exit from object 1 at t if at t the unit bidder

is active on object 2; otherwise all types of the bundle bidder stay on both

objects until x; Out-of-equilibrium-path beliefs: if at time t > 0 the bundle bidder is activeon object 1 then the unit bidder holds that Pr(x > u1) = 1


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the unit bidders strategy, at t = 0 the bundle bidders expected payoff fromstaying on both objects untilx is equal to 2

x0 F(u)du, whereas if she exits object

1 at t = 0, her payoff is x. Let us now check that1

0 udF(u) 1/2 is a necessary

and sufficient condition for 2 x0

F(u)du x to hold for allx [0, 1]. First, noticethat

2

x

0

F(u)du x x(F(x) 1/2)

x

0

udF(u).

To see necessity (only if), note that x(F(x) 1/2) is bounded above by 1/2.

Thus the inequality implies that1

0 udF(u) 1/2. To see sufficiency (if), notethatx(F(x)1/2) is positive for allx > x, where x is such that F(x) = 1/2. More-over, the first derivative ofx(F(x) 1/2) is greater than the one of

x0 udF(u)

for all x > x. Thus1

0 udF(u) 1/2 is a sufficient condition for x(F(x) 1/2) x0 udF(u) to hold for all x > x.

Finally, suppose that, contrary to the equilibrium strategy, the bundle bidderis active on both objects at t > 0. The stated out-of-equilibrium-path beliefsensure that it is weakly optimal for the unit bidder to stay on both objects atmost until u1 as long as the bundle bidder does not exit object 1. Indeed, givensuch out-of-equilibrium beliefs, the unit bidder expects the bundle bidder towin object 2 for sure.

As is often typical in such retaliatory equilibria, the retaliating bidder (herethe unit bidder) obtains a higher ex ante payoff than in the competitiveequilibrium2 (1/2 vs. 1/6, for F uniform), while the other agents (here the bun-dle bidder and the auctioneer) are both worse off (1/2 vs. 2/3 and 0 vs. 1/3,respectively, for F uniform).

The above example relies on the fact that the unit bidder has some extrainformation about the bundle bidders valuation of object 1 relative to object2 (the values are perfectly correlated) and on the fact that in expectation his

valuation for object 1 is sufficiently large to make the threat of retaliation credi-ble. Without this information and for a more general distribution ofu1, the unitbidder needs to resort to a more refined threat.

Example 2 Consider the case of a unit bidder competing against an arbitrarynumber M of bundle bidders with private values distributed according to ageneral distribution function F defined on [0, 1]. Take l (0, 1] and let c < lbethe unique solution to the equation

l

z G(z)dz = c G(l), (1)


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where G = FM is the distribution function for the bundle bidders highest val-uation for object 1 (and also for 2). Then, for any l (0, 1], the following is aPBE of the JAMO:

All types of unit bidder with u1 l bid only on object 1 and stay until u1;all types of unit bidder with u1 > lbid on both objects and stay on object 1until u1 and on object 2 until c;

If the unit bidder is active on the two objects, then all types of bundle bidderwith v1 < lbid on both objects and stay on object 1 until c and on object 2until v2; all types of bundle bidder with v1 l bid on both objects alwaysstaying until v1, v2, respectively;

If the unit bidder is active only on object 1 then all types of bundle bidderstay until v1, v2 on object 1 and 2 respectively;

Out-of-equilibrium-path beliefs: If a bundle bidder quits object 1 at t < c when the unit bidder is active

on both objects, the other bidders hold that Pr(v2 > c) = 1; If the unit bidder is active on both objects and quits object 2 at t < c, the

bundle bidders hold that Pr(1 u1 1) = 1, for small.

This characterizes a family of retaliatory equilibria indexed by the parameterl that are PBE of the JAMO. Note that the equilibria are not in undominatedstrategies, since the unit bidder always has a (weakly) dominant strategy to drop

from object 2 whenever it is the only object he is bidding on. If the unit bidderis active on both auctions this signals that his valuation is above the threshold l,that is, u1 > l; if he bids only on object 1, then u1 l, and all bidders bid up totheir valuations and only on the objects they value. Unlike the equilibrium ofExample 1, here, to ensure incentive compatibility for the unit bidder, he mustbid on object 2 up to the threshold c. This is costly for the unit bidder as he winsthe object he does not value (object 2) with positive probability. The thresholdc is chosen such that only a unit bidder with u1 > l has an incentive to paythis cost. Thus, by remaining active on both objects, the unit bidder provides a

credible signal that he actually has a high valuation for object 1. Note that as thenumber Mof bundle bidders increases, the probability that the unit bidder winsobject 2 during the retaliatory phase decreases. As a consequence the thresholdc must increase with M.

When l = 1 we get the competitive equilibrium, since with probability onethe unit bidder will not be active on object 1. When l 0 we almost getthe competitive equilibrium, since c 0, that is, the unit bidder enters bothauctions but almost immediately exits object 2.

These examples are related to the collusive equilibria of Brusco and Lopomo(2002) in the sense that bundle and unit bidders have overlapping interests onobject 1; the unit bidder threatens to retaliate (i.e., to be active) on object 2 if


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triggers the beliefs that sustain the retaliatory equilibrium. Unlike Brusco andLopomo, these equilibria are implementable in the JAMO and thus do notrequire actual rounds of information exchanges. In particular, bidders do not(and cannot) raise prices on most preferred objects as in Brusco and Lopomo.

In the JAMO, in order to signal the retaliatory strategy the unit bidder has tobe active on both objects, thus having to adopting a more costly strategy, whichis also likely to be overall less costly for the auctioneer.

Proof of Example 2 We first check optimality for any bundle bidder, then wecheck it for the unit bidder. If the unit bidder is active on both objects, bundlebidders infer that u1 > l. Hence, a bundle bidder with v1 lknows that she willnever pay for object 1 less than what she values this object. Therefore, exiting

object 1 at time c and exiting object 2 at v2 is a (weak) best reply for such abundle bidder. If, however, v1 > l, then a bundle bidder is better off remainingon each object so long as her expected continuation payoff remains strictlypositive. Namely, she will stay on objects 1 and 2 until v1 and v2, respectively.Suppose now that, contrary to the equilibrium strategy, the bundle bidder wereto exit object 1 before c in order to induce the unit bidder to immediately exitobject 2. In this case, if the unit bidder believes (out of the equilibrium path)that v2 > c, then he nonetheless finds it weakly optimal to exit object 2 at c.Hence, exiting object 1 before c is not a profitable deviation for the bundle

bidder. This proves that the bundle bidders strategy is a best reply to the unitbidders strategy.To prove optimality for the unit bidder, we need to show that the unit bid-

ders strategy is a best reply and that it is profitable for the unit bidder to bidon both objects if and only ifu1 > l, that is, that the equilibrium is incentivecompatible, so that being active on both objects gives a credible signal thatu1 > l. Let yk [0, 1] denote the bundle bidders highest valuation for objectk {1, 2}, and let G denote the corresponding distribution function. Recall thatthe bundle bidder with the highest valuation for object 2 will exit auction 2before c only ify2 < c. When the unit bidder is not active on object 2 and stillactive on object 1, the bundle bidder with the highest valuation for object 1 willexit auction 1 at y1. When the unit bidder is active on object 2, ify1 l, thenthe bundle bidder with the highest valuation for object 1 will exit auction 1 atc;whereas ify1 > land the unit bidder is still active on object 1, then the bundlebidder with the highest valuation for object 1 will exit auction 1 at y1.

Ifu1 < c,thenitisclearlynotoptimalfortheunitbiddertobidonbothobjectssince he will have to pay at leastc forobject1.Hencewefocusonthecase u1 c.Suppose that u1 [c, l]. If the unit bidder decides to implement the retaliatory

strategy, then his expected payoff is l0

(u1 c) G(y1)dy1 c

0

y2 G(y2)dy2.

The first integral is the unit bidders payoff from object 1: the unit bidder winsobject 1 only ify1 < l, (recall that u1 l) and he pays c. The second integral


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At equilibrium we want the unit bidder to bid only on object 1 when u1 l,that is, the following needs to be satisfied

l0

(u1 c) G(y1)dy1

c0

y2 G(y2)dy2

u10

(u1 y1) G(y1)dy1,

which is always satisfied when c solves Eq. (1); (recall that y1 and y2 have thesame distribution function G).

Suppose now u1 > l. Then, at t = 0, the unit bidders expected payoff fromthe retaliatory strategy must be greater or equal than the payoff from biddingonly on object 1, that is,

l

0

(u1 c) G(y1)dy1 +

u1

l

(u1 y1) G(y1)dy1

c

0

y2 G(y2)dy2

u1

0

(u1 y1) G(y1)dy1.

It is easy to check that the above inequality is satisfied for any l (0, 1]. Con-

sider now any time t < c. It must be optimal for the unit bidder to insist on hisretaliatory strategy, that is, to exit object 2 at c rather than before c. Supposethat the unit bidder deviates and exits object 2 at t (0, c). Given the bundlebidders out-of-equilibrium-path beliefs, a weakly dominant continuation strat-egy for a bundle bidder with typev1 following a deviation of the unit bidder is toexit object 1 at = max{1 , v1}. Indeed, even ifv1 < 1 , the bundle bidderexpects not to have to buy object 1 at 1 since she anticipates that the unitbidder will exit at u1 > L. Therefore the expected payoff for the unit bidderfrom quitting the retaliatory strategy at t (0, c) and continuing optimally on

object 1 can be fixed arbitrarily close to 0 by choosing sufficiently close to 0.This expected payoff must be smaller than the unit bidders expected payofffrom insisting with the retaliatory strategy, that is,

l

0

(u1 c) G(y1)dy1 +

u1

l

(u1 y1) G(y1)dy1

c

t

y2 G(y2|y2 > t)dy2 > 0 (2)


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we need to show that

(l c)G(l)(1 G(t))

c

t

z G(z)dz > 0. (3)

By integrating by parts the left-hand side of Eq. (1) and after rearranging terms

we have cG(c) = (c l)G(l) +l

c G(z)dz. Substituting this last expression into(3), integrating by parts and simplifying, we have that the left-hand side of (3)is equal to

l

t

G(z)dz (l c)G(t)G(l) + tG(t) >

l

c

G(z)dz (l c)G(c) > 0,

where the two inequalities follow from t < c < l and G being positive, strictlyincreasing and bounded by 1.

Acknowledgments We are grateful to Philippe Jehiel for useful discussions. Germano acknowl-edges financial support from the Spanish Ministry of Science and Technology grant SEJ2004-03619and in form of a Ramn y Cajal Fellowship, as well as from BBVA grant Aprender a jugar. Thesupport of the Economic and Social Research Council (ESRC) is also gratefully acknowledged.The work was part of the programme of the ESRC Research for Economic Learning and Social

Evolution.

References

Albano GL, Lovo S, Germano F (2001) A Comparison of standard multi-unit auctions with syner-gies. Econ Lett 71:5560

Albano GL, Lovo S, Germano F (2006) Ascending auctions for multiple objects: the case for theJapanese design. Econ Theory 28:331355

Brusco S, Lopomo G (2002) Collusion via signaling in open ascending auctions with multiple objectsand complementarities. Rev Econ Stud 69:130

Cramton P, Schwartz J (2000) Collusive bidding: lessons from the FCC spectrum auctions. J RegulEcon 17:229252

Cramton P, Schwartz J (2002) Collusive bidding in the FCC spectrum auctions. Contrib Econ AnalPolicy 1:111

Engelbrecht-Wiggans R, Kahn C (2005) Low revenue equilibria in simultaneous ascending priceauctions. Manag Sci 51:356371

Salmon T (2004) Preventing collusion between firms in auctions. In: Janssen MCM (ed) Auctioningpublic assets: analysis and alternatives. Cambridge University Press, Cambridge


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Albano2006-Japanese Eng Ascending Auction B JOURNAL