Simultaneous Auctions for Complementary Goods
Wiroy Shin
June 21, 2015
Abstract
This paper studies simultaneous and separate first-price auctions for two complementary goods. Agents observe private values of each good before making bids,
and the complementarity between goods is explicitly incorporated in their utility. We show that monotone pure-strategy Bayesian Nash equilibria exist in this
environment. Such equilibria also exist in an environment with arbitrary many simultaneous first-price auctions if bidders obtain complementarity by acquiring all
of the objects on the auctions.
Keywords : simultaneous auctions, multiple-object auctions, first-price auction, private value auctions, equilibrium existence, monotone pure-strategies.
JEL Classification : D44, C72
∗ Department of Economics, The Pennsylvania State University, University Park, PA, 16802, USA.
† E-mail address: wus130@psu.edu
‡ I thank to Edward J. Green for his guidance and support throughout this project. I am also grateful
for the comments of James Jordan, Leslie Marx, Ed Coulson, Paul Grieco, and seminar participants at
Penn State and the 25th International Conference on Game Theory at Stony Brook University. I gratefully acknowledge the Human Capital Foundation, (http://www.hcfoundation.ru/en/) and particularly
Andrey P. Vavilov, for research support through the Center for the Study of Auctions, Procurements,
and Competition Policy (CAPCP, http://capcp.psu.edu/) at the Pennsylvania State University. All
errors are mine.
♯ An earlier version of this paper had been posted on arXiv.org (http://arxiv.org/abs/1312.2641)
in December 2013 and had been presented at the 25th International Conference on Game Theory in
July 2014. Independently, Gentry et al. (2014), a working paper in simultaneous first-price auctions,
was posted on the authors’ websites in October 2014.
1
1
Introduction
This paper studies simultaneous and separate first-price auctions for two complementary goods. Agents observe private values of each good before making bids, and the
complementarity between goods is explicitly incorporated in their utility. We show that
monotone pure-strategy Bayesian Nash equilibria exist in this environment. Such equilibria also exist in an environment with arbitrary many simultaneous first-price auctions
if bidders obtain complementarity by acquiring all of the objects on the auctions. This
model, in which the auction mechanism has an exogenous (albeit widely used) form and
in which sellers do not interact strategically with one another, is intended to provide a
tractable framework for applied research focusing on competition among buyers. Each
buyer is trying to acquire a portfolio of objects that must be purchased by auction from
several unrelated sellers. EBay’s various sealed-bid auctions under same collectibles category which are run by different sellers with similar closing time can be an example of
this environment. Also, less explicit, but equally compelling, auctions for acquisition of
firms manufacturing a final good and an intermediate good input can be another example. Here, a buyer intends to create a vertically integrated firm and the two acquired
firms can be viewed as complements when they are merged.
There are several branches of existing auction research that are similar to, but distinct from, what is being done here. Let us comment briefly on those contributions,
and on how the present research is distinguished from them. A literature on multi-unit
auctions, including Bresky (1999), McAdams (2003), and Reny (2011) studies sale of
multiple identical goods in an auction with specific pricing rules. As modeled, multiple
identical objects resemble substitutable distinct objects rather than complementary ones
modeled here. Package auctions(see Bernheim and Whinston (1986) and Ausubel and
Milgrom (2002)) model multiple goods without assuming substitutability only. They
suppose that an explicit bid can be offered for a package of objects, which is not possible when various objects are auctioned by distinct and unrelated sellers.
There are a few prior research contributions on simultaneous auctions, including
Krishna and Rosenthal (1996) and Bikhchandani (1999). They either assume a single
parameter for private values of the multiple objects or complete information on bidders’ valuations. However, such parameterization is too rigid to model heterogeneity of
bidders’ preferences on each item and complementarity. In this paper, we relax those
restrictions so that the environment in which each bidder receives private signals for
each heterogeneous object is permitted.
We follow McAdams (2003) to prove the existence of a monotone pure-strategy
Bayesian Nash equilibrium. Along with basic assumptions on the Bayesian game (Assumption12
3, McAdams (2003)), single crossing differences and quasisupermodularity are required
in his paper.1
The remainder of this paper is as follows. In the next section, we present a model
of simultaneous first-price auctions for complementary goods in detail with two objects
and two agents, and propose a theorem for the existence of a monotone pure-strategy
Bayesian Nash equilibrium. Section 3 extends the section 2’s results to multiple (more
than two) objects and multiple bidders cases assuming a specific type of complementarity. Section 4 shows examples of complementarity that the generalization to a multiple
objects case is limited. Section 5 discusses an extension of the study to a continuum bid
space setting.
2
Simultaneous first-price auctions for complementary goods
In this section, we define simultaneous first-price auctions for complementary goods
with two non-identical objects and two bidders. At the end of the section, we propose
a theorem for the existence of a monotone pure-strategy Bayesian Nash equilibrium.
2.1
Environment
Bidder i ∈ {1, 2}, i ̸= j ; i’s opponent is denoted by j.
Object k ∈ {1, 2}, Values of objects(or bidder’s type) are represented by vector xi =
(xi1 , xi2 ). ∀xik ∈ [0, 1] = X, and the vector xi is distributed by F (·) on the
support X 2 . Let f (·) be a density function of the cumulative distribution function
F (·), and let F (·) be atomless and common knowledge among bidders.
Bid bi = (bi1 , bi2 ) ∈ {0, n1 , n2 , ..., ū} × {0, n1 , n2 , ..., ū} = A2 , where n ∈ Z+ and ū ∈ R+ .
Each bidder i chooses bids bi1 and bi2 simultaneously from a two dimensional bid
space A2 . A2 is a finite subset of R2 . Therefore, A2 , the action space of this game
is a Euclidean lattice with a coordinatewise partial order.
Utility Bidder i has a quasilinear ex-post utility Ui (bi , bj , xi ).



