Abstract I analyze an auction environment where two units of an object are sold at twosimultaneous, sealed-bid, first-price auctions to bidders who have a one-dimensionaltype space, where a type represents the value a bidder places on each of the two units.All bidders have an identical budget constraint that binds their ability to spend in theauctions. I show that if the valuation distribution is atom-less then there does not existany equilibrium in behavioral strategies in this auction game.
Sales of offshore drilling rights and spectrum licenses are examples of auctions wheremultiple items are sold simultaneously. Invariably, a bid a participating player placeson an item/unit is affected by her bids on the other items. This could be due to, eitherlinks in the valuation of the items, or budget constraints that the players face. The aimof this paper is to explore the effect of the latter on the existence of equilibria underincomplete information in a multi-unit auction setting.
Consider the following auction: two units of a good are being sold in two simulta-neous first-price auctions. Bidders have one-dimensional types, which represent their
G. Ghosh (B)Department of Economics, Steven G. Mihaylo College of Business and Economics,California State University, Fullerton, CA 92834, USAe-mail: [email protected]
G. GhoshDepartment of Economics, Copenhagen Business Schools, 2000 Frederiksberg, Denmark
123
G. Ghosh
value for the two units. In the absence of budget constraints, this auction game hasa pure strategy equilibrium, where each bidder bids equal amounts (though possiblydifferent from other bidders) in the two auctions, according to some bid function.1
Now suppose there is a budget constraint that each bidder faces. This budgetlevel is the same for all bidders.2 In this paper, I show that the above game hasno equilibrium in behavioral strategies. The intuition behind this result is the fol-lowing. In any equilibrium the bidder with the highest valuation will place thehighest bid (which will be greater than half the budget) in one of the auctions.However, this would imply that she is bidding a small amount in the other auc-tion. As such, even though she is winning one auction for sure, she is losing theother with high probability. But then, she could increase her payoff by reducingher high bid and increasing her low bid, which shows that there is no equilib-rium.
From the examples cited at the start of the introduction, it is clear that there is aconsiderable need to study simultaneous auctions with budgets. Bulow et al. (2009)document the existence of budget constraints in the US spectrum licenses. At a the-oretical level, multi-unit auctions with budgets has not received a lot of attentionin the literature. Palfrey (1980) and Benoit and Krishna (2001) study the effect ofbudget-constraints on various properties of simultaneous auctions such as equilibriumand revenue. In contrast to the current paper, both these papers considered a com-plete information environment. A similar environment to the one in my paper wasstudied by Brusco and Lopomo (2009). However their setting was one of simultane-ous ascending price auctions, where as I study the case of simultaneous sealed-bidauctions.
Similar games in which players expend limited resources in simultaneous con-tests, such as the Colonel Blotto Game, have been analyzed extensively in thecontest theory literature. Most of these papers study complete information envi-ronments and use mixed strategy equilibria to analyze player behavior. Since thecurrent paper analyzes an incomplete information environment and shows non-existence of any equilibria, it is markedly different from the papers in this litera-ture.
My paper also relates to the literature on existence of equilibria in discon-tinuous games. Since this is an involved discussion, I have dedicated Sect. 5 toit.
The paper is organised as follows. The auction environment is set up in Sect. 2. InSect. 3 I describe the action sets available to players, as well their payoff functions.Section 3.1 gives a brief explanation of the final result as well as how the argumentproceeds in the remaining sections to prove the said result. In Sect. 4 I establish aseries of properties any equilibrium will posses and finally show that an equilibriumsatisfying the said properties does not exist. In Sect. 5 I discuss the connections betweenthe current paper and the literature on the existence of equilibria in discontinuousgames. Section 6 concludes.
1 This bidding function is the pure strategy equilibrium of a single-unit first price auction.2 For a discussion of the case of known but different budget levels, see Sect. 5.2.2. The case of privatelyknown budgets is analyzed in a working paper, Ghosh (2012).
123
Non-existence of equilibria in simultaneous auctions
2 Environment
Two units of a good are being sold via two simultaneous first-price auctions (FPA)to two bidders. Each bidder has a single-dimensional type, v ∈ [0, v]. The biddersindependently draw these types from a common, atom-less distribution F with densityf , which is positive everywhere. Bidder i’s valuation of unit k is vk = v. All biddershave a budget w < 2
∫ v
0 xd F(x).3 No bidder is able to spend more than her budget.A notational convention: i and j will be used to identify bidders and k and l the
auctions. If i and j are both mentioned in a context then i �= j . Similarly, for k and l.
3 Strategies and payoffs
Without loss of generality number the auctions 1 and 2. Bidders place two bids, {b1, b2}simultaneously in the two auctions. The highest bid in each auction wins the unit. Theex-post payoff from winning auction k is (v − bk). Ties are broken via coin-flips.
The feasible set of bid pairs (action space) for any bidder is B = [{b1, b2}|b1 +b2 ≤ w]. A behavioural strategy for a player i is a probability transition functionσ i : B × [0, v] → [0, 1] where B is the Borel sigma algebra on B. In the absenceof the budget constraint, the unique equilibrium is σ i ({C(v), C(v)}|v) = 1 for alli , where C(v) = 1
F(v)
∫ v
0 xd F(x). Given the upper bound on the budget level, thisstrategy is not feasible for some bidder types.
Suppose bidder i follows the strategy σ i . Then her strategy in auction k is given byσ i
k = ∫bl∈[0,w] σ
i where k �= l. Note, σ ik : Bk × [0, v] → [0, 1] where Bk is the Borel
sigma algebra on [0, w]. This strategy generates cumulative distributions of bids givenby
Hik (b) =
∫
[0,v]σ i
k ([0, b]|x) f (x)dx
Bidder i �= j faces the bid distribution H jk (·) in auction k. A bidder’s interim
payoff, �i depends on her valuation of the units, her bids and the strategy followedby player j .
�i (v, b1, b2, σj ) = (v − b1)
(
H j1 (b1) − 1
2�
j1(b1)
)
+(v − b2)(H j2 (b2)− 1
2�
j2(b2))
where �(·) represent mass points in the bid distributions and hence the probabilitiesof ties.
�jk (b) =
∫
[0,v]σ i
k ({b}|x) f (x)dx
3 A higher budget level will not be binding for any bidder type, as explained in the next section.
123
G. Ghosh
3.1 Final result
As mentioned previously, the main result of this paper is the non-existence of equilibriain behavioral strategies. In the following two sections I will establish a number ofproperties, through a series of lemmas, that any equilibrium must posses, and finallyshow that the equilibrium can not satisfy them. In this section I provide a brief sketchof how this will be achieved.
In any equilibrium, the bidder with the highest valuation will submit the over allhighest bid in the auctions (Lemma 4.6). This high bid will be greater than w
2 , (Lemma4.4). Also, since the highest valued bidder is expending all her budget (Lemma 4.6),her bid in the other auction will be strictly smaller than w
2 . This implies that the highestvalued bidder’s ex-post payoff in the auction in which she places the high bid will belower than her ex-post payoff in the other auction.
Using a ‘local’ argument I will show that the high valued bidder will want to deviatefrom bidding the highest bid, if it is the case that the loss in probability (of winning)from bidding a little less than the high bid was made up by the probability gain frombidding more in the other auction. This is indeed the case as shown in Lemma 4.7.
4 Equilibrium charecterization
σ = (σ 1, σ 2) is an equilibrium if and only if σ 1 is a best response to σ 2 and vice-versa.This implies that X ∈ B is in the support of σ i if and only if there exist {b1, b2} ∈ Xsuch that {b1, b2} ∈ β i (v), where
β i (v) = arg max{b1,b2}∈B
�i (v, b1, b2, σj )
The following lemma establishes some properties of best-response sets. These areessentially technical in nature and are used repeatedly in subsequent lemmas. Theproofs can be found in the appendix.
Lemma 4.1 Properties of β i (·). Suppose, {b1, b2} ∈ β i (v), where v > 0.
(i) If b1 + b2 < w, then, for v < v, b ∈ (b2, w − b1] and b′ ∈ (b1, w − b2],{b1, b}, {b′, b2} /∈ β i (v).
(ii) If b1 + b2 = w and H j1 (·) is differentiable at b1 then H j
2 (·) is differentiable at b2.
(iii) If {b1, b2} ∈ β i (v), where v �= v, then {b1, b2} ∈ β i (v) for all v ∈ (v, v).
Now, I will charecterize properties that any equilibrium will possess. In the finaltheorem I will show that an equilibrium with the said properties does not exist.
Let bik be the supremum bid in the support of bidder i’s equilibrium bids in auction
k.
Lemma 4.2 bik = b j
k for all k. Also, Hik (b + ε) − Hi
k (b) > 0 if and only if H jk (b +
ε) − H jk (b) > 0, where ε > 0
Proof The first part of the lemma is obvious with two bidders. From now on, letbi
k = b jk = bk .
123
Non-existence of equilibria in simultaneous auctions
Hik (b + ε) − Hi
k (b) > 0 implies there exists v such that σ ik ((b, b + ε]|v) > 0. So,
there exists bk ∈ (b, b+ε] and bl such that {bk, bl} ∈ β i (v). If H jk (b+ε)−H j
k (b) = 0,then bidder i type v is better off reducing her bid in auction k as her probability ofwinning auction k does not change with this deviation but the amount she has to payconditional on winning, decreases. This increases her expected payoff. Hence, it mustbe the case that H j
k (b + ε) − H jk (b) > 0. ��
The following lemma establishes the continuity of the bid distributions almosteverywhere. It is an important lemma as it shows that the payoff function is continuousalmost everywhere on the support of equilibrium bids, and hence permits the use ofthe Theorem of the Maximum (in a local sense) in Lemma 4.7. The continuity of thebid distributions also allows me to prove their differentiability at certain points, as Ido in Lemma 4.7.
Lemma 4.3 Hik (·) is continuous at all b �= w
2 and for all i and k. That is, the equilib-rium bid distributions can have at most one mass point. Also, if there is a mass pointat w
2 in auction k then there must be a mass point of equal measure at w2 in auction l,
k �= l.
Proof Without loss of generality, suppose there exists b ∈ [0, b1] such that �i1(b) > 0.
This implies, there exists a positive measure of bidder i types denoted by Ai1(b) who
bid b1 = b with positive probability. Let Ai2(w − b) be the set of bidder types who
bid b2 = w − b with positive probability. Suppose v ∈ Ai1(b) but v /∈ Ai
2(w − b).Then there must exist {b, b′
2}, where b + b′2 < w and {b, b′
2} is in the support of σ i
for bidder type v. Lemma 4.1(i) implies v = inf{v|v ∈ Ai1(b)}. All other bidder types
v ∈ Ai1(b) also belong to Ai
2(w −b). Since bidder type v has measure zero in the typespace, it must be the case that the measure of the two sets, Ai
1(b) and Ai2(w − b), are
equal.If it is the case that for all but a countable set of v ∈ Ai
1(b), σ i1({b}|v) = σ i
2({w −b}|vi ), then �i
2(w −b) = �i1(b). Suppose, without loss of generality, there exists v ∈
Ai (b) such that σ i1({b}|v) > σ i
2({w − b}|v). This implies there must exist b2 < w − bsuch that {b, b2} is in the support of σ i for v. Again using Lemma 4.1(i), this impliesthat v = inf{v|v ∈ Ai
1(b)}. For all v ∈ Ai (b)−{v}, σ i1({b}|v) = σ i
2({w−b}|v). Sincethe bidder type v has measure zero, �i
2(w − b) = �i1(b).
Now, suppose b �= w2 . I will show a contradiction to �i
2(w−b) = �i1(b). Consider
the bidder j type v, for whom {b, w − b} belongs in the support of σ j .4 Her payoff is
� j (v, b, w − b, σ i ) = (v − b)(Hi1(b) − 1
2�i
1(b))
+(v − w + b)(Hi2(w − b) − 1
2�i
2(w − b))
4 If such a bidder type does not exist, then using a similar argument we can show there will be a profitabledeviation for a bidder type bidding in the neighborhood of {b, w − b}.
123
G. Ghosh
Any deviations from {b, w − b} must give weakly lower payoffs
� j (v, b, w − b, σ i ) ≥ � j (v, b + ε,w − b − ε, σ i )
⇒ 1
2(v − b)�i
1(b) ≤ 1
2(v − w + b)�i
2(w − b)
� j (v, b, w − b, σ i ) ≥ � j (v, b − ε,w − b + ε, σ i )
⇒ 1
2(v − b)�i
1(b) ≥ 1
2(v − w + b)�i
2(w − b)
The implications are drawn as ε → 0 and use the fact that any cumulative distri-bution can only have a countable set of discontinuities. The two equations imply(v − b)�i
1(b) = (v − w + b)�i2(w − b). Since b �= w
2 it must be the case that�i
1(b) �= �i2(w − b), which is a contradiction. ��
The next lemma shows that the highest bid submitted in the auctions is higher thanw2 . This lemma implies that the bidder type who submits the highest bid, must begetting a lower ex-post payoff, i.e. (v − b), in the auction in which she submits thehighest bid, than the other auction. This is true since in the second auction she mustbe bidding less than half the budget. This plays a crucial role in final result, Theorem4.1, as it is used in showing that the bidder submitting the highest bid will chose todeviate and submit a lower bid, if the loss in probability is not ‘too great’.
With two auctions, the first part of the lemma is reasonably intuitive. The only com-plication arises when there is a mass point at w
2 . In this case I show that either thebidder types bidding {w
2 , w2 } will deviate or those bidding close to w
2 will deviate. Theincentive to deviate is always the positive probability gain from winning against themass point in the bid distribution. The second part is used in the subsequent lemmas.The proof can be found in the appendix.
From now on, let b1 = max{b1, b2}. So, b2 ≥ w − b1.In most auction environments, the equilibrium bid distributions is usually strictly
increasing. In the auction game studied in this paper, the presence of a budget con-straint, and hence a (possible) mass point at w
2 , complicates matters a little. The nextlemma takes care of these complications and is used in Lemmas 4.6 and 4.7 to ensurethat we can always find bidder types who submit a bid in any interval which is a subsetof [0, bk].
Lemma 4.5 Hik (·) is strictly increasing, for all i and k
Proof Suppose to the contrary, there exist b′ and b′′, where b′ < b′′ such that Hi1(b
′′)−Hi
1(b′) = 0. Without loss of generality, let b′′ = sup{b1 > b′|σ i
1([b, b1]|v) =0 for all v}. The proof depends on the values of b′ and b′′.
First, suppose b′, b′′ �= w2 . If Hi
1(b′′) − Hi
1(b′) = 0, then Lemma 4.2 implies
H j1 (b′′) − H j
1 (b′) = 0. Lemma 4.3 implies there is no mass point at b′′. Therefore,
123
Non-existence of equilibria in simultaneous auctions
there clearly exist bidder i types who prefer b′ to b′′ + ε where ε > 0, in auction 1.This contradicts the definition of b′′.5
Second, suppose b′′ = w2 . If there is no mass point at w
2 then this case is similar tofirst. If �i
1(w2 ) > 0, then the technique to show Hi
k (w2 ) − �i
k(w2 ) − Hi
k (w2 − ε) > 0
in the proof of Lemma 4.4 can be applied to show a contradiction.Last, suppose b′ = w
2 . If there is no mass point at w2 then this case is similar to
first. Suppose, �i1(
w2 ) > 0. Then Lemma 4.3 implies �i
2(w2 ) > 0. Since there is a
positive measure of bidder i types bidding w2 in auction 2, it must be the case that for
all ε > 0 there exist bidder j type v who for whom {b1, b2} is in the support of σ j ,where b2 ∈ (w
2 − ε, w2 ). If the bidder bids b2 < w
2 , then she can afford to bid b1 > w2 ,
which would be profitable due to �i1(
w2 ) > 0.
Since there is a positive measure of bidder j types bidding b2 ∈ (w2 − ε, w
2 ), therewill be a positive measure of bidder types bidding b1 ∈ (b′, b′ + ε) ⊂ (b′, b′′), whichimplies H j
1 (b′′) − H j1 (b′) > 0. Lemma 4.2, would then imply Hi
1(b′′) − Hi
1(b′) > 0.
��The following lemma shows that only the highest valued bidder will submit the
highest bid, and that this bidder type will expend her entire budget. This lemma isused in showing that bidder types close to the highest bidder will also expend theirentire budget, as shown in Lemma 4.7.
Lemma 4.6 If for some i and some v, {b1, b2} ∈ β i (v) then b2 = w − b1 and v = v.Also, b2 > w − b1.
The proof of this lemma is involved and can be found in the appendix. The proof forthe first part proceeds through three steps. First, I will prove that if {b1, b2} ∈ β i (v)
then b2 = w − b1. Second, I will show {b1, w − b1} ∈ β i (v). And finally, I will provethat only the highest valued bidder may submit the highest bid.
Lemma 4.7 There exists ε > 0 such that if {b1, b2} ∈ β i (v), where b1 ∈ (b1 − ε, b1)
then b2 = w − b1.
Proof Suppose to the contrary, such an ε does not exist. Consider a sequence εn → 0.Due to Lemma 4.5, for each n, there exists vn such that there exists {b1n, b2n} ∈ β i (vn)
where b1n ∈ (b1 − εn, b1) and b2n < w − b1n .By construction, b1n → b1. Since {b2n} is a bounded sequence, it has a convergent
subsequence b2m → b2 Consider the sequence {vm} such that {b1m, b2m} ∈ β i (vm).Again, {vm} is a bounded sequence hence it has a convergent subsequence, vn′ → v.Also, {b1n′ , b2n′ } → {b1, b2}.
Clearly, b1n′ > w2 and b2n′ < w
2 . Since H jk (·) is continuous at all bids other than w
2 ,we know that the payoff function of bidder i types vn′ is continuous along bid-pairswhich maximize it. Therefore I can invoke the Theorem of Maximum, at least in alocal sense to establish the upper hemi-continuity of β i (·) along the sequence vn′ .Since {b1n′, b2n′ } ∈ β i (vn′) for all n′, therefore, {b1, b2} ∈ β i (v). By Lemma 4.6,b2 = w − b and v = v.
5 Bidders will prefer to reduce their bid since they will not lose out too much in terms of probability ofwinning but will gain considerably in terms of ex-post payoffs.
123
G. Ghosh
The bid distributions H j1 (·) and H j
2 (·) are continuous at all points along the
sequence {b1n′ , b2n′ }. Since b1n′ +b2n′ < w, I can show that H j1 (·) and H j
2 (·) are in factcontinuously differentiable along the sequence {b1n′ , b2n′ }. This is done using standardtechniques, which shows that at points of non-differentiability, bidder’s always have aprofitable deviation. Since the sequence of bidder types {vn′ } are not expending theirbudgets, such deviations are always feasible.
The first-order conditions for {vn′ } are
(vn′ − b1n′)h j1(b1n′) − H j
1 (b1n′) = 0
(vn′ − b2n′)h j2(b2n′) − H j
2 (b2n′) = 0 (4.1)
This is true since the bidders are not expending their budgets. Therefore, the slopeof the payoff function with respect to either of the bids must equal zero, a necessarycondition for optima. Taking limits as n′ → ∞ in Eq. (4.1)
(v − w + b1)hj2(w − b1) − H j
2 (w − b1) = 0 (4.2)
Suppose w − b1 = 0, then Eq. (4.2) implies that h j2(w − b1) = 0, which implies that
no bidder types bid 0 in auction 2, a contradiction to the fact that bidder type v = 0would bid zero in both auctions since bidding higher would lead to negative payoffssince the bid distributions are strictly increasing (Lemma 4.5). For w − b1 > 0 andv < v, Eq. (4.2) implies
(v − w + b1)hj2(w − b1) − H j
2 (w − b1) < 0
Therefore, any type of bidder i who does not have the highest valuation strictlyprefers to bid lower than w − b1 in auction 2. As a result, there exists δ > 0 such thatno bidder type will submit a bid b2 ∈ (w − b1, w − b1 + δ). From Lemma 4.4, weknow that b2 > w − b1. Since the bid distributions are strictly increasing (Lemma4.5) we have reached a contradiction. ��Corollary 4.1 Hi
1(b1) − Hi1(b1 − ε) ≤ Hi
2(w − b1 + ε) − Hi2(w − b1) for all i
Proof
Hi1(b1) − Hi
1(b1 − ε) =∫
[0,v]
(σ i
1([0, b1]|x) − σ i1([0, b1 − ε]|x)
)f (x)dx
=∫
[0,v]σ i
1((b1 − ε, b1]|x) f (x)dx
≤∫
[0,v]σ i
2((w − b1, w − b1 + ε]|x) f (x)dx (4.3)
123
Non-existence of equilibria in simultaneous auctions
=∫
[0,v]
(σ i
2([0, w − b1 + ε]|x) − σ i2([0, w − b1]|x)
)f (x)dx
= Hi2(w − b1 + ε) − Hi
2(w − b1) (4.4)
Equation (4.3) is implied by Lemma 4.7. This corollary also implies, Hi1(b1 − ε) ≥
Hi2(w − b1). ��
Theorem 4.1 There does not exist any equilibrium in behavioral strategies.
Proof Since, {b1, w − b1} ∈ β i (v)
(v − b1)H j1 (b1) + (v − w + b1)H j
2 (w − b1)
≥ (v − b1 + ε)H j1 (b1 − ε) + (v − w + b1 − ε)H j
2 (w − b + ε)
where ε > 0 and small. Rearranging the above equation,
(v − b1)(H j1 (b1) − H j
1 (b1 − ε)) − ε(H j1 (b1 − ε) − H j
2 (w − b1))
≥ (v − w + b1 − ε)(H j2 (w − b1 + ε) − H j
2 (w − b1))
Since b1 > w − b1 + ε and Hi1(b1 − ε) ≥ Hi
2(w − b1), the above equation implies
H j1 (b1) − H j
1 (b1 − ε) > H j2 (w − b1 + ε) − H j
2 (w − b1)
which contradicts Corollary 4.1. ��Remark This non-existence result applies to any equilibrium in behavioral strategies,including strategies which call for bidders to randomize over a continuum of bids. So,even if some bidders follow ‘strictly mixed strategies’, Lemmas 4.6 and 4.7 will leadto a breakdown of equilibrium. As an example, consider the following strategy profile.
Example 6 Bidders with valuations below some critical value, say v, bid according tothe usual no-budget-constraints equilibrium strategy,7 on each of the goods. Thereforeeach bid < w
2 . A bidder with a valuation > v follows a strictly mixed strategy placingbids {b1, b2} such that b1 + b2 = w. There is a bound b such that w − b ≤ bk ≤ b.The distribution of bk on [w − b, b] is continuous and smooth.
This example violates Lemma 4.6 and therefore cannot be an equilibrium.
5 Discussion
In some sense, the main result of non-existence of equilibria established in this paperis a surprising one. Especially, given the simplicity of the environment and the number
6 I thank an anonymous referee for providing this example.7 First-price auction strategy.
123
G. Ghosh
of papers that deal with the existence of equilibria in auctions8 and more generally,discontinuous games.
The presence of a budget constraint distinguishes the current paper with most ofthe auction literature dealing specifically with the existence of equilibria in multi-unitauctions. Despite this crucial difference, the main result of the current paper needs tobe put in perspective with respect to other existence results. In this section I will tryto do precisely this.9
First, I will show why a well-known existence result from Reny (1999) does notapply in the current setting. The reason for this is a failure of a condition calledbetter-reply security. This failure occurs due to a restriction of the bidders’ actionset by the budget constraint. Second, I will discuss two ways in which equilib-ria can obtained in the current auction environment. The first of these is to usean endogenous tie-breaking rule à la Simon and Zame (1990) and Jackson et al.(2002). The second is to change the environment to include a continuum of budget-constraints.
5.1 Counter-example to better-reply security
In a widely influential paper, Reny (1999) showed that a class of discontinuous gamesalways posses a Nash equilibrium as long as these games are better-reply secure. Agame is better-reply secure if for every non-equilibrium strategy x∗ and every payoffvector limit u∗ resulting from strategies approaching x∗, some player has a strategyyielding a payoff strictly above u∗ even if the others deviate slightly from x∗. First-priceauction games, including asymmetric multi-unit pay-your-bid auctions fall within theabove class of games. In this section I show, via a counter-example, the failure ofbetter-reply security in the auction game we have studied. Using a specific counter-example makes the argument straightforward, though the intuition behind the failureof better-reply secure is rather general as I will explain in last paragraph of this section.
The environment remains the same as laid out in Sect. 2, with a few specifications.v is equal to 1 and F is the uniform distribution over [0, 1].
β I (v) = v2 , is the equilibrium strategy of a single-unit first-price auction. Consider
a joint strategy x∗ as defined below, which is not an equilibrium. I will show that thereexist sequence of strategies xε converging to x∗ such that no bidder i can secure apayoff strictly above u∗
i = limε→0 ui (xε).
x∗i =
{{ v
2 , v2 } if v ≤ v
{w2 , w
2 } otherwise, for all i (5.1)
8 For a survey, see de Castro and Karney (2012)9 I thank Vijay Krishna and two anonymous referees who encouraged me to investigate the link betweenthe result of my paper and current existence result in auctions.
123
Non-existence of equilibria in simultaneous auctions
where v is the bidder type who is indifferent between submitting { v2 , v
2 } and {w2 , w
2 },when the other bidder is playing according to x∗.10 We find that v = w
2−w. All bidder
types below v prefer the bid pair { v2 , v
2 } to {w2 , w
2 } and the reverse applies to all biddertypes above v.This strategy leads to an ex-ante expected payoff vector, u(x∗), for the bidders
ui (x∗) =2
v∫
0
(v − v
2
)F(v) f (v)dv + 2
1∫
v
(v − w
2
) (
F(v) + 1 − F(v)
2
)
f (v)dv
The strategy x∗ is not an equilibrium. To see this, consider the strategy xδi ,
xδi =
⎧⎪⎨
⎪⎩
{ v2 , v
2 } if v ≤ v
{ v2 , w
2 + δ} if v < v ≤ v
{ v2 , w
2 + δ} otherwise
(5.2)
For a small and positive δ, the strategy xδi gives bidder i a higher ex-ante expected
payoff than x∗i when the other bidder is playing according to x∗. In fact, as δ → 0,
xδi gives the supremum payoff. The strategy xδ
i was constructed using the deviationsfor different bidder types which gave them the highest payoff (in the limit, as δ → 0).Bidder type v is indifferent between submitting { v
2 , v2 } and { v
2 , w2 + δ} as δ → 0.11
The valuation v solves the equation w = 2v − v2.Notice, that if a bidder plays according to xδ and the other bidder follows x∗, then
she would choose a δ as small as possible. So, her ex-ante expected payoff is decreasingin δ. Let usup
i = lim supδ→0 ui (xδi , x∗−i ). usup
i is the upper bound on ex-ante expected
10 I can find v by solving for it in �i (v, { v2 , v
2 }, x∗−i ) = �i (v, {w2 , w
2 }, x∗−i ). That is,
2
(
v − v
2
)
F(v) = 2(v − w
2
)(
F(v) + 1 − F(v)
2
)
where in the right hand side of the above equation I use the fact that there is are mass points in the biddistributions and that ties are broken with equi-probability.11 We can find v by solving for it in
2
(
v − v
2
)
F(v) =(v − w
2
)+
(
v − v
2
)
F(v)
.
123
G. Ghosh
payoffs that bidder i can achieve if the other bidder is following x∗.
usupi = 2
v∫
0
(v − v
2
)F(v) f (v)dv +
v∫
v
((v − w
2
)+
(v − v
2
)F(v)
)f (v)dv
+1∫
v
((v − w
2
)+
(
v − v
2
)
F(v)
)
f (v)dv
= v3 + v3
6+ 1 − v
2(1 + v − w) + v(1 − v)(1 + 2v)
2(5.3)
In the “Appendix” I show that usupi > ui (x∗). This inequality implies that there exist
δ > 0 such that ui (xδi , x∗−i ) > ui (x∗).
Now, consider a strategy xε given by
xεi =
{{ v
2 , v2 } if v ≤ v
{w2 − ( 1−v
v
)ε, w
2 − ( 1−vv
)ε} otherwise
(5.4)
where ε > 0 and small. Notice, limε→0 xε = x∗.Better-reply security requires that there must exist a bidder who can secure a payoff
higher than u∗ = limε→0 u(xε). I will show presently, that there exist parameter values(budget levels) for which u∗
i > usupi . Since usup
i bounds the payoff a bidder can get ifthe other bidder is playing according to x∗, the above inequality will lead to a failureof better-reply security.
To see this, note that ui (xδi , x−i ) is continuous in x−i , where x−i is within a small
neighborhood of x∗−i . The reason for this continuity is that since the probability of ties(ex-ante) between bidder i , who plays xδ
i , and the others, who play according to x∗, iszero, there exist neighborhoods of x∗, where this probability will remain zero. Now,since usup
i > ui (xδi , x∗−i ), there exist a neighborhood of x∗−i where this inequality will
still hold. And since u∗i > usup
i , bidder i can not secure a payoff higher than u∗i in a
neighborhood of x∗−i . Formally,
ui (xε) = 2
v∫
0
(v − v
2
)F(v) f (v)dv + 2
1∫
v
(
v − w
2+ 1 − v
vε
)
F(v) f (v)dv
u∗i = lim
ε→0ui (xε) = 2
v∫
0
(v − v
2
)vdv + 2
1∫
v
(v − w
2
)vdv
= 2
3− v3
3− w
2(1 − v2) (5.5)
123
Non-existence of equilibria in simultaneous auctions
Suppose, w = 13 . Then v = 1
5 and v ≈ 0.1835.12 Substituting the two valuesin Eqs. 5.3 and 5.5 we get usup
i ≈ 0.4561 and u∗i = 0.504 which completes the
counter-example.There are several points worth noting about the above counter example. Milgrom
(2004) (p. 132) shows that the analogue of x∗ in single-unit first-price auction,
x∗i =
{v2 if v ≤ v
w otherwise, for all i
is an equilibrium. The reason it is not an equilibrium in the current auction game isthat even when bidders are constrained in our game, they still have feasible, possiblyprofitable, strategies available to them by way of changing their split of budgets acrossthe two auctions. Hence the dimensionality of the action space is a crucial aspect inthe failure of better-reply security.
However, dimensionality of action space can not be the sole reason behind thefailure. The results in Reny (1999) apply to multi-unit auctions as well. In his paper,a requirement for showing that multi-unit pay-your-bid auctions satisfy better-replysecurity is that the action sets available to bidders are the space of nondecreasingfunctions, each from the type space to [0, v]. The upper bound is the highest valuation,in order to preserve individual rationality. This is the crucial difference between myset up and that of Reny (1999). In my auction game, a bidder may not be able toplay strategies which give her a strictly better payoffs, as they may be infeasible. Forexample, in the counter-example I constructed, if a bidder can submit the bid pair{w
2 + δ, w2 + δ}, then she can secure a payoff higher than u∗. This is permitted under
the model studied by Reny (1999) but not in the current auction game.
5.2 Existence of equilibria
If we were to consider a finite version (finite bidding grids and finite valuation space) ofthe auction game studied in the current paper then equilibrium existence is guaranteed.However, as we move towards the limiting game with a continuum of actions andtypes, we no longer have existence. In this section I ask under what conditions ormodel changes, can equilibrium existence be restored in the limiting game. As suchwe focus on two model specifications (1) tie-breaking rule and (2) single budget level.
5.2.1 Changes in the tie-breaking rule
Technically, the a.e. continuity of the bid distribution (Lemma 4.3), played a majorrole in proving non-existence in the auction game. The absence of ‘mass points’ meantthat a deviation from submitting the highest bid would be profitable for the highesttype bidder, since her loss in probability would not be ‘too much’ and she could gainsubstantially by bidding more in the other auction (since it provides a higher ex-postpayoff). It follows that one of the requirements for restoring equilibrium is a change
12 The equation w = 2v − v2 has two roots; 0.1835 and 1.8165. The latter is not a feasible value for v.
123
G. Ghosh
in the model specification which will allow the existence of discontinuities in the biddistribution at points other than w
2 . One possible way of achieving this would be tochange the tie-breaking rule from the uniform one to something else.
There are other exogenous tie-breaking rules which have been studied in literature.Maskin and Riley (2000) and Araujo et al. (2008) use a second price auction and anall-pay auction, respectively to resolve ties.13 These secondary auctions ‘order’ thebidders types correctly in the case of ties and hence can resolve issues of existencein non-standard auction environments. While these techniques are useful in single-unit auctions,14 they do not aid in restoring equilibrium in our game. To see this,recall that positive probability ties must occur as a ‘pair’ in our auction game (tie atb implies tie at w − b).15 Now, suppose a secondary auction is used to break ties.Notice, the lowest type bidder would lose to all the other tieing bidders in both theauctions. This bidder could do better by bidding slightly more in the auction in whichshe is submitting the lower bid. Hence she would actually prefer not to tie at all whichimplies positive probability ties can not exist under these tie-breaking rules also. Themulti-dimensional action space is what prevents these techniques from working in oursetting.
A tie-breaking rule which might allows for discontinuities and hence restore equi-librium in our auction game is an endogenous tie breaking rule as introduced by Simonand Zame (1990) and later expanded to cover incomplete information by Jackson etal. (2002). The tie-breaking rule is found by looking at the limiting auction games offinite auction game. Locating such a tie-breaking rule is outside the purview of thecurrent paper. For a similar discussion, I refer the interested reader to a discussion inJackson (2009).
5.2.2 Different budget levels
If we change the model to accommodate known but different budget levels, I do nothave a general existence or non-existence result. It will depend on the values of thebudgets and the distribution of valuations. As an example, consider the following. If the‘richer’ bidder was rich enough that she could always submit the unconstrained optimalbid and the poorer bidder splits her budget (up to a permutation) between the twoauctions once her valuation is high enough, under the right distribution assumptions,we may get an equilibrium. However, if the two budget levels were arbitrarily close,then a similar reasoning to the one in this paper could imply non-existence. Thesevariations of the model, while important, do not detract from the main point of thecurrent paper, which is to show how budgets and dimensionality of type-space andaction-space affect equilibrium existence.
Equilibrium existence may be possible in an environment where there is incompleteinformation about budget-levels. In Ghosh (2012) I am attempting to use a recent
13 I thank an anonymous referee for bringing the latter to my attention.14 Araujo et al. (2008) also covers the case of multi-unit auctions with unitary demands.15 I conjecture that this must be true independent of tie-breaking rules. It certainly is the case for tie-breakerswhich use a secondary auction.
123
Non-existence of equilibria in simultaneous auctions
existence result from Barelli et al. (2014) to get existence of equilibria in the case wherebudgets are also private information and are drawn from a continuous distribution.
6 Conclusion
In this paper I showed that in a simple class of multi-unit auctions, if a single budgetconstraint is introduced, equilibrium may fail to exist. This result adds to the currentliterature in auctions in two ways.
First, it adds to the literature in auctions with budget constraints, which have beenhistorically hard to analyze. In fact the current paper is one of the first to study theeffects of budget constraints in multi-unit environment with incomplete information.The non-existence result can serve as a signpost for modeling choices for futureresearch in this area.
Second, the paper also speaks to the literature on existence of equilibria in auctions.In Sect. 5 I discussed a few of the known results and showed by way of a counter-example, why existence of equillibria in auctions with budgets is not readily impliedby some of these results.
Acknowledgments I am grateful to Srihari Govindan, Ayça Kaya, Vijay Krishna (editor), Heng Liu,Clinton Levitt, the associate editor and two anonymous referees for their comments and suggestions.
Appendix
Proof of Lemma 4.1
(i) Without loss of generality b1 ≥ b2 as shown in Fig. 1. Note that H j1 (b1) > 0 and
H j2 (b2) > 0 if b1 > 0 and b2 > 0 respectively.16
When playing {b1, b2}, bidder type v does not expend her entire budget. Hence, v
must weakly prefer b2 to any b ∈ (b2, w − b1], since the bid pair {b1, b} is affordable.
(v − b2)H j2 (b2) ≥ (v − b)H j
2 (b)
Note that if the above inequality is an equality, then H j2 (b2) < H j
1 (b). In either case,whether it is a strict inequality or an equality, all bidder types below v strictly preferb2 to b in auction 2. A similar argument shows that all bidder types v < v strictlyprefer b1 to b′ ∈ (b1, w − b2] in auction 1.
If either b1 = 0 or b2 = 0, then all higher bids would give negative payoffs forbidder type v since {b1, b2} ∈ β i (v). The same would be the case for all bidder typesbelow v.
16 Suppose H j1 (b1) = 0. Let b = lim infn→∞{bn |H j
1 (bn) = 0}. Then there is a positive measure ofbidder types with v ∈ [0, b] ⊂ [0, v] who are submitting bids higher than b in auction 1, winning withpositive probability and hence making negative payoffs .
123
G. Ghosh
Fig. 1 Lemma 1(i)
(ii) In order to prove differentiability of H j2 (·) at w − b1, I will first show continuity.
Suppose to the contrary, limε→0 H j2 (w − b1 + ε) − limε→0 H j
2 (w − b1 − ε) =� > 0.17 The payoff to bidder i of type v is,
�i (v, b1, w − b1, σj ) = (v − b1)H j
1 (b1) + (v − w + b1)(H j2 (w − b1) − 1
2�)
Clearly, by changing the bids to {b1 −ε,w− b2 +ε}, where ε > 0 and small, bidderi would be better off. Her payoff will approximately increase by 1
2 (v − w + b1)�
because she will not lose too much by bidding less in auction 1 due the differentiabilityof H j
1 (·) at b1. Hence H j2 (·) cannot have a mass point at w − b1. What remains to
be shown is differentiability. Since v is playing best responses, for any ε > 0 thefollowing are true,
Expanding the above inequalities, dividing by ε and taking limits as ε → 0 we get,
(v − b1)hj1(b1) − H j
1 (b1) ≥ (v − w + b1) limε→0
H j2 (w − b1 + ε) − H j
2 (w − b1)
ε
− H j2 (w − b1)
(v − b1)hj1(b1) − H j
1 (b1) ≤ (v − w + b1) limε→0
H j2 (w − b1) − H j
2 (w − b1 − ε)
ε
− H j2 (w − b1)
17 Since H j2 (·) is a monotone function, we know the left-hand and right-hand limits at w − b1 exist.
123
Non-existence of equilibria in simultaneous auctions
From the above, notice
limε→0
H j2 (w − b1 + ε) − H j
2 (w − b1)
ε= lim
ε→0
H j2 (w − b1) − H j
2 (w − b1 − ε)
ε
Therefore H2k (·) is differentiable at b2 = w − b1.
(iii) Without loss of generality, let v < v. For any v ∈ (v, v), there exists α < 1, suchthat v = αv + (1 − α)v. Then,
�i (v, b1, b2, σj ) = α�i (v, b1, b2, σ
j ) + (1 − α)�i (v, b1, b2, σj
≥ α�i (v, b1, b2, σj ) + (1 − α)�i (v, b1, b2, σ
j )
= �i (v, b1, b2, σj ), (6.1)
for any {b1, b2} ∈ B. Therefore, {b1, b2} ∈ β i (v). ��
Proof of Lemma 4.4
Without loss of generality, let max{b1, b2} = b1. I need to disprove b1 ≤ w2 . First,
suppose, b1 < w2 . Then, Lemma 4.3 implies Hi
k (·) is continuous everywhere for all iand k. Then there exists a unique equilibrium which is symmetric, in pure strategiesand differentiable.18 This equilibrium should also be an equilibrium of the auction(s)without a budget constraint. However the first-price auction game without a budgetconstraint has a unique equilibrium, which is not possible in the current environmentwith a budget constraint w < 2
∫ 10 xd F(x).
Now, suppose, b1 = w2 . If there is no mass point at w
2 , then the argument in thepreceding paragraph applies. Suppose there is a mass point, �i
1(w2 ). Lemma 4.3 shows
�i1(
w2 ) = �i
2(w2 ). The rest of the proof proceeds as follows. I will show that there are
bidder types bidding arbitrarily close to w2 , who prefer to bid more than w
2 to beat theprobability mass �i
1(w2 ). This contradicts b1 = w
2 .First, let us establish that there are bidder types bidding arbitrarily close to w
2 .Suppose, to the contrary, there exists ε > 0 such that Hi
1(w2 )−�i
1(w2 )−Hi
1(w2 −ε) = 0.
Consider bidder j with type v for whom {w2 , w
2 } belongs in the support of σ j .19 This
18 This equilibrium can be found using standard arguments which use ordinary differential equations as inLebrun (1996). In this case the auction game becomes two single-unit first price auctions, since no bidderis hitting her constraint.19 Since Hi
1( w2 ) − Hi
1( w2 − ε) > 0, Lemma 4.2 implies H j
1 ( w2 ) − H j
1 ( w2 − ε) > 0. If there does not
exist v for whom {w2 , w
2 } belongs in the support of σ j , then there must exist a positive measure of bidder jtypes who bid in the interval ( w
2 − ε, w2 ). This implies there must exist positive measure of bidder i types
who bid in ( w2 − ε, w
2 ).
123
G. Ghosh
bidder’s payoff is given by
� j (v,w
2,w
2, σ i ) = (v − w
2)(1 − 1
2�i
1(w
2)) + (v − w
2)(1 − 1
2�i
1(w
2))
= 2(v − w
2) − (v − w
2)�i
1(w
2)
Suppose this bidder submits the bid pair {w2 − ε, w
2 + εN }, where N is some large
number. With this strategy, the bidder will win auction 2 for sure and the auction 1with probability Hi
1(w2 − ε) = (1 − �i
2(w2 )). The payoff from this deviation is
� j (v,w
2− ε,
w
2+ ε
N, σ i ) = (v − w
2+ ε)(1 − �i
2(w
2))+(v − w
2− ε
N)
= 2(v − w
2)−(v − w
2)�i
2(w
2)+ε
(N − 1
N− �i
2(w
2)
)
> � j (v,w
2,w
2, σ i )
The strict inequality (true for N large enough) is a contradiction. Therefore, for allε > 0, Hi
k (w2 )−�i
k(w2 )− Hi
k (w2 − ε) > 0. Using Lemma 4.2, this implies, H j
k (w2 )−
�jk (
w2 ) − H j
k (w2 − ε) > 0.
Now, I show that a bidder type bidding close to w2 prefers to deviate. Consider bidder
j type v for whom {b1, b2} belongs to the support of σ j , where b1 ∈ (w2 − ε, w
2 ) andb1 ≥ b2.20 Therefore w − b1 − b2 > w
2 − b1 = ε′. Even as ε → 0 I can always findsuch a bidder. Suppose she bids {b1 + ε′+, b2}, where ε′+ > ε′ but very close to ε′.
� j (v, b1 + ε′+, b2, σi ) = (v − w
2− ε′+) + (v − b2)Hi
2(b2)
> (v − w
2− ε′+) + (
w
2+ ε′+ − b1) − (v − b1)�
i1(
w
2)
+ (v − b2)Hi2(b2)
= (v − b1) − (v − b1)�i1(
w
2) + (v − b2)Hi
2(b2)
= (v − b1)(1 − �i1(
w
2)) + (v − b2)Hi
2(b2)
> (v − b1)Hi1(b1) + (v − b2)Hi
2(b2)
= � j (v, b1, b2, σi )
The first inequality is true since (w2 + ε′+ − b1) ≈ 2ε′ < (v − b1)�
i1(
w2 ) as ε
gets small enough. The second inequality is true since Hi1(b1) < 1 − �i
1(w2 ). Since
w − b1 − b2 > ε′, the bid pair {b1 + ε′+, b2} is affordable. Therefore {b1, b2} /∈ β j (v)
which contradicts {b1, b2} being in the support of σ j .
20 An identical proof applies if b1 ≤ b2, since there is also a mass point in Hi2(·) at w
2 .
123
Non-existence of equilibria in simultaneous auctions
Fig. 2 Step 1
Finally, suppose without loss of generality, b1 = max{b1, b2}. Then b2 =min{b1, b2}. If b2 < w − b1, then the sum of every equilibrium bid-pair is less thanthe budget. Therefore no bidder type is hitting her constraint. Applying the same rea-soning as in the first paragraph of this proof, this can not be case since the single-unitfirst price auctions have a unique equilibrium. ��
Proof of Lemma 4.6
Step 1Suppose, there exists v and b2, such that {b1, b2} ∈ β i (v) where b2 < w − b1.
Lemma 4.5 implies there must exist a positive measure of bidder i types bidding inthe interval (b2, w − b1) in auction 2, since b2 ≥ w − b1 due to Lemma 4.4. Considerv such that {b1, b2} ∈ β i (v) where b2 ∈ (b2, w − b1) as shown in Fig. 2.
First, I will show that v > v. To the contrary, suppose v < v. Since {b1, b2} ∈β i (v), it must be the case that (v − b2)H j
2 (b2) ≥ (v − bi2)H j
2 (b2). This implies
(v − b2)H j2 (b2) > (v − b2)H j
2 (b2). Since {b1, b2} is affordable, it must be the casethat �i (v, b1, b2, σ
j ) > �i (v, b1, b2, σj ). This contradicts {b1, b2} ∈ β i (v).
Suppose v = v. We know that {b1, b2}, {b1, b2} ∈ β i (v). Since both these bid pairsare affordable, this implies no bidder would submit b′
2 ∈ (b2, b2), a contradiction toLemma 4.5.21 Therefore, v > v.
The final part in this step is to show b1 > b1 which is a contradiction. Supposeb1 < b1. Since {b1, b2} is affordable and {b1, b2} ∈ β i (v), it must be the case that(v − b1)H j
1 (b1) ≥ (v − b1)H j1 (b1). Since v > v, the above inequality implies
j ),which is a contradiction to {b1, b2} ∈ β i (v). The only case remaining is b1 = b1.This would apply to all the bidder types for whom b2 ∈ (b2, w − b1) belongs inthe support of σ i
2, of which there is positive measure. Therefore it must be the case
2(b2) = �i1(b1) > 0. That is there is positive mass at b1 in
Hi1(·), which is a contradiction, due to Lemmas 4.4 and 4.3. Hence the only remaining
possibility is b1 > b1, a contradiction in itself. Therefore, b2 = w − b1.Step 2Suppose, {b1, w − b1} /∈ β i (v). Also, {b1, w − b1} ∈ β i (v) for some v. Let,
{b1, b2} ∈ β i (v), where b1 < b1.22 Also note, b1 + b2 = w.23 Without loss ofgenerality let b1 ≥ b2. Since v and v are playing best responses and {b1, w − b1} /∈β i (v),
(v − b1)H j1 (b1) + (v − w + b1)H j
2 (w − b1) ≥ (v − b1)H j1 (b1) + (v − b2)H j
2 (b2)
(v − b1)H j1 (b1) + (v − w + b1)H j
2 (w − b1) < (v − b1)H j1 (b1) + (v − b2)H j
2 (b2)
Substituting b2 = w − b1 and then subtracting the first equation from the second weget
H j1 (b1) + H j
2 (w − b1) < H j1 (b1) + H j
2 (w − b1) (6.2)
Since H j1 (b1) = 1, the above implies H j
1 (b1) > H j2 (w − b1).
Now, again, since {b1, w − b1} ∈ β i (v)
(v − b1)H j1 (b1) + (v − w + b1)H j
2 (w − b1)
≥ (v − b1)H j1 (b1) + (v − w + b1)H j
2 (w − b1)
⇒ (v − b1)(H j1 (b1) − H j
1 (b1)) − (b1 − b1)(H j1 (b1) − H j
2 (w − b1))
≥ (v − w + b1)(H j2 (w − b1) − H j
2 (w − b1))
⇒ H j1 (b1) − H j
1 (b1) > H j2 (w − b1) − H j
2 (w − b1)
The final implication follows from H j1 (b1) > H j
2 (w−b1), b1 > b1 and b1 ≥ w−b1.24
The final implication is contradictory to (6.2). Therefore {b1, w − b1} ∈ β i (v).Step 3Finally, suppose, there exists v < v, such that {b1, w− b1} ∈ β i (v). Lemma 4.1(iii)
implies that for any v′ ∈ (v, v), it must be the case that {b1, w−b1} ∈ β i (v′). However,note that if {b1, w − b1} = β i (v′) for some v′ ∈ [v, v], then for all v′′ ∈ (v′, v),{b1, w − b1} = β i (v′′), leading to mass points in Hi
1(·) and Hi2(·) at b1 and w − b1
respectively.25 Therefore, β i (v′) − {b1, w − b1} �= φ, for all v′ ∈ [v, v].
22 Since b1 > b1 is a contradiction to the definition of b1 and bi1 = b1 will lead to the conclusion of the
previous step.23 Suppose, b1+b2 < w. Since {b1, b2} ∈ βi (v), (v−b1)H j
1 (b1) ≥ (v−b′1)H j
1 (b′1), where b′
1 ∈ (b1, w−b2). This implies, (v−b1)H j
1 (b1) > (v−b′1)H j
1 (b′1) for all v < v. Therefore, Hi
1(w−b2)− Hi1(b1) = 0,
a contradiction, since b1 < b1.24 This is true since, b1 = max{b1, b2} ≥ b2 = w − b1.25 If {b1, w − b1} = βi (v′),then a similar argument as in (6.1) with strict inequalities, can show that{b1, w − b1} = βi (v′′) for all v′′ ∈ (v′, v).
123
Non-existence of equilibria in simultaneous auctions
Now, I will show that β i (v′) = β i (v′′) for any v′, v′′ ∈ (v, v). I will prove thatβ i (v′) ⊆ β i (v′′). Suppose, this is not true for some v′, v′′, where v′ < v′′, without lossof generality. Then, there exists {b′
1, b′2} �= {b1, w−b1}, such that {b′
1, b′2} ∈ β i (v′) and
{b′1, b′
2} /∈ β i (v′′). The latter implies �i (v′′, b1, w − b1, σj ) > �i (v′′, b′
1, b′2, σ
j ).Since there exists α such that v′ = αv + (1 − α)v′′, I can use a strict version ofthe argument in Lemma 4.1(iii), to show �i (v′, b1, w − b1, σ
j ) > �i (v′, b′1, b′
2, σj )
which will contradict {b′1, b′
2} ∈ β i (v′). Therefore β i (v′) ⊆ β i (v′′). Similarly I canshow, β i (v′′) ⊆ β i (v′).
Recapping, for all v′, v′′ ∈ [v, v], {b1, w − b1} ∈ β i (v′), β(v′)−{b1, w − b1} �= φ
and β i (v′) = β i (v′′). Suppose, there exists {b′1, b′
2} ∈ β i (v), such that b′1 + b′
2 <
w. Lemma 4.1(i) will imply, that Hi1(b) − Hi
1(b′1) = 0 which is a contradiction.26
Therefore if {b′1, b′
2} ∈ β i (v′) where v′ ∈ [v, v], then b′1 + b′
2 = w.Consider, {b′
1, w − b′1} ∈ β i (v′) where b′
1 < b1. So, �i (v′, b′1, w − b′
1, σj ) =
�i (v′, b1, w − b1, σj ). If b′
1 > w − b1, then using a similar argument used in Step2 of the proof, we can show a contradiction. Formally, �(v′, b′
1, w − b′1, σ
j ) =�(v′, b1, w − b1, σ
j ) and �(v, b′1, w − b′
1, σj ) = �(v, b1, w − b1, σ
j ) togetherimply,
H j1 (b1) + H j
2 (w − b1) = H j1 (b′
1) + H j2 (w − b′
1) (6.3)
Since H j1 (b1) = 1, the above implies H j
2 (w − b1) ≤ H j1 (b′
1).Then, similar to the procedure in Step 2 we can show that �(v′, b′
1, w − b′1, σ
j ) =�(v′, b1, w − b1, σ
j ) implies,
(v′ − b1)(H j1 (b1) − H j
1 (b′1)) − (b1 − b′
1)(H j1 (b′
1) − H j2 (w − b1))
= (v′ − w + b′1)(H j
2 (w − b′1) − H j
2 (w − b1))
⇒ H j1 (b1) − H j
1 (b′1) > H j
2 (w − b′1) − H j
2 (w − b1)
where the final inequality follows from b1 > b′1, b′
1 > w − b and H j2 (w − b1) ≤
H j1 (b′
1). This is contradiction to equation 6.3.So, the only remaining possibility is b′
1 = w − b. This would apply to any {b′1, w −
b′1} ∈ β i (v′). Therefore for all v′ ∈ [v, v], β i (v′) = {w − b, b}, {b, w − b}, leading
to mass points, which is a contradiction.Now, I can prove the second part of the lemma. We know that b2 ≥ w − b1 from
Lemma 4.4. If b2 = w − b1 < w2 , then Lemma 4.3 implies that Hi
2(·) is continuousfor all i . This implies Hi
1(·) is continuous. Also, b2 = w − b1 along with Lemma4.6 implies that only the highest valued bidder is at the budget constraint. However,since this bidder type is winning both units with probability one, she is not necessarily‘constrained’. All other bidder types submit bid-pairs which are strictly inside thebudget set. Since no bidder type is ‘constrained’ per se, and the bid distributions arecontinuous, we are led to the conclusion of the first paragraph in the proof of Lemma
26 If b = b′1 then Hi
2(w − b) − Hi2(b′
2) = 0, also a contradiction.
123
G. Ghosh
4.4. Namely, that these strategies should also form an equilibrium of the auction(s)without a budget constraint, which is not possible. ��
Proof of usupi > ui (x∗)
usupi − ui (x∗) =
v∫
v
((v − w
2) + (v − v
2)F(v)
)f (v)dv − 2
v∫
v
(v − v
2)F(v) f (v)dv
+1∫
v
(
(v − w
2) + (v − v
2)F(v)
)
f (v)dv − 2
1∫
v
(v − w
2)(
1 + F(v)
2) f (v)dv
=v∫
v
((v − w
2) − (v − v
2)F(v)
)f (v)dv +
1∫
v
(w
2− v
2)F(v) f (v)dv > 0
The first term is positive since bidder types v ∈ (v, v), strictly prefer bidding { v2 , w
2 +δ}to { v
2 , v2 }. That is the payoff from bidding w
2 +δ, which is (v− w2 −δ) > (v− v
2 )F(v),where the second term is the payoff from bidding v
2 . The second term is positive sincev = w
2−w< w ��
References
Araujo A, de Castro L, Moreira H (2008) Non-monotonicities and the all-pay auction tie-breaking rule.Economic Theory 35(3):407–440
Barelli P, Govindan S, Wilson R (2014) Competition for a majority. Econometrica 82(1):271–314Benoit J-P, Krishna V (2001) Multiple-object auctions with budget constrained bidders. Review of Economic
Studies 68(1):155–179Bulow J, Levin J and Milgrom P (2009) Winning play in spectrum auctions, mimeoBrusco S, Lopomo G (2009) Simultaneous ascending auctions with complementarities and known budget
constraints. Economic Theory 38:1de Castro LI, Karney DH (2012) Equilibria existence and characterization in auctions: Achievements and
open questions. Journal of Economic Surveys 26(5):911–932Ghosh G (2012) Simultaneous First-Price Auctions with Budget Constrained Bidders. mimeoJackson MO (2009) Non-Existence of Equilibrium in Vickrey. Second Price and English Auctions, Review
of Economic Design 13:137–145Jackson MO, Simon LK, Swinkels JM, Zame WR (2002) Communication and Equilibrium in Discontinuous
Games of Incomplete Information. Econometrica 70(5):1711–1740Lebrun B (1996) Existence of an equilibrium in first price auctions. Economic Theory 7:421–443Maskin E, Riley J (2000) Equilibrium in sealed high bid auctions. Review of Economic Studies 67:439–454Milgrom PR (2004) Putting Auction Theory to Work. Cambridge University Press, CambridgeMilgrom PR, Weber RJ (1985) Distributional strategies for games with incomplete information. Mathemat-
ics of Operation Research 10(4):619–632Palfrey TR (1980) Multiple-object, discriminatory auctions with bidding constraints: A game-theoretic
analysis. Management Science 26(9):935–946Reny PJ (1999) On the existence of pure and mixed-strategy Nash equilibria in discontinuous games.
Econometrica 67(5):1029–1056Simon LK and Zame WR (1990) Discontinuous Games and endogenous sharing rules. Econometrica
