Journal of Mathematical Economics 48 (2012) 51–58

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Journal of Mathematical Economics

journal homepage: www.elsevier.com/locate/jmateco

Defensive sniping and efficiency in simultaneous hard-close proxy auctionsGreg Taylor ∗

Oxford Internet Institute, University of Oxford, 1 St Giles, Oxford, OX1 3JS, UK

a r t i c l e i n f o

Article history:Received 30 March 2011Received in revised form23 September 2011Accepted 14 November 2011Available online 8 December 2011

Keywords:Hard-closeSimultaneous proxy auctionsEfficiencyMyopic biddingBelief-free

a b s t r a c t

A well-known myopic bidding strategy fails to support an equilibrium of simultaneous ascending proxyauctions for heterogeneous items when a hard-close rule is in place. This is because, in common withthe single-auction case, last minute bidding (sniping) is a best response to naive behaviour. However,a modification to the myopic strategy in which all bidders submit an additional bid in the closingstages of the auction – a practice I call ‘defensive sniping’ – is shown to yield an efficient, belief-freeequilibrium of such environments. This equilibrium is essentially unique within the class of belief-free,efficient equilibria.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

This paper is concerned with the question of how bidders cancoordinate in a process of price discovery, whilst still discouragingopportunistic behaviour in hard-close simultaneous ascendingproxy auctions.1 This kind of auction has been popularised bythe Internet auction site eBay, which uses such auctions toconcurrently sell millions of items at any given time. Hard-closeproxy auctions have an obvious appeal for selling goods online—allowing a decentralised and asynchronous bidding process,whilst guaranteeing a timely end to the auction. There are alsoother contexts in which a fixed end time might be appealingsuch as when some bidders have a strategic incentive toprolong the auction, when the items for sale are perishable, orwhen uncertainty over the ending time causes complementaryinvestments to be delayed.

Auctions that have a hard-close rule often induce bidders to bidvery close to the ending time (a practice known as sniping),2 whichhas been linked to a general failure to attain efficiency.3 Given the

∗ Tel.: +44 01865 287201.E-mail address: greg.taylor@oii.ox.ac.uk.

1 An ascending proxy auction is an auction in which bidders submit a reservationprice to a proxy agent that then bids on their behalf as if it were participating in anEnglish auction—increasing the standing price by the minimum amount necessaryto win the auction (provided that this does not exceed the entered reservationvalue). Such auctions are said to be ‘hard-close’ when there exists a fixed deadlineafter which no new or revised reservation bids are accepted.2 See, for example, Ariely et al. (2005), Bajari and Hortaçsu (2003), Ely and

Hossain (2009), and Roth and Ockenfels (2002) for empirical and experimentaldemonstrations of this phenomenon.3 See Bajari and Hortaçsu (2003) and Ely and Hossain (2009).

0304-4068/$ – see front matter© 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.jmateco.2011.11.006

increasingly prominent role of this kind of auction in consumer-to-consumer and business-to-consumer electronic commerce, itis therefore important to understand if and how the efficiencyproblem can be remedied. One way to mitigate this problem inthe case of a single-item auction is to introduce a proxy biddingsystem that submits incremental bids up to some reservationprice on the bidder’s behalf: provided that all bidders submit aproxy bid equal to their true value at the start of the auction,no bidder can profitably snipe and the result will be the efficientallocation.4 The proxy solution is, however, problematic in the casethat there are multiple, heterogeneous items for sale. Bidders thenface a trade-off between winning the most desirable item (theone to which they attribute the highest value) and winning thecheapest one. Moreover, bidders will not generally agree on therelative desirability of the items. Unlike the bidding problem in thesingle-auction case, a simultaneous heterogeneous good auctiontherefore presents a non-trivial coordination problem and largepremature proxy bids risk leaving bidders committed to winningitems that subsequently turn out to be sub-optimal (in the sensethat there are alternative items that offer a better value-pricemargin). Moreover, attempts to hedge by bidding on several itemssimultaneously may result in a bidder winning multiple items forwhich he has no use.

The main contribution of this paper is to show that there existsa strategy – which I call defensive sniping – that both allowsbidders to coordinate in price discovery and deters sniping. This

4 With independent values, using proxy bids in this way ostensibly converts theprocedure into a second price auction. This notwithstanding, there are equilibria ofthe single-unit proxy auction that involve sniping.

52 G. Taylor / Journal of Mathematical Economics 48 (2012) 51–58

strategy builds upon a well-known myopic bidding rule providedby Demange et al. (1986). In particular I show that, whilst themyopic strategy is vulnerable to sniping, a modification in whichall bidders submit an additional, large bid in the closing stages ofthe auction is not. Besides using an intuitive and straightforwardbidding rule, the resulting equilibrium has the desirable propertiesof being efficient and belief-free. I show that any equilibrium withthese attractive properties is equivalent to the defensive snipingstrategy and, in particular, must have some kind of defensiveelement.

The results presented here suggest that the observed ineffi-ciency in environments such as eBay might be remedied with bet-ter bidder education. Currently, eBay advises buyers to use theproxy system as follows5:

‘‘. . .we suggest that you bid the maximum amount that you’rewilling to pay for an item. . . ’’

Left unqualified, this is problematic to the extent that it rendersprice discovery and coordination around the efficient outcomealmost impossible. Indeed, there is a sizeable body of evidencethat suggests bidders are eschewing eBay’s advice (and the proxysystem) altogether, instead bidding ‘naively’ in an incrementalfashion.6 I show that this behaviour creates opportunities forprofitable sniping. However, the message that bidders shouldaugment this naive strategy with defensive bidding is relativelystraightforward and should be easy to communicate—even torelatively unsophisticated auction participants.

This paper is most closely related to work on biddingin simultaneous, heterogeneous item auctions. In addition toDemange et al. (1986), who provide an algorithmic bidding rulethat is a starting point for the analysis, results from the broaderassignment literature are also of direct relevance. In particular,Leonard (1983) and Shapley and Shubik (1971) provide usefulresults on the properties of core allocations in such markets. Guland Stacchetti (2000) establish that the algorithmofDemange et al.(1986) forms a perfect Bayesian equilibrium of a simultaneousascending auctionwith unit demandbidderswhen there is no fixedending time.

Bajari and Hortaçsu (2004) provide a survey of a numberof theoretical and empirical results on various issues connectedto online auctions, many of which use a hard-close rule. Moreclosely related to the present paper, recent work has recognisedthe significance of concurrency in Internet auctions. Peters andSeverinov (2006) provide a theoretical model of online auctionmarkets that have concurrent auctions for a homogeneousgood. The authors model agents as arriving sequentially andincrementally bidding in a uniform-price fashion until they aredeclared the high bidder for one of the available items. In aprimarily experimental paper, Ely and Hossain (2009) modelbidding in a similar fashion. Unlike Ely and Hossain (2009) andPeters and Severinov (2006), I allow the various items for sale tobe heterogeneous, which significantly complicates the allocationand equilibrium determination problem.7

A number of empirical studies speak to the importance of thedynamic and simultaneous structure of online proxy auctions.Zeithammer and Adams (2010) provide an empirical analysis ofbidding patterns in eBay proxy auctions. They reject the sealedbid model of bidding in such auctions (which holds that these

5 See http://pages.ebay.co.uk/help/buy/bidding-overview.html (accessed 4thAugust 2011).6 See, for example, Roth and Ockenfels (2002) for survey evidence and Ely and

Hossain (2009) and Zeithammer and Adams (2010) for empirical results.7 Even where items are homogeneous, de facto heterogeneity may be introduced

by differences in seller reputation or the terms of sale.

auctions can be modelled in the abstract as a second price sealed-bid auction), and show that the data is more consistent witha mixed model that has both sealed bidding and incrementalbidding. In order to resolve the coordination problem inmymodel,bidders submit small bids across a number of items in a processof price discovery. This process also calls upon bidders to submitmore than one bid per-item. Empirical evidence from Anwar et al.(2006) supports the prediction that agents in simultaneous auctionenvironments bid across concurrent competing auctions in sucha fashion, whilst Chiang and Kung (2005) and Wilcox (2000) findevidence for multiple-bidding in proxy auctions.

2. Model

Let I = {1, 2, . . . , I} be bidders for items K = {1, 2, . . . , K}.The items are sold by simultaneous proxy auction during aninterval of time denoted t ∈ [0, T ], where T is a fixed ending time.More precisely, denote by bi (t) =

b1i (t), . . . , b

Ki (t)

the vector of

i’s bids at time t . At each t , players chose somebi(t) ∈ RK+satisfying

bi(t) ≥ bit ′

∀t ′ < t .8 Player i is named provisional winner for kat time t if bki (t) > maxj=i

bkj (t)

. Ties are broken instantaneously.

Denote k’s (unique) provisional winner at time t by wk(t). Letxi(t) =

x1i (t), . . . , x

Ki (t)

be the vector whose kth element is

equal to 1 ifwk(t) = i, and 0 otherwise. A (provisional) allocation,X(t), is the matrix formed by concatenating the xi(t) together.

Players observe the game’s history, H(t), comprised of theprovisional winner and the second highest bid for each k at each t .9The second highest bid serves as a provisional price, which I denoteby b

k(t)—that being the kth element of the vector b(t). Provisional

winners’ bids are not directly observed. Ultimately, each k ∈ K isallocated towk(T ). Prices are determined according to p = b(T ).10

Every i ∈ I has a vector of (private) valuations, vi = (v1i , . . . ,

vKi ). Each vki is drawn from some distribution F ki on a non-

degenerate, bounded support, ϕk⊂ R+. Values are allowed to be

correlated across bidders, across items, or both.11 For themost part,and unless stated otherwise, the correlation structure of values isallowed to take an arbitrary form, but on occasion it will prove use-ful to invoke one of the following assumptions.

Assumption 1. (i) vki > infϕk⇒ vki > inf

supp F k

j

vkj |vi

∀j

= i; and (ii) vki < supϕk⇒ vki < sup

supp F k

j

vkj |vi

∀j = i.

Assumption 2. supp F kj

vkj

vi = supp F kj

vkj

vmj = ϕk

∀j = i∀m = k.

In words, both of these assumptions state that bidders’ valuesare imperfectly correlated12 in the sense that (conditional onobservation of his own vi) a bidder, i, cannot typically be certainthat his value for k is above or below that of a given rival bidder.Note that Assumption 2 implies Assumption 1 and therefore

8 I use ≥ and ≤ as element-wise relations for vectors.9 The second highest bid is to be understood as meaning the highest reservation

bid submitted by someone other than the provisional winner.10 A (hidden) reserve price, rk , can be easily incorporated into the model byviewing the seller s ∈ I as a bidder who bids bks (0) = rk . A posted reserve price

can be modelled by viewing the seller as two such bidders, resulting in bk(0) = rk .

Provided that the reserve price is set in a manner consistent with efficiency, theresults below transfer straightforwardly to such a set-up.11 This paper is primarily concerned with the existence of an equilibrium that isbelief-free and is therefore quite robust to variations in the precise distributionalproperties of bidders’ values.12 Note, in particular, that Assumptions 1 and 2 encompass independent values asa special case.

G. Taylor / Journal of Mathematical Economics 48 (2012) 51–58 53

represents a stronger restriction on the correlation structure—requiring that whilst vi and vmj may be informative about therelative likelihood of different vkj , they contain no new informationabout the range of values vkj can take.

A strategy maps H(t) and vi into bids bi(t) for each t ∈

[0, T ]. Utility is quasi-linear in money, and players are not budgetconstrained. Each bidder is assumed to have unit demand so thatfinal utility is Ui = maxk

xki (T )v

ki

− p · xi(T ) ≡ ui − p · xi(T ).13

Unit demand is a natural fit under a number of circumstancessuch as when bidders are consumers bidding for durable goods, orwhen the bidders are firms bidding for indivisible inputs (such asoperating licences) and the regulator has imposed that no firmmaybuy-up more than one input in order to stimulate after-marketcompetition.

3. Bidding

Define by γ ki (t) the indicator function having value 1 for

precisely one randomly chosen element of the surplus-maximisingset of items:

Si(t) =

k : vki > b

k(t) and k ∈ argmax

vmi − b

m(t)

,

and value 0 otherwise. If Si(t) = ∅ then γ ki (t) = 0 ∀k. Thus, γi(t)

selects one of the ‘‘most appealing’’ items to i at prices b(t), wheremost appealing means having the largest difference between thebidder’s value and the current second highest bid.

Now consider the myopic bid updating strategy given bybki (0) = 0, and the following rule:

bki (t + dt)

=

bk(t)+ αk

i (t)dt

if γ ki (t)

k∈K

1 − xki (t)

= 1

bki (t) otherwise,(1)

with αki (t) > 0 a function that controls the rate of bid updating for

each i, k, t . I let dt → 0, and assume that theαki (t) are large enough

to ensure that (1) achieves terminal prices before T . In words, (1)has i increase his bid on some k provided (i) he is not currently aprovisional winner, and (ii) at the current bid level he extracts apositive surplus from k that is at least as great as that from anyother item.

At this juncture it is helpful to spend some time consideringthe form of a Walrasian (or competitive) equilibrium within thepresent framework. I use the following definition:

Definition 1 (Walrasian Equilibrium). An allocation X = x1, . . . ,xI , and a price vector p constitute a Walrasian equilibrium if (i)

i∈I xki ≤ 1 ∀ k, (ii) pk = 0 whenever

i∈I x

ki = 0, and (iii)

maxkxki (T )v

ki

− (xi · p) ≥ maxk

yki (T )v

ki

− (yi · p) for all i, and

any yi with yki ∈ {0, 1} ∀k.

These conditions respectively require that no item is allocatedmore than once, that unallocated items have a price of zero, andthat – taking the Walrasian price as given – no buyer can dobetter than his Walrasian allocation.14 Sellers play no part in thisdefinition and, consequently, condition (ii) is required in order torule out the sort of unattractive allocations in which all prices are

13 Simultaneous ascending auctions cannot implement efficiency for generalsubstitute preferences (Gul and Stacchetti, 2000).14 Note that, when bidders have unit demand, condition (iii) requires that nobidder is allocated more than one item with pk > 0. Condition (iii) also demandsthat bidders prefer their Walrasian allocation and price to the outside option ofreneging on the agreement to buy and receiving the null allocation.

prohibitive and no player has (or wants to have) any item. Withthis definition in mind, it becomes possible to introduce Lemma 1as follows:

Lemma 1. In the dt → 0 limit, the algorithm proposed in (1) termi-nates in a Walrasian allocation with corresponding supporting pricevector.

Proof. See Appendix. �

There will generally be a multitude of price vectors that arecapable of supporting a Walrasian equilibrium. For example, ifI = K = 2 and v1 = (2, 1) , v2 = (1, 2) then any p such thatp1, p2 ∈ [0, 2] and

p1 − p2 ≤ 1 supports the Walrasian equi-

librium allocation with w1= 1, w2

= 2. Given this multiplic-ity, one might ask: is there any logic to the means by which (1)chooses an equilibriumprice vector? The answer is affirmative: the(Walrasian) allocation reached by (1) is supported by a price vec-tor which is small in the strong sense that each item price, pk, isno greater than the corresponding price from any other Walrasianprice vector.15 I denote this minimum Walrasian price vector by pwhere, in particular, if there exists a k such that pk < pk then pmust produce excess demand for some k.

Lemma 2. The competitive equilibrium reached by (1) is supportedby p, the smallest price vector capable of supporting a Walrasianequilibrium in the market.

Proof. See Appendix. �

A version of this convergence result was first demonstratedby Demange et al. (1986) in the context of standard (open-ended) English auctions. This result is of special interest since p,and the corresponding Walrasian allocation have been shown byLeonard (1983) to be identical to the Vickrey–Clarke–Groves (VCG)outcome. Demange et al. (1986) did not consider the incentiveproperties of their bidding rule, but Gul and Stacchetti (2000)established that symmetric myopic bidding in a similar fashionto (1) is a perfect Bayesian equilibrium (PBE) of a simultaneousEnglish auction whenever p and the VCG price vector coincide inthis fashion.

The main implication of instituting a fixed end time, T , is thatagentsmay place a final bid (at time T ) towhich no rival has time torespond. In Lemma 4, I establish that this undermines the ability ofmyopic bidding to support an equilibrium in such an environment.I begin, though, with the following result:

Lemma 3. If p−i

is the minimum Walrasian price in a market withbuyers I \ i then p ≥ p

−i.

Proof. Ifk : pk < pk

−i

= ∅ then, with set of players I \ i, p

must produce excess demand since p−i

is a minimum Walrasianprice. Since i’s demand is non-negative anddepends only on i’s ownvaluations, p also produces excess demand for I. But then p is notan equilibrium price vector. �

Lemma 4. Under Assumption 1, the strategy defined in (1) does notconstitute an equilibrium of the hard-close proxy auction.

Proof. If all players use (1) then Ui = max0,maxk

vki − pk

.

If, instead, i abstains from bid updating during [0, T ) then (1)converges to p

−i. Thus, i can win any k at price pk

−iwith a single

15 Shapley and Shubik (1971) provide an existence proof for such a ‘smallestWalrasian price vector’ in the unit demand setting.

54 G. Taylor / Journal of Mathematical Economics 48 (2012) 51–58

snipe bid at time T to which rivals cannot respond. Lemma 3implies that abstention followed by a snipe bid is never worse than(1). To establish that (1) is not an equilibrium, it therefore sufficesthat there is some type, vi, of bidder that sometimes strictly gainsby abstaining and then sniping. The existence of such a bidder typecan be demonstrated by construction of a generic example. Fixvalues according to the following conditions:

vkl < vkj ≤ vki ∀l = j, i; vml > vmj ∀m = k ∀l = j, (2)

where k ∈ argmaxm∈Kvmi − pm

(k is i’s efficient item). It is

clear that pk = vkj and (1) thus induces Ui = vki − vkj , whilstpk

−i≤ maxl=i,j v

kl < vkj so that abstention followed by a snipe bid

is a strictly profitable deviation for iwhen (2) is satisfied. Assump-tion 1 implies that, given vki > infϕk for some k, (2) is satisfiedwith positive probability, and every i with vki > infϕk for some kobtains a strictly higher expected payoff from sniping.16 �

Players that use a myopic bidding rule in a hard-close auctionare typically described as naive since they behave as one mightexpect them to do in a standard English auction, withoutacknowledging the strategic considerations introduced by thefixed ending time. Such naive behaviour is open to exploitation, asdemonstrated in Lemma 4; indeed, Ockenfels and Roth (2006) andEly and Hossain (2009) have shown in a less general setting thatsniping (last-minute bidding) is a best response to such behaviour.It is not immediately obvious that this should be true in theheterogeneous good environment. In particular, bidders using (1)adjust their bidding pattern in response to shifts in the relativeprices of the items in a manner that depends upon their vi, andit is not clear whether i can construct some sequence of bidsthat manipulates the algorithmic substitution across items andthe evolution of prices so that he is better off than when sniping.Proposition 5 establishes that he cannot. Specifically, let puresniping be the strategy that has a bidder update his bid only at timeT , bidding bki (T ) = vki on one k ∈ Si(T − dt). The following resultobtains:

Proposition 5. Pure sniping is a best response to the naive myopicbidding given in (1).

Proof. When using a pure sniping strategy against rivals who use(1), i is able to buy any k ∈ K at a price of pk

−iso that his payoff

is max0,maxk

vki − pk

−i

. Suppose that i uses some other

strategy. Clearly, if i abstains from updating his bid during [0, T ),he can do no better than max

0,maxk

vki − pk

−i

, regardless of

his choice of time T bid. Suppose, instead, that i uses some strategythat has him bid during [0, T ) (possibly in addition to bidding attime T ), whilst his rival bidders continue to use (1). Denote byX the allocation that results when i uses his deviation strategyduring [0, T ) but abstains from bid updating at time T , and bypthe associated price vector. Thus i extracts a surplus that is, atmost,equal to max

0,maxk∈K

vki −pk.17

16 Note that if the vki are atomlessly distributed on intervals of R+ then vki >

infϕk∀m = k is satisfied with probability 1 and every bidder finds that sniping

yields higher expected utility than does compliance with the putative myopicbidding equilibrium.17 The actual surplus may be somewhat less than this if, for example, i uses astrategy that calls upon him to make a large and early bid—leaving him committedto winning an item that is ultimately sub-optimal.

Now, consider the valuation vectors given by

vkω(k) =

0 ifxki = 1supϕk

+ 1 ifxki = 1,

vmI+k =

min

bmi (T ), b

m(T )

ifm = k

0 ifm = k,

with {ω(1), . . . , ω(K), I + 1, . . . , I + K} ∩ I = ∅. If bidders{ω(1), . . . , ω(K), I + 1, . . . , I + K}, having such utility functions,were to bid in i’s place, and do so according to (1) then the outcomeobtained at time T would be similar to that produced by the arbi-trary strategy used by i during [0, T ); having the same price vector,p, with i’s (provisionally allocated) items assigned instead to theω(k)s. Since all members of {ω(1), . . . , ω(K), I + 1, . . . , I + K} ∪

I \ i are using (1),p must be the minimum Walrasian equilibriumprice for the market with this set of bidders. It must, then, be truethatpk ≥ pk

−ibecause adding players can never lower the mini-

mumWalrasian price (Lemma 3). �

An important caveat attached to the behaviour described inProposition 5 is that the surplus maximising item for the sniperneed not be the same item that he would otherwise have beenallocated by the mechanism viz. his socially efficient allocation.Example 1 demonstrates how the incentive to snipe can induce thiskind of inefficiency.

Example 1. Consider valuation vectors given as follows

v1 = (10, 10) ; v2 = (5, 0) ; v3 = (5, 0) ;v4 = (0, 15) .

The efficient assignment isw1(T ) = 1 andw2(T ) = 4 with a totalsocial surplus of 25 (minimum Walrasian prices are p1 = 5, p2 =

5). However, if 1 uses the pure sniping strategy against rivals using(1) then he is faced with b (T − dt) = (5, 0) and prefers to winitem 2 at a price of zero yielding w1(T ) ∈ {2, 3}, w2(T ) = 1, anda total social surplus of 15.

Thus, bidding of the form described by (1) is vulnerableto opportunistic sniping behaviour. The behaviour described inProposition 5 and illustrated in Example 1 does not, however,constitute an equilibrium of the hard-close auction game when allplayers are fully-rational. In particular, as Example 1 shows, thoseplayers that bid algorithmically face a positive probability of beingoutbid by the sniper, evenwhen their valuation for their respectiveefficient items is higher than that of all of their rivals. The riskof surplus loss due to usurpation can be partially mitigated if thealgorithmic bidders also submit a late bid equal to their value onthe item to which they are allocated by the algorithm. I call thiskind of bidding ‘defensive’ and refer to the algorithmic strategyaugmented with a defensive element as ‘defensive sniping’18:

Definition 2. Defensive sniping is a strategy that has i bid using (1)during [0, T ), and then according to

bki (T ) =

vki ifwk(T − dt) = ibki (T − dt) otherwise

at time T .

18 It should be emphasised that the ‘late’ bid submitted as part of a defensivesniping strategy may be submitted at any t after the algorithm has converged onterminal prices. For concreteness, I take the time of this bid to be T . Introducingstochastic bidder arrival can encourage bidders to delay their defensive snipe bidby creating the possibility that a new bidder arrival will cause algorithmic biddingto restart (and the relative item prices to change).

G. Taylor / Journal of Mathematical Economics 48 (2012) 51–58 55

Returning to Example 1, if player 4 uses the defensive snipingstrategy and bids b24(T ) = 15 then, in the face of pure sniping byplayer 1, he extracts a surplus of 5 from item 4—still worse thanthe efficient allocation at p that leaves him with a surplus of 10,but non-the-less better than the outcome if he bidsmyopically andreceives zero surplus.

The adoption of the defensive sniping strategy by players j ∈ I\iundermines the advantage of the pure sniping strategy for i sincethe latter strategy exploits the general differential between thehighest algorithmic bid placed by the provisional winner for eachitem and the valuation of that same bidder. The snipe portion ofthe defensive sniping strategy has the effect of eliminating thisdifferential. In Example 1, sniping against naive rivals produces apositive increase in surplus of 5 for player 1, but using the samestrategy when all rivals switch to defensive sniping results in a fallin surplus of 5 for player 1 vis-à-vis the efficient outcome.

More generally, once players adopt the defensive snipingstrategy, i must take as given the fact that he will face a valuationbid on every k for which wk(T − dt) ∈ I \ i. Since otherplayers’ valuations are invariant to observed bidding behaviour,i’s primary strategic concern is to ensure that he is allocated hissurplus maximising item before T . There is thus hope that anefficient equilibriumwith symmetric, algorithmic strategies mightexist and indeed the existence of such an equilibrium is confirmedin Proposition 6. The proof of Proposition 6 establishes that anydeviation from defensive sniping is not only non-profitable inexpectation, but also non-profitable for any realised profile of rivaltypes. Aside from being efficient, the equilibrium characterised inProposition 6 is therefore belief-free in the following sense:

Definition 3. An equilibrium is belief-free if, starting from anarbitrary price vector, b(t), each i has a continuation strategy thatis optimal against every

vj ∈ ×k ϕ

kj∈I\i given the other bidders’

continuation strategies.

Thus, in a belief-free equilibrium, each bidder would never wishto change his strategy, even given complete knowledge of theother bidders’ types (provided that those bidders continue to playaccording to the equilibrium).

Proposition 6. Symmetric use of the defensive sniping strategyconstitutes a belief-free equilibrium of the hard-close auction game.

The proof of Proposition 6 is divided across two sub-casesdetailed in the following two lemmas. Firstly, any deviation by ithat results in

K xki = 0 is clearly weakly worse than compliance

with the defensive sniping equilibrium. Now, for concreteness,consider a deviation by i in which w1(T ) = i. Moreover, supposethat argmaxk

vki : wk(T ) = i

= 1 so that when i wins multiple

items the unit he consumes is that labelled 1. This is without lossof generality. Use a tilde to denote deviation variables, and a hat todenote the outcome that would prevail if i followed his deviationstrategy during [0, T ), but abstained from bid updating at time T .

Lemma 7. Suppose that players j ∈ I \ i use a defensive snipingstrategy and consider a deviation by i that results in w1(T ) = i. Ifw1(T ) = i or if p1 = 0 then i is no better off than if he had also useda defensive sniping strategy.

Proof. If w1(T ) = i, or ifp1 = 0, then,p1 = p1 and v1j −p1 ≤

max0,maxk=1

vkj −pk ∀j = i. Begin by considering the case

with xi(T ) = (1, 0, 0, . . . , 0). The deviation payoff for i is thenv1i −p1. Consider an artificial player, ψ , having valuations:

vkψ =

0 if k = 1supϕk

+ 1 if k = 1. (3)

Note that p is a Walrasian price in the market with ψ ∪ I \ i.Moreover, when i imitates ψ in myopic bidding against I \ i, hewins item1 at itsminimumWalrasian price in thismarket,p

ψ≤ p.

Since the VCG mechanism is incentive compatible, it follows thatmax

0,maxk

vki − pk

≥ v1i − p1

ψ≥ v1i −p1 = v1i −p1.

If

Kx ki > 1 and w1(T ) = i orp1 = 0 then the deviation

payoff for i is v1i − p1 −

K\1x ki pk, so it suffices to show that

v1i − p1 = v1i − p1 ≤ max0,maxk∈K

vki − pk

. Introduce a

collection of new bidders with valuations given by

vmI+k =

bmi (T ) ifm = k0 ifm = k, (4)

and consider the market consisting of players ψ ∪ {I + 2, . . . , I +

K} ∪ I \ i. Note thatp is a Walrasian price in this market. The proofis then completed using the same logic as above and noting thatimitating ψ in symmetric defensive sniping against rival bidders{I + 2, . . . , I + K} ∪ I \ i yields weakly lower payoffs than doesdoing so against I \ i (by Lemma 3). �

Lemma 8. Suppose that players j ∈ I \ i use a defensive snipingstrategy and consider a deviation by i that results in w1(T ) = i. Ifw1(T ) = i andp1 > 0 then i is no better off than if he had used adefensive sniping strategy.

Proof. The deviation payoff for i is v1i −p1−K\1 x

kipk. Ifw1(T ) = i

andp1 > 0 then it must be thatp1 = v1j , j = i. Now consider themarket consisting of players ψ ∪ {I + 2, . . . , I + K} ∪ I \ i (whereψ and I + k have values given in (3) and (4) respectively), who allbid according to (1). Player ψ wins item 1 at the VCG price in thismarket so that ψ ’s payment is

maxallocating K

l=ψ

ul A

− maxallocating K\1

l=ψ

ul B

= v1j + maxallocating K\1

l=ψ,j

ul C

− maxallocating K\1

l=ψ

ul B

≤ v1j ,

where the equality follows from the fact that j is assigned item 1by (1) under set of players {I + 2, . . . , I + K} ∪ I \ i. The inequalityfollows from the fact that one possibility for calculating B has

k xkj = 0, and all other player’s allocations unchanged from those

inA—whenceB = C . Itmay, however, be better to allocate to j somek ∈ K \ 1, so that C ≤ B. Now we have

max0,max

k∈K

vki − pk

≥ i′s surplus when imitating ψ in VCG against I \ i≥ i′s surplus when imitating ψ in VCG against players

{I + 2, . . . , I + k} ∪ I \ i

≥ v1i − v1j −

K\1

xkipk= i′s deviation surplus,

where the first inequality follows from the VCG mechanism’sincentive compatibility, the second from Lemma 3, and the thirdfrom the argument above. �

Thus, regardless of the configuration of rival values, anydeviation for i is outperformed by the defensive sniping strategywhen those rivals also defensively snipe. The mechanics of theproof are as follows: any deviation by a bidder that causes theauction’s outcome to change results in a surplus for that bidderthat can be bounded above by the surplus obtained by imitating an

56 G. Taylor / Journal of Mathematical Economics 48 (2012) 51–58

artificial bidder who bids according to (1) against an enlarged setof rivals. The incentive compatibility of the VCGmechanism, alongwith Lemma 3 then ensures that the deviation is non-profitable.Since p and the corresponding allocation are the products of aVCG mechanism, efficiency of the defensive sniping equilibrium isimmediate.

Corollary 9. The equilibrium described in Proposition 6 is efficient.

The belief-free nature of the defensive sniping equilibriumimplies that it is not necessary to restrict belief updating tobe consistent with H(t) in a Bayesian fashion (or, indeed, toplace any requirements on beliefs at all). This not only makesthe bidder’s strategic problem extremely simple, but also makesthe entire equilibrium more robust to mistakes by other players.Moreover, this equilibrium does not require that bidders knowthe distributions from which others’ values are drawn. The lowinformation requirements and intuitive bidding strategymake thisequilibrium particularly attractive for practical implementation.

One might ask whether there are other, substantively differentbidding behaviours that also support an efficient, belief-freeequilibrium. Proposition 10, below, and the ensuing discussionestablishes that, subject to some mild additional assumptions,every efficient, belief-free equilibrium is essentially equivalent tothe defensive sniping equilibriumdiscussed above in the sense thatit deterministically converges to the same VCG prices via myopic,incremental bidding. I say ‘essentially equivalent’ because thereare a lot of slight variations on the defensive sniping strategy thatare substantively equivalent to it. For example, if K = 2 and i’svalues are v1i > v2i then, at the start of the auction, algorithmicbidding as defined in (1) calls upon i to repeatedly incrementhis bid on the first item so long as its price is below v1i − v2i .This is essentially equivalent to submitting a single proxy bid ofb1i (0) = v1i − v2i , and allowing the proxy system to implement theincremental bidding on his behalf. I refer to such bids as pseudo-incremental since they correspond to a concise implementationof incremental bidding. These kinds of non-substantive variationsnotwithstanding, all efficient, belief-free equilibria fall into thetemplate of the defensive sniping strategy.

To establish this result it will be useful to define t ≡

inft : b(t) = b(t +∆) ∀∆ > 0

, the time at which price discov-

ery/algorithmic bidding terminates. Note that after t prices do notchange and therefore neither to the wks. It follows that, in an effi-cient equilibrium, the efficient allocation and a supporting pricemust have been reached by t and that bki (t + ∆) > b

k(t) ⇒

wk(T ) = i. The following will also be useful:

Assumption 3. Each ϕk is an interval of R+, and at least one of thefollowing two conditions is satisfied for each k: (i) infϕk

∈ ϕk, or(ii) b

k(0) = infϕk.

In words Assumption 3 says that it must be the case for eachitem either (i) that the support of the buyers’ value distributiondoes not include its lower boundary, or (ii) that there is a reserveprice equal to the minimum possible buyer valuation. Note thatthe second of these alternatives encompasses the standard case inwhich ϕk

= [0, vk] and there is no reserve price (i.e. bk

= 0).It is now possible to demonstrate that the defensive snipingequilibrium is essentially unique within the class of efficient,belief-free equilibria as follows:

Proposition 10. Under Assumptions 2 and 3, any efficient, belief-freeequilibrium of a simultaneous ascending proxy auction must (a) havep ≥ vi whenever

k x

ki = 0; (b) bemyopic in the sense that wk(t) =

i only if k ∈ Si(t); (c) be pseudo-incremental in the sense that, for allt < t, each player’s bids satisfy vki − vmi ≥ bki (t)− b

m(t − dt) ∀k,m;

and (d) be algorithmic in the sense that it deterministically convergesto price p.

Proof. To prove part (a) suppose that the converse were true,which implies that ∃k such that bki (T ) ≤ pk < vki ,

m∈K xmi = 0,

wherepk is the resulting price for item k. Moreover, if i’s strategy isbelief-free, it must always be true that bkj (T ) ≥ vki > bkh(T ) ∀h = jfor one j = i—otherwise, i would sometimes be better off withbki (T ) = vki . Now suppose that we replace jwith bidder J, where

vmJ

∈

pk, vki ifm = k< min

Ivmi ifm = k and infϕm < min

Ivmi

= infϕm ifm = k and minIvmi = infϕm

and consider the outcomewhen allJ∪I\j complywith the putativeequilibrium. Since the equilibrium is efficient by hypothesis, Jmust either be allocated no item or be allocated item m for whichvmJ = infϕm

= bm(0) (by Assumption 3), and therefore receives

a payoff of zero. However, J’s strategy cannot then be belief-freesince he can obtain a payoff of vkJ −pk > 0 against bidders I \ j byimitating j’s strategy.

Part (b) is implied by part (c). To prove part (c) suppose thatvki − bki (t) < vmi − b

m(t − dt) for some m ∈ K \ k, t < t . In

a belief-free equilibrium, every continuation bi(t)must be optimalagainst every possible {vi}I\i. Fix values so that vh ≤ b(t−dt) ∀h =

i, j; vlj ≤ bl(t − dt) ∀l = k; and vkj = min

supϕk, bki (t)

− ϵ.

Since the equilibrium price of each l = k cannot then rise andthe equilibrium price of k cannot exceed bki (t), i’s strategy is onlyoptimal if b

kis and remains below vki − vmi + b

m(t − dt), which in

turn places a ceiling on how high bkj can get. However, j’s strategycan then not be consistent with part (a) for ϵ → 0.

Now, consider the three parts of Definition 1. As in the proof ofLemma 1, parts (i) and (ii) of the definition are always satisfied inthe ascending proxy auction. Given that, by part (b) of this proof,wk(t) = i ⇒ k ∈ Si(t), every winning bidder prefers his allocationto any alternative at price p. Moreover, by part (a) of this proofall unsuccessful bidders can do no better than accepting the nullallocation at price p. Thus, part (iii) of Definition 1 is also satisfiedand an efficient, belief-free equilibriummust result in a Walrasianoutcome. The proof of Lemma 2 can now be applied to establishpart (d). �

Proposition 10 establishes that any efficient, belief-free equilib-rium bidding must be myopic in the sense that bidders only everhold the provisional winner position for items in their myopicallysurplus-maximising set, and pseudo-incremental in the sense thatthe increments in bidders’ bids are bounded above in magnitudeby the amount of surplus those bidders are able to achieve fromthe next best alternative. This result applies to simultaneous proxyauctions with or without a fixed ending time.

Of course, once the price discovery phase is over and efficientitems have been identified, defensive bidding need not beincremental. In particular, note that part (c) of Proposition 10applies only for t < t; after t bidders can use a coordination devicesuch as ‘if item prices have remained unchanged for a period oflength1t then only bid defensively on items forwhich I amalreadythe high bidder’.19 If such a strategy is in use by other bidders theni can bid up to vi on his efficient item in a belief-free fashion withimpunity after t +1t has elapsed.

Indeed, part (d) of Proposition 10 also implies that any pu-tative equilibrium bid profile that is belief-free, implements theefficient allocation, and has no defensive element (i.e. that hasbkwk(T−dt)(T ) = pk) is vulnerable to sniping in the fashion made

clear by Lemma 4 and thus cannot be an equilibrium. To yield an

19 An alternative coordination device, used in Definition 2 above, is to only biddefensively at T .

G. Taylor / Journal of Mathematical Economics 48 (2012) 51–58 57

efficient, belief-free equilibrium, any such strategy profile musttherefore be augmented with a defensive element (must havebkwk(T−dt)(T ) > pk whenever vk

wk(T−dt) > pk) when a hard-closerule is in effect. Proposition 6 showed that, for any deviation andany rival valuation profiles, a defensive bid of bk

wk(T−dt)(T ) =

vkwk(T−dt) can serve this purpose.More generally, write wk for the efficient winner of item k and

wk−for the bidder who would win k if wk were not to participate.

The surplus accruing towk is Uwk = vkwk − pk, where

pk ≡ vkwk

−

+ maxallocating K\k

l=wk,wk

−

ul − maxallocating K\k

l=wk

ul

≤0

(5)

is the minimum Walrasian (VCG) price for item k.20 If, on theother hand, wk does not participate in the equilibrium, but ratheruses the pure sniping strategy then he obtains a surplus of atleast vk

wk − bkwk

−

(T ), where bkwk

−

(T ) is the defensive bid of wk−.

For the putative equilibrium to be belief-free, it must be true thatcompliance with the equilibrium by wk is at least as profitable asthe pure sniping strategy for every configuration of rival values,which requires, in particular, that bk

wk−

(T ) ≥ pk always holds;

pk is therefore a lower bound on an effective defensive bid forwk

−in such an equilibrium, and a similar argument can be made

for the other k. However, it will typically be impossible for wk−

to calculate pk based only on observations of the b and X thatresult when wk abstains. More to the point – in keeping with thespirit of the belief-free solution concept – bidders’ defensive bidsshould not be based on calculations involving (beliefs about) otherbidders’ values. It is therefore interesting to ask: what is the lowestbkwk

−

(T ) that is guaranteed to always satisfy bkwk

−

(T ) ≥ pk, regardless

of rivals’ values? It is fairly clear from (5) that this ‘minimumdefensive bid’ is equal to vk

wk−

. Thus, whilst a defensive bidding

strategy of bkwk

−

(T ) = vkwk

−

is always sufficient to deter sniping,

any lower defensive bid would sometimes leave open profitableopportunities for such opportunistic behaviour.

Remark 11. Under Assumptions 2 and 3, every efficient, belief-free equilibrium must involve defensive bidding. The smallestdefensive bid that is always sufficient to deter sniping isbkwk(T−dt)(T ) = vk

wk(T−dt).

More generally, since any efficient equilibrium has to be payoffequivalent to the defensive sniping equilibrium (see Krishnaand Perry, 2000; Williams, 1999), the expected price in such anequilibriummust be equal to p. Thus, if winning bidders incrementthe price of the item theywin by just enough to ensure victory thenthe expected high bidwill also be equal to p, which creates gains inexpectation from sniping (more precisely, from imitation of a typev = (infϕ1, . . . , infϕK ) bidder followed by a last-minute snipebid). It follows that any efficient equilibrium that is not vulnerableto sniping must have some kind of defensive element. As in thebelief-free case, defensive bids of bk

wk(T−dt)(T ) = vkwk(T−dt) always

suffice to render this kind of deviation unprofitable in expectation.

4. Discussion and conclusion

There exists an efficient, belief-free equilibrium of simultane-ous, hard-close proxy auctions when bidders have unit demand.

20 The VCG price can be written in this way because we know thatmaxallocating K

l=wk ul = vk

wk−

+ maxallocating K\k

l=wk,wk−ul .

Thus, fixed ending times need not always be regarded with suspi-cion. Indeed, if certainty regarding the end time is of value, if theseller fears that some bidders have an interest in unduly prolong-ing the auction, or if circumstances impose a natural deadline uponproceedings, then a hard-close rule may be a viable alternativeto minimum bid increments or rules on the minimum rate of bidupdating in ensuring a timely conclusion to bidding. The equilib-rium is also robust to entry by new bidders, provided that biddersdo not arrive too late. Being belief-free, equilibrium bids followa simple rule that requires minimal knowledge and calculation,and can therefore be easily implemented. This is especially com-pelling because the leading practical implementation of ascend-ing proxy auctions is in the consumer-to-consumer e-commercespace, where bidders are less likely to undertake a full-scale strate-gic analysis of the auction. In particular, it should be fairly straight-forward for an auction operator such as eBay to communicate theefficient, belief-free strategy – and the need for defensive biddingin particular – to its participants.

It has been shown that when some bidders naively pursuea myopic strategy others will have an incentive to snipe, whichis not generally efficient. This, again, serves to emphasise theimportance of properly explaining the strategic nature of theauction to bidders. The environment considered in this article isalso relatively friction-free. Where market frictions do exist theyclearly have the potential to interfere with efficiency. For example,if some bidders arrive at the auction very close to its endingtime then they may not have sufficient time to coordinate aroundthe efficient allocation. Likewise, if bidding is costly then biddersmay not be willing to submit enough bids to induce proper pricediscovery. In either case, knowledge that this may happen distortsthe incentives of all other participants.

Acknowledgements

I am especially grateful to Robin Mason for many usefuldiscussions. Maksymillian Kwiek, David Myatt, and an anonymousreferee also provided invaluable comments and suggestions thathave materially enriched this paper. I thank the participants ofthe 63rd European Meeting of the Econometric Society (UnivesitàBocconi, Milan) and various seminars for their feedback on anearlier draft of this manuscript. The Economic and Social ResearchCouncil provided generous financial support for this project.

Appendix A. Omitted proofs

Proof of Lemma 1. Consider the three conditions in Definition 1:(i) is always satisfied by the auction mechanism since wk(t) isdefined as a singleton; to see that (ii)must be true, note that pk > 0implies bki (T ) > 0 for some subset of I and, in turn, thatwk(T ) ∈ I.Thus, by definition,

i∈I x

ki = 0. Finally, for condition (iii): if

wk(T ) = i then, from (1), when i last increased his bid on k, γ ki

was equal to 1. Since the high bid for k has not since risen, and thehigh bid for everym ∈ K \ k cannot have fallen, it must remain thecase that

maxm∈K

vmi − b

m(T )

=

vki − b

k(T )

≥ 0.

Moreover, ifwk(T ) = i ∀k ∈ K then (1) implies that γ ki = 0 ∀k—no

item can yield positive surplus to player i. Thus, with unit demand,xi can be no worse for i than any other allocation – includingthe null allocation – at prices p = b(T ), and condition (iii) issatisfied. �

Proof of Lemma 2. Let p denote the minimum Walrasian price

vector and suppose that the lemma is false. Note that bk(0) ≤ pk.

Lemma 1 tells us that (1) always produces a competitive equi-librium outcome. Thus, if (1) does not lead to an allocation sup-

58 G. Taylor / Journal of Mathematical Economics 48 (2012) 51–58

ported by p, there must be somet such that (i) bt ≤ p and (ii)

bk t + dt

> pk for some k. Denote by A the set of all such k ∈ K.

Since bk t + dt

is the second highest bid for k at timet + dt this

implies that, for each k ∈ A, at least two members of I have valua-tions such that vki − b

k t + dt

≥ vmi − bm t + dt

∀ m ∈ K \ k,

and, more specifically, such that

vki − bk t + dt

≥ vmi − b

m t + dt

∀ m ∈ K \ A. (6)

Since (6) is true for at least two i for every k ∈ A and playersnever bid (or hold the provisional winner position) on more thanone item simultaneously, it must be true for no fewer than 2 |A|

members of I.Now compare b

t + dtto p. We know that pk < b

k t + dt

∀k ∈ A and thatpk ≥ bk t + dt

∀k ∈ K\A. Thus, there are at least

2 |A| bidders with vki − pk > vmi − pm ∀m ∈ K \ A for some k ∈ A,but only |A| such items. Theremust, then, be at least one k ∈ A thatis in excess demand at prices p, and hence p is not an equilibriumprice. The necessary contradiction is thus produced. �

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