w(xi1 , xi2 ) − (bi1 + bi2 ) if bi1 > bj1 and bi2 > bj2 , where i ̸= j



u(x ) − b
if bi1 > bj1 and bi2 < bj2
i1
i1
Ui (bi , bj , xi ) =
(1)

u(xi2 ) − bi2
if bi1 < bj1 and bi2 > bj2




0
otherwise.
1
Theorem 1, McAdams (2003)
3
Note that when ties occur, the object is allocated randomly to one of the two bidders with probability 12 . The ex-post utility Ui can be interpreted as follows. First,
an agent realizes his utility from winning objects through a function u : R → R
when he wins a single item(e.g. identity function), and w : R2 → R when he wins
both items. w(xi1 , xi2 ) transforms the independent values of goods to utility that
counts a synergy effect between two objects(e.g. w(xi1 , xi2 ) = xi1 + xi2 + α, where
α > 0). And then, the bidding values for the winning objects are subtracted from
u(·) or w(·, ·). Additional assumptions on u(·) and w(·, ·) are listed below.
Let ≥ be a coordinatewise partial order on the values of objects and bids. For
example, when xi , yi ∈ R2 , [xi ≥ yi ⇔ (xi1 ≥ yi1 ) and (xi2 ≥ yi2 )].
(A1) Monotone utility: u(·) and w(·, ·) are weakly increasing in their arguments.
(A2) Superadditivity: Let λ be the complementary value from winning both goods.
λ(xi1 , xi2 ) = w(xi1 , xi2 ) − [u(xi1 ) + u(xi2 )] ≥ 0.
; This assumption implies that regardless of the values of objects, agents realize
that utility from winning both objects is greater than or equal to sum of utilities
of each good. (e.g. Goods are complementary, Acquiring all bike shops in New
York City yields monopolistic positioning benefit in the business)
(A3) Monotone complementarity: [xi ≥ yi ⇒ λ(xi ) ≥ λ(yi )]. That is, the complementary value λ is weakly increasing in xi = (xi1 , xi2 ).
In summary, bidders expect extra synergy by obtaining both items, and the value
of synergy is weakly increasing with the values of obtained goods.
In addition, since it is reasonable not to bid more than the maximum of utility that an
agent can earn, we assign w(1, 1) to the maximum bid on one good, ū.
2.2
Monotone pure-strategy Bayesian Nash equilibrium
A pure-strategy for bidder i is a function si : X 2 → A, and it is monotone if x′i ≥ xi
implies si (x′i ) ≥ si (xi ), for all x′i , xi ∈ X 2 . Let Si be a set of monotone pure strategies
of i. Based on the environment in section 2.1, bidder i’s expected utility is written as
follows.
Let µsj be an opponent’s bids distribution constructed by the opponent’s purestrategy sj (xj ) and the type distribution F (xj ). Define three probabilities in terms of
s
s
winning cases. P1 j (bi ) and P2 j (bi ) denote probabilities for winning object 1 only and
4
s
object 2 only, respectively, when the opponent plays sj . P3 j (bi ) indicates a probability
that bidder i wins both objects with bi .
1
s
P1 j (bi ) = µsj ([0, bi1 ) × (bi2 , ū]) + µsj ([0, bi1 ) × [bi2 , bi2 ])
2
1 sj
1
+ µ ([bi1 , bi1 ] × (bi2 , ū]) + µsj ([bi1 , bi1 ] × [bi2 , bi2 ]); 2
2
4
1 sj
sj
sj
P2 (bi ) = µ ((bi1 , ū] × [0, bi2 )) + µ ([bi1 , bi1 ] × [0, bi2 ))
2
1
1 sj
+ µ ((bi1 , ū] × [bi2 , bi2 ]) + µsj ([bi1 , bi1 ] × [bi2 , bi2 ]);
2
4
1 sj
sj
sj
P3 (bi ) = µ ([0, bi1 ) × [0, bi2 )) + µ ([bi1 , bi1 ] × [0, bi2 ))
2
1 sj
1
+ µ ([0, bi1 ) × [bi2 , bi2 ]) + µsj ([bi1 , bi1 ] × [bi2 , bi2 ]).
2
4
(2)
Then, if an agent with type xi bids bi , the bidder i’s expected utility is
s
s
s
Vi (bi , xi , sj ) = P3 j (bi )[w(xi1 , xi2 )−(bi1 +bi2 )]+P1 j (bi )[u(xi1 )−bi1 ]+P2 j (bi )[u(xi2 )−bi2 ].
(3)
To make proofs tractable in section 3, we modify the representation of the expected
s
s
s
utility. First, we define Q1j , Q2j and Q3j as follows.
1
s
Q1j (bi1 ) = µsj ([0, bi1 ) × [0, ū]) + µsj ([bi1 , bi1 ] × [0, ū]);
2
1
sj
Q2 (bi2 ) = µsj ([0, ū] × [0, bi2 )) + µsj ([0, ū] × [bi2 , bi2 ]);
2
s
s
Q3j (bi1 , bi2 ) = P3 j (bi ).
(4)
s
Q1j (bi1 ) denotes a probability that bidder i with a bid bi1 wins the object 1 regardless
s
of i’s acquiring object 2. Similarly, Q2j (bi2 ) describes a probability that bidder i with a
s
bid bi2 wins the object 2. Note that Q1j only depends on bi1 and is weakly increasing
s
s
in bi1 , and Q2j only depends on bi2 and is weakly increasing in bi2 . Q3j , the probability
of winning both objects is weakly increasing in (bi1 , bi2 ) with the coordinatewise partial
order.
s
s
s
The redefined winning probability functions (4) imply [P3 j (bi ) = Q3j (bi ), P1 j (bi ) =
s
s
s
s
s
Q1j (bi1 ) − Q3j (bi ), P2 j (bi ) = Q2j (bi2 ) − Q3j (bi )]. Substituting these to the expected
utility function (3), we have
µsj ([0, bi1 )×(bi2 , ū]) = probability(0 ≤ bj1 < bi1 , bi2 < bj2 ≤ ū). The terms with coefficients 12 and
count the probabilities of events that ties occur for one and both of the two objects, respectively.
2
5
1
4
s
s
Vi (bi , xi , sj ) = Q1j (bi1 )[u(xi1 ) − bi1 ] + Q2j (bi2 )[u(xi2 ) − bi2 ]
s
+Q3j (bi1 , bi2 )[w(xi1 , xi2 ) − (bi1 + bi2 ) − (u(xi1 ) − bi1 ) − (u(xi2 ) − bi2 )]
s
s
= Q1j (bi1 )[u(xi1 ) − bi1 ] + Q2j (bi2 )[u(xi2 ) − bi2 ]
=
s
+Q3j (bi1 , bi2 )[w(xi1 , xi2 ) − u(xi1 ) − u(xi2 )]
s
s
Q1j (bi1 )[u(xi1 ) − bi1 ] + Q2j (bi2 )[u(xi2 ) − bi2 ]
(5)
s
+ Q3j (bi1 , bi2 )λ(xi ).
Under the auction environment in section 2.1, (s1 , s2 ) ∈ S1 × S2 is a monotone purestrategy Bayesian Nash equilibrium, if for every bidder i ∈ {1, 2} and every xi ∈ X 2 ,
Vi (si (xi ), xi , sj ) ≥ Vi (bi , xi , sj )
(6)
for all bi ∈ A2 .
Theorem 1. A monotone pure-strategy Bayesian Nash equilibrium exists in simultaneous first-price auctions for complementary goods, which are specified in section 2.1.
2.3
Proof of Theorem 1
Under the simultaneous auctions environment in section 2.1, we have a finite common
action set A2 and an atomless cumulative distribution function for object values with
a two-dimensional compact support. Therefore, by employing the coordinate partial
order and usual Euclidean topology and measure, Assumption 1-2 in McAdams (2003)
are satisfied. The assumption 3 is also satisfied by the fact that the ex-post utility Ui
defined in (1) is bounded and jointly measurable in bi ∈ A2 for every xi ∈ X 2 . Therefore,
by the theorem 1 in McAdams (2003), if the interim utility Vi (bi , xi , sj ) satisfies single
crossing differences and quasisupermodularity for all monotone pure-strategies of the
opponent sj , Theorem 1 is proved. Lemma 1 and Lemma 2 in the following sections
show that the two conditions are satisfied for all pure-strategies of the opponent in the
simultaneous auctions environment of section 2.1.
2.3.1
Single crossing differences of Vi (bi , xi , sj )
Definition 1. Vi (bi , xi , sj ) satisfies single crossing differences, if for all pure-strategies
sj of the opponent, for all bhi ≥ bli , and for all xhi ≥ xli ,
Vi (bhi , xli , sj ) ≥ (>)Vi (bli , xli , sj ) implies Vi (bhi , xhi , sj ) ≥ (>)Vi (bli , xhi , sj ).
(7)
(Weak inequality implies weak inequality, and strict inequality implies strict inequality.)
6
l
h
Lemma 1. With bli , b′h
i , xi and xi , (7) holds. That is,
Vi (bhi1 , bhi2 , xli , sj ) − Vi (bli1 , bli2 , xli , sj ) ≥ (>)0
(8)
Vi (bhi1 , bhi2 , xhi , sj ) − Vi (bli1 , bli2 , xhi , sj ) ≥ (>)0.
(9)
implies
Proof. By (5), (8) gives
s
s
Q1j (bhi1 )[u(xli1 ) − bhi1 ] + Q2j (bhi2 )[u(xli2 ) − bhi2 ]
s
+Q3j (bhi1 , bhi2 )[w(xli1 , xli2 ) − u(xli1 ) − u(xli2 )]
s
≥ (>)
(10)
s
Q1j (bli1 )[u(xli1 ) − bli1 ] + Q2j (bli2 )[u(xli2 ) − bli2 ]
s
+Q3j (bli1 , bli2 )[w(xli1 , xli2 ) − u(xli1 ) − u(xli2 )]
s
⇔
s
[Q3j (bhi1 , bhi2 ) − Q3j (bli1 , bli2 )]λ(xli )
s
≥ (>)
s
[Q1j (bli1 )[u(xli1 ) − bli1 ] − Q1j (bhi1 )[u(xli1 ) − bhi1 ]]
s
+[Q2j (bli2 )[u(xli2 )
−
bli2 ]
−
s
Q2j (bhi2 )[u(xli2 )
−
(11)
bhi2 ]]
We want to show that (11) gives (9). Rewriting (9),
s
s
[Q3j (bhi1 , bhi2 ) − Q3j (bli1 , bli2 )]λ(xhi )
s
≥ (>)
s
[Q1j (bli1 )[u(xhi1 ) − bli1 ] − Q1j (bhi1 )[u(xhi1 ) − bhi1 ]]
s
(12)
s
+[Q2j (bli2 )[u(xhi2 ) − bli2 ] − Q2j (bhi2 )[u(xhi2 ) − bhi2 ]]
To prove the lemma, it is sufficient to show that [(LHS(12) ≥ LHS(11)) and (RHS(11) ≥
RHS(12))].
s
s
LHS(12) − LHS(11) = [Q3j (bhi1 , bhi2 ) − Q3j (bli1 , bli2 )] · [λ(xhi ) − λ(xli )] ≥ 0.
(13)
s
The inequality comes from the fact that both terms are nonnegative. Q3j is weakly
increasing in its argument. Also, λ(xi ) is weakly increasing in xi by (A3) of section 2.1,
and xhi ≥ xli .
3
3
If bidders are risk averse, the argument in the proof of Lemma 1 is not applicable. For example,
suppose u(xik ) = xik and λ(xi ) = α > 0. Let r(·) : R+ → R+ be a concave function, where r′ > 0 and
s
r′′ < 0. Then, the expected utility function can be written as Vi (bi , xi , sj ) = Q1j (bi1 )[r(xi1 − bi1 )] +
sj
sj
Q2 (bi2 )[r(xi2 − bi2 )] + Q3 (bi1 , bi2 )[r(α + xi1 − bi1 + xi2 − bi2 ) − r(xi1 − bi1 ) − r(xi2 − bi2 )]. Following
the same procedure from (8) to(14), the inequality (13) is undetermined in the risk averse utility setting
, while (14) still holds.
7
s
s
RHS(11) − RHS(12) = Q1j (bli1 )[u(xli1 ) − u(xhi1 )] − Q1j (bhi1 )[u(xli1 ) − u(xhi1 )]
s
s
+Q2j (bli2 )[u(xli2 ) − u(xhi2 )] − Q2j (bhi2 )[u(xli2 ) − u(xhi2 )]
s
s
= [Q1j (bli1 ) − Q1j (bhi1 )][u(xli1 ) − u(xhi1 )]
s
+[Q2j (bli2 )
−
s
Q2j (bhi2 )][u(xli2 )
(14)
− u(xhi2 )]
≥ 0
s
s
The inequality comes from the fact that Q1j , Q2j and u are weakly increasing in their
arguments. ♦
By Lemma 1, Vi (bi , xi , sj ) satisfies single crossing differences.
2.3.2
Quasisupermodularity of Vi (bi , xi , sj )
Definition 2. Vi (bi , xi , sj ) is quasisupermodular if for all pure strategies sj of the opponent, all bi , b′i ∈ A, and every xi ∈ X 2 ,
Vi (bi , xi , sj ) ≥ (>)Vi (bi ∧ b′i , xi , sj ) implies Vi (bi ∨ b′i , xi , sj ) ≥ (>)Vi (b′i , xi , sj ).
(15)
If b′i ≥ bi , where ≥ is a coordinatewise partial order, (15) holds trivially. Therefore,
we focus on a case [(bi1 > b′i1 ) and (bi2 < b′i2 )]4 . Denote bi = (bhi1 , bli2 ) and b′i = (bli1 , bhi2 ),
where bhik > blik for k ∈ {1, 2}.
Substitute bi , b′i , to (15). That is,
Vi (bhi1 , bli2 , xi , sj ) − Vi (bli1 , bli2 , xi , sj ) ≥ (>)0
(16)
Vi (bhi1 , bhi2 , xi , sj ) − Vi (bli1 , bhi2 , xi , sj ) ≥ (>)0.
(17)
implies
(16) gives
s
s
Q1j (bhi1 )[u(xi1 ) − bhi1 ] + Q2j (bli2 )[u(xi2 ) − bli2 ]
s
+Q3j (bhi1 , bli2 )[w(xi1 , xi2 ) − u(xi1 ) − u(xi2 )]
≥ (>)
s
(18)
s
Q1j (bli1 )[u(xi1 ) − bli1 ] + Q2j (bli2 )[u(xi2 ) − bli2 ]
s
+Q3j (bli1 , bli2 )[w(xi1 , xi2 ) − u(xi1 ) − u(xi2 )]
s
s
s
s
⇔ Q1j (bhi1 )[u(xi1 ) − bhi1 ] − Q1j (bli1 )[u(xi1 ) − bli1 ] ≥ (>)[Q3j (bli1 , bli2 ) − Q3j (bhi1 , bli2 )]λ(xi ).
(19)
4
A proof for weak quasisupermodularity of this case similarly applies to the opposite case [(bi1 <
b′i1 ) and (bi2 > b′i2 )].
8
Rewriting (17),
s
s
s
s
Q1j (bhi1 )[u(xi1 )−bhi1 ]−Q1j (bli1 )[u(xi1 )−bli1 ] ≥ (>)[Q3j (bli1 , bhi2 )−Q3j (bhi1 , bhi2 )]λ(xi ). (20)
Since LHS of (19) and (20) are identical, a sufficient condition for quasisupermodularity
is [RHS (19) ≥ RHS (20)].
RHS(19) − RHS(20) =
λ(xi ) ·
(21)
s
s
s
s
[[Q3j (bhi1 , bhi2 ) − Q3j (bhi1 , bli2 )] − [Q3j (bli1 , bhi2 ) − Q3j (bli1 , bli2 )]].
With the assumption(A2) λ(xi ) ≥ 0 in section 2.1 , for quasisupermodularity, it is
sufficient to show that the next term of λ(xi ) in (21) is positive. That is, we want to
show that
s
s
s
s
[Q3j (bhi1 , bhi2 ) − Q3j (bhi1 , bli2 )] − [Q3j (bli1 , bhi2 ) − Q3j (bli1 , bli2 )] ≥ 0.
(22)
Lemma 2. Recall X 2 = [0, 1] × [0, 1] and A2 = {0, n1 , n2 , ..., ū} × {0, n1 , n2 , ..., ū}, n ∈ Z+
in the auction environment of section 2.1. In this environment, the inequality (22),
which is a sufficient condition of quasisupermodularity, holds for all pure-strategies of
the opponent.
Proof. Let xj ∈ X × X be object values of the opponent j and f (xj ) = probability(xj ).
First, we define qi (bi , bj ), i’s ex-post winning probability function for two objects, as
follows.



1
if bi > bj




1/2 if [(b > b ) and (b = b )] or [(b = b ]) and (b > b )]
i1
j1
i2
j2
i1
j1
i2
j2
qi (bi , bj ) =


1/4 if bi = bj




0
otherwise.
(23)
Suppose that the opponent j plays an arbitrary pure-strategy sj (xj ) = (sj1 (xj ), sj2 (xj )).
s
Then, given sj , the bidder i’s interim probability for winning both objects Q3j (bi ) is
s
Q3j (bi ) =
∫
xj ∈X 2
f (xj ) · qi (bi , sj (xj )) dxj .
9
(24)
As a result, the LHS of (22) can be expressed as follows.
s
s
s
s
[Q3j (bhi1 , bhi2 ) − Q3j (bhi1 , bli2 )] − [Q3j (bli1 , bhi2 ) − Q3j (bli1 , bli2 )]
∫
=
f (xj )[{qi (bhi1 , bhi2 , sj (xj )) − qi (bhi1 , bli2 , sj (xj ))}
xj ∈X 2
∫
=
xj ∈X 2
−{qi (bli1 , bhi2 , sj (xj )) − qi (bli1 , bli2 , sj (xj ))}] dxj
f (xj )[H(bhi1 , bli2 , bhi2 , sj (xj )) − D(bli1 , bli2 , bhi2 , sj (xj ))] dxj ,
(25)
where H(bhi1 , bli2 , bhi2 , sj (xj )) = qi (bhi1 , bhi2 , sj (xj )) − qi (bhi1 , bli2 , sj (xj ))
and D(bli1 , bli2 , bhi2 , sj (xj )) = qi (bli1 , bhi2 , sj (xj )) − qi (bli1 , bli2 , sj (xj )).
Note that when the agent i increases the second object’s bid from bli2 to bhi2 , H represents the increase in the ex-post winning probability for both objects, while bidding
high for the first object. D describes the same effect, when i bids low for the first object.
For the proof, we need to show that the second term of the integral in (25) is always
positive. That is, ∀sj (xj ), ∀b⃗i ∈ {(bl , bl , bh , bh )| bh > bl , k ∈ {1, 2}},
i1
i2
i1
i2
ik
ik
H(bhi1 , bli2 , bhi2 , sj (xj )) − D(bli1 , bli2 , bhi2 , sj (xj )) ≥ 0
(26)
.
Let βj = (βj1 , βj2 ) be an arbitrary two-dimensional bid vector generated by sj (xj )
and xj ∈ X 2 . Table 1 shows values of H(·) − D(·) for every 25 cases determined by the
values of (blik , bhik , βjk )k∈{1,2} . In the table, every cell is composed of a difference between
two brackets. The first bracket corresponds to the value of H(·), and the second bracket
corresponds to the value of D(·). For instance, a cell at the 3rd row and the 3rd column
in the table contains [ 14 − 0] − [0 − 0]. In this cell, the first bracket [ 41 − 0] corresponds
to H(bhi1 , bli2 , bhi2 , sj (xj )) = qi (bhi1 , bhi2 , sj (xj )) − qi (bhi1 , bli2 , sj (xj )), when [bli1 < bhi1 = βj1 ]
and [bli2 < bhi2 = βj2 ].
From the table, we observe that every value of cells is positive. Therefore, with
f (xj ) ≥ 0, the value of the integral in (25) is positive and the inequality (22) holds. ♦
By Lemma 2, the interim utility Vi (bi , xi , sj ) is quasisupermodular. Together with
Lemma 1, this completes the proof of Theorem 1.
3
Extension to multiple objects and agents environment
In this section, we show that the arguments in section 2 can be extended to multiple (more than two) objects and multiple agents cases by assuming a special type of
10
Table 1: H(bhi1 , bli2 , bhi2 , sj (xj )) − D(bli1 , bli2 , bhi2 , sj (xj ))
H −D
bli2 < bhi2 < βj2
bli2 < bhi2 = βj2
bli2 < βj2 < bhi2
bli2 = βj2 < bhi2
βj2 < bli2 < bhi2
bli1 < bhi1 < βj1
[0 − 0] − [0 − 0]
[0 − 0] − [0 − 0]
[0 − 0] − [0 − 0]
[0 − 0] − [0 − 0]
[0 − 0] − [0 − 0]
bli1 < bhi1 = βj1
[0 − 0] − [0 − 0]
[ 14 − 0] − [0 − 0]
[ 12 − 0] − [0 − 0]
[ 12 − 41 ] − [0 − 0]
[ 21 − 21 ] − [0 − 0]
bli1 < βj1 < bhi1
[0 − 0] − [0 − 0]
[ 12 − 0] − [0 − 0]
[1 − 0] − [0 − 0]
[1 − 21 ] − [0 − 0]
[1 − 1] − [0 − 0]
bli1 = βj1 < bhi1
[0 − 0] − [0 − 0]
[ 12 − 0] − [ 14 − 0]
[1 − 0] − [ 21 − 0]
[1 − 21 ] − [ 21 − 14 ]
[1 − 1] − [ 21 − 12 ]
βj1 < bli1 < bhi1
[0 − 0] − [0 − 0]
[ 12 − 0] − [ 12 − 0]
[1 − 0] − [1 − 0]
[1 − 12 ] − [1 − 21 ]
[1 − 1] − [1 − 1]
complementarity: bidders obtain complementarity by acquiring all objects on auctions.
Assumptions on bid and type spaces are same as section 2.1. That is, for each k, bik ∈ A,
xik ∈ X, and xi ∼ F (·) where F (·) is atomless. Therefore, for the extension, it is sufficient to argue whether the proofs of Lemma 1 and Lemma 2 are maintained in multiple
objects and multiple agents case. In the following sections, we discuss such extendability
under two objects and multiple agents setting; multiple objects and two agents setting;
and multiple objects and multiple agents setting.
3.1
Two objects and multiple agents case
Suppose k ∈ {1, 2} and i ∈ {1, 2, ..., I}. Then, the winning probabilities defined in (4)
change reflecting first-order statistics of the opponents’ bids on the two objects. Define
yk as the bidder i’s opponents’ highest bid on the object k. That is, yk = maxi′ ̸=i bi′ k .
Let µs−i be distributions of y1 and y2 constructed by the opponents’ pure-strategies s−i
and the type distribution F . Also, let τk (bik , j) be a probability that there are j number
of bi′ k s.t. bi′ k = yk when bidder i bids bik . Similarly, τ3 (bi1 , bi2 , j, m) is defined for
a probability that j and m ties occur for the object 1 and 2 respectively. Then, the
winning probabilities in (4) are modified as follows.
11
s
I−1
∑
s
I−1
∑
Q1−i (bi1 ) = µs−i ([0, bi1 ) × [0, ū]) + µs−i ([bi1 , bi1 ] × [0, ū]) ·
j=1
Q2−i (bi2 ) = µs−i ([0, ū] × [0, bi2 )) + µs−i ([0, ū] × [bi2 , bi2 ]) ·
j=1
s
Q3−i (bi1 , bi2 ) = µs−i ([0, bi1 ) × [0, bi2 )) + µs−i ([bi1 , bi1 ] × [0, bi2 )) ·
1
· τ1 (bi1 , j);
j+1
1
· τ2 (bi2 , j);
j+1
I−1
∑
j=1
+µs−i ([0, bi1 ) × [bi2 , bi2 ]) ·
I−1
∑
j=1
+µs−i ([bi1 , bi1 ] × [bi2 , bi2 ]) ·
1
· τ2 (bi2 , j)
j+1
I−1 ∑
I−1
∑
j=1 m=1
s
s
1
· τ1 (bi1(27)
, j)
j+1
1
1
·
· τ3 (bi1 , bi2 , j, m).
j+1 m+1
s
Given s−i , Q1−i , Q2−i and Q3−i are weakly increasing in their arguments. Therefore,
the proof for Lemma 1 (single crossing differences) is valid under the multiple agents
assumption. The proof for Lemma 2 (quasisupermodularity) is also maintained by
replacing values of qi in Table 1. For the change of the table, instead of βj1 and βj2 , cases
are divided by inequalities between (y1 , y2 ) and (bli1 , bli2 , bhi1 , bhi2 ). Under the multiple
agents assumption, qi in (23) is newly defined as follows.



1




 1
qi (bi , b−i ) =
j+1
1



 j+1


0
In (28),
1
j+1
if bik > yk , ∀k ∈ {1, 2};
if bi(3−k) > y(3−k) , and the number of ties for the object k is j, ∀k ∈ {1, 2};
·
1
m+1
if j and m ties occur for the object 1 and 2;
otherwise.
and
1
m+1
are bounded above by 21 . Calculating H(·) and D(·) based on
the new qi in (28), every value in Table 1 is positive. Therefore, Lemma 2 also holds
under the two objects and multiple agents setting.
3.2
Multiple objects and two agents case
Suppose k ∈ {1, 2, ..., N } = K and i ∈ {1, 2}. In addition to superadditivity (A2) and
monotonicity in a complementary value (A3) in section 2.1, we assume that bidder i
obtains the complementarity value by winning a whole set K.5 Let Γ be an arbitrary
proper subset of K. Then, the ex-post utility defined in (1) is modified as follows.
5
As the number of objects (auctions) increases, the number of ways to assume complementarity
between objects also exponentially increases.
12
(28)

∑


w(xi ) − N

k=1 (bik ) if bik > bjk , ∀k ∈ K

∑
Ui (bi , bj , xi ) =
if bik > bjk , ∀k ∈ Γ and bi2 < bj2 , ∀k ∈ K\Γ(29)
k∈Γ u(xik ) − bik



0
otherwise.
Note that when ties occur, the object is allocated randomly to one of the two bidders
with probability 12 . The assumptions on u(·) and w(·), (A1), (A2), and (A3) in section
∑
2.1, are maintained for this extension with λ(xi ) = w(xi ) − k∈K u(xik ). Probability
functions for winning at least each k ∈ K and winning a whole set K are denoted by
s
s
Qkj (bik ) and QKj (bi ) respectively. Then, the expected utility function can be defined as
follows.
Vi (bi , xi , sj ) =
N
∑
s
s
Qkj (bik ) · [u(xik ) − bik ] + QKj (bi ) · λ(xik )
(30)
k=1
Conditions for single crossing differences, (31) and (32), which are modified versions
of (13) and (14), hold for the multiple objects case as below with the monotonicity
s
s
assumptions of Qkj , QKj , u, and λ.
s
[QKj (bhi ) − QK (bli )] · [λ(xhi ) − λ(xli )] ≥ 0.
N
∑
s
s
[Qkj (blik ) − Qkj (bhik )] · [u(xlik ) − u(xhik )] ≥ 0.
(31)
(32)
k=1
To extend the quasisupermodularity proof in Lemma 2, let bi , b′i be bidder i’s two
arbitrary N -dimensional bids. Define J to be a subset of K, where ∀k ∈ J, bik ≥ b′ik
and let J ′ = K\J. Then, ∀k ∈ J ′ ⊂ K, bik < b′ik . Denote bi = (bhJ , blJ ′ ) and b′i =
(bliJ , bhiJ ′ ), where for ∀k ∈ K, bhik ≥ blik , and [bhiJ = {bhik }k∈J ; bliJ ′ = {blik }k∈J ′ ; bliJ =
{blik }k∈J ; and bhiJ ′ = {bhik }k∈J ′ ]. Let y be the opponent’s bids for N objects generated
by arbitrary sj and xj . Given (bhik , blik )k∈K , the opponent’s bids y can be categorized
into four cases.
1. y ∈ ϕ1 ⊂ AN ⇒ ∃k ∈ K, yk > bhik .
2. y ∈ ϕ2 ⊂ AN \ϕ1 ⇒ ∃k s.t. yk = bhik .
3. y ∈ ϕ3 ⊂ AN \(ϕ1 ∪ ϕ2 ) ⇒ ∃k s.t. yk > blik .
4. y ∈ ϕ4 ⊂ AN \(ϕ1 ∪ ϕ2 ∪ ϕ3 ) ⇒ ∀k, yk ≤ blik .
To satisfy quasisupermodularity, we need to show that the inequality (26) holds for
all of the four cases above. That is, for all bi , b′i , y ∈ AN ,
13
H(bhiJ , bliJ ′ , bhiJ ′ , y) − D((bliJ , bhiJ ′ , bhiJ , y)
(33)
= [qi (bhiJ , bhiJ ′ , y) − qi (bhiJ , bliJ ′ , y)] − [qi (bliJ , bhiJ ′ , y) − qi (bliJ , bliJ ′ , y)] ≥ 0.
The following part shows that given arbitrary bi and b′i , the inequality (33) holds
∀y ∈ AN .
1. Let y ∈ ϕ1 .
In this case, every qi (·) = 0. Therefore, LHS (33) = [0 − 0] − [0 − 0] = 0
2. Let y ∈ ϕ2 .
2(a). Suppose that for some k ∈ J, [yk = bhik > blik ]. Then, D((bliJ , bhiJ ′ , bhiJ , y) =
[qi (bliJ , bhiJ ′ , y) − qi (bliJ , bliJ ′ , y)] = 0. H(bhiJ , bliJ ′ , bhiJ ′ , y) ≥ 0 as qi (·) is increas-
ing in bidder i’s bids. Therefore, LHS (33) ≥ 0. If such k ∈ J ′ , then
qi (bhiJ , bliJ ′ , y) = 0 and LHS (33) ≥ 0.
2(b) Suppose that 2(a) is not the case. Then, there are some k ∈ J ∪ J ′ s.t.
[yk = bhik = blik ]. Let j be a set of objects k ∈ J s.t. [yk = bhik = blik ]. Similarly, let j ′ be a subset of objects k ∈ J ′ s.t. [yk = bhik = blik ]. If |j| = |J| and
1
|j ′ | = |J ′ |, then LHS (33) =[ 2( |j|+|j
′ |) −
1
1
] − [ 2( |j|+|j
′ |)
2( |j|+|j ′ |)
−
1
]
2( |j|+|j ′ |)
= 0.
Assume |j| < |J| or |j ′ | < |J ′ |. Suppose that ∃yk s.t. k ∈ J\j = j c and bhik >
yk > blik . Then, D(·) = 0. Therefore, LHS (33) ≥ 0. Suppose that such
k ∈ J ′ \j ′ = j ′c , then qi (bhiJ , bliJ ′ , y) = 0 and LHS (33) ≥ 0. Now, assume
∀k ∈ j c or j ′c , bhik > blik ≥ yk . Let η be a subset of j c s.t. ∀k ∈ η, blik = yk ,
and let η ′ be a subset of j ′c s.t. ∀k ∈ η ′ , blik = yk . Then, for all |η|, |η ′ |
1
s.t. 0 ≤ |η| ≤ |j c | and 0 ≤ |η ′ | ≤ |j ′c |, LHS (33) = [ 2|j|+|j
′| −
−[ 2|j|+|η| ·
1
1
2|j
′|
−
1
2|j|+|η|
·
1
2|j
′ |+|η ′ |
] ≥ 0.
1
2|j|
·
1
2|j
′ |+|η ′ |
]
3. Let y ∈ ϕ3 .
Suppose that for some k ∈ J, yk > blik . Then, qi (bliJ , bhiJ ′ , y) = 0. Therefore,
qi (bliJ , bhiJ ′ , y) − qi (bliJ , bliJ ′ , y) = 0. Since [qi (bhiJ , bhiJ ′ , y) − qi (bhiJ , bliJ ′ , y)] ≥ 0, LHS
(33) ≥ 0. If such k ∈ J ′ , qi (bhiJ , bliJ ′ , y) = 0 and LHS (33) ≥ 0.
4. Let y ∈ ϕ4 .
Suppose that ∀k ∈ K, yk < blik . Then, LHS (33) = [1 − 1] − [1 − 1] = 0.
Otherwise, let j be a subset of J s.t. ∀k ∈ j, yk = blik , and let j ′ be a subset of J ′
s.t. ∀k ∈ j ′ , yk = blik . Then, for all |j|, |j ′ |, s.t. 0 ≤ |j| ≤ |J|, 0 ≤ |j ′ | ≤ |J ′ |, and
j + j ′ ≥ 1, LHS (33) = 1 −
1
2|j|
−
1
′
2|j |
+
14
1
2|j|
·
1
′
2|j |
≥ 0.
3.3
Multiple objects and multiple agents case
s
s
s
Qkj and QKj in section 3.2 (multiple objects and two agents case) can be rewritten as Qk−i
s
s
s
s
and QK−i assuming multiple opponents, similarly to Q1−i (bi1 ), Q2−i (bi2 ), and Q3−i (bi ) in
(34). Then, the inequalities (31) and (32) still hold by the monotonicity assumptions
s
s
of Qk−i , QK−i , u, and λ. The argument for quasisupermodularity in section 3.2 is also
extendable to the multiple agents case by defining y to be a vector of first-order statistics
in the opponents’ bids and adjusting the probabilities of ties. Therefore, a monotone
pure-strategy Bayesian Nash equilibrium exists in the simultaneous first-price auctions
with N objects and I bidders when complementarity is realized by obtaining all objects.
4
Discussion
In this section, we provide two counterexamples showing that the results of section 2
cannot be generalized very far beyond the special case that was considered in section
3. Especially, the second example will show that quasisupermodularity of the expected
utility function is quite a fragile property. In each of these examples, three objects are
auctioned. In the first example, complementarity exists only between any two of the
three objects. In the second example, the payoff function that gives complementarity is
symmetric in the three objects’ private values.
First, let us specify a utility function and state the conditions of single crossing differences and quasisupermodularity in three objects cases. Suppose k ∈ {1, 2, 3} = K
and i ∈ {1, 2}. Let zi ∈ Z = {0, 1}3 , where zik = 1 if bidder i wins object k. Otherwise,
zik = 0. Then, the ex-post utility function u(·)is a function of zi and xi . All other
assumptions on bid, private values (e.g. bounds, distributions), and notations are same
as section 2.1. Also, each equation below with numbering [♯] corresponds to the equation
(♯) as a three objects version.
The bidder i’s expected utility is
[3] Vi (bi , xi , sj ) =
∑
s
Pzij (bi ) · [u(zi , xi ) − zi · bi ].
zi ∈Z
Let e1 = (1, 0, 0), e2 = (0, 1, 0), and e3 = (0, 0, 1). Then, we can modify the expected
utility as follows.
15
[5] Vi (bi , xi , sj ) =
+
+
+
+
∑
s
Qejk (bik ) · [u(ek , xi ) − bik ]
k=K
sj
Q(1,1,0)
(bi1 , bi2 ) · [u((1, 1, 0), xi ) − u((1, 0, 0), xi ) − u((0, 1, 0), xi )]
sj
Q(1,0,1) (bi1 , bi3 ) · [u((1, 0, 1), xi ) − u((1, 0, 0), xi ) − u((0, 0, 1), xi )]
sj
Q(0,1,1)
(bi2 , bi3 ) · [u((0, 1, 1), xi ) − u((0, 1, 0), xi ) − u((0, 0, 1), xi )]
sj
Q(1,1,1) (bi1 , bi2 , bi3 ) · [u((1, 1, 1), xi ) + u((1, 0, 0), xi ) + u((0, 1, 0), xi )
+ u((0, 0, 1), xi )
− u((1, 1, 0), xi ) − u((1, 0, 1), xi ) − u((0, 1, 1), xi )].
For notational simplicity, define β(xi ) and α(bi , xi ) as follows.
β(xi ) = u((1, 1, 1), xi ) + u((1, 0, 0), xi ) + u((0, 1, 0), xi ) + u((0, 0, 1), xi )
−u((1, 1, 0), xi ) − u((1, 0, 1), xi ) − u((0, 1, 1), xi ).
s
j
(bi1 , bi2 ) · [u((1, 1, 0), xi ) − u((1, 0, 0), xi ) − u((0, 1, 0), xi )]
α(bi , xi ) = Q(1,1,0)
s
j
(bi1 , bi3 ) · [u((1, 0, 1), xi ) − u((1, 0, 0), xi ) − u((0, 0, 1), xi )]
+ Q(1,0,1)
s
j
+ Q(0,1,1)
(bi2 , bi3 ) · [u((0, 1, 1), xi ) − u((0, 1, 0), xi ) − u((0, 0, 1), xi )]
s
j
(bi1 , bi2 , bi3 ) · β(xi ).
+ Q(1,1,1)
Then,
Vi (bi , xi , sj ) =
∑
s
Qejk (bik ) · [u(ek , xi ) − bik ]
(34)
k=K
+ α(bi , xi ).
Let bhi ≥ bli , and xhi ≥ xli . Based on (34), inequalities (11) and (12) in section 2.3.1,
statements for the proof of single crossing differences of Vi (bi , xi , sj ), can be rewritten
for a three objects environment. The following are the modified ones.
[11]
α(bhi , xli ) − α(bli , xli ) ≥ (>)
∑
s
s
[Qejk (blik ) · [u(ek , xli ) − blik ] − Qejk (bhik ) · [u(ek , xli ) − bhik ]]
k∈K
[12]
α(bhi , xhi ) − α(bli , xhi ) ≥ (>)
∑
s
s
[Qejk (blik ) · [u(ek , xhi ) − blik ] − Qejk (bhik ) · [u(ek , xhi ) − bhik ]]
k∈K
16
Define b̌i = bi ∨ b′i and b̂i = bi ∧ b′i . Then, inequalities (19) and (20) in section 2.3.2,
statements for the proof of quasisupermodularity of Vi (bi , xi , sj ), can be rewritten as
follows.
[19]
∑
s
s
Qejk (bik ) · [u(ek , xik ) − bik ] − Qejk (b̂ik ) · [u(ek , xik ) − b̂ik ] ≥ (>)α(b̂i , xi ) − α(bi , xi )
k∈K
[20]
∑
Qejk (b̌ik ) · [u(ek , xik ) − b̌ik ] − Qejk (b′ik ) · [u(ek , xik ) − b′ik ] ≥ (>)α(b′i , xi ) − α(b̌i , xi )
s
s
k∈K
4.1
Complementarity from winning two objects
Suppose that every pair of the objects is complementary with one another, but winning
all three objects does not add any further complementary value. We consider a specific
utility function below.
• u((1, 0, 0), xi ) = xi1 , u((0, 1, 0), xi ) = xi2 , u((0, 0, 1), xi ) = xi3 .
• u((1, 1, 0), xi ) = xi1 + xi2 + xi1 xi2 ,
u((1, 0, 1), xi ) = xi1 + xi3 + xi1 xi3 ,
u((0, 1, 1), xi ) = xi2 + xi3 + xi2 xi3 .
• u((1, 1, 1), xi ) = xi1 + xi2 + xi3 + max{xi1 xi2 , xi1 xi3 , xi2 xi3 }.
Naturally, it satisfies superadditivity (A2) and monotone complementarity (A3) in section 2.1. Superadditivity (A2) in the three objects case can be explicitly written as
follows. ∀zi , zi′ , zi′′ ∈ Z s.t. zi = zi′ + zi′′ , u(zi , xi ) ≥ u(zi′ , xi ) + u(zi′′ , xi ).
Then, the expected utility function is
Vi (bi , xi , sj ) =
+
+
+
+
∑
s
Qejk (bik ) · [xik − bik ]
k∈K
sj
Q(1,1,0)
(bi1 , bi2 ) · xi1 xi2
sj
Q(1,0,1) (bi1 , bi3 ) · xi1 xi3
sj
Q(0,1,1)
(bi2 , bi3 ) · xi2 xi3
sj
Q(1,1,1) (bi1 , bi2 , bi3 ) · β(xi ).
Without loss of generality, we consider a case where max{xi1 xi2 , xi1 xi3 , xi2 xi3 } =
xi1 xi2 . Then,
β(xi ) = −xi1 xi3 − xi2 xi3 < 0.
17
4.1.1
Single Crossing Differences of Vi (bi , xi , sj )
As noted in the proof of Lemma 1, we want to show that LHS[12] ≥ LHS[11] and RHS[11] ≥
RHS[12]. Like the two objects case in section 2, in this example, RHS[11]−RHS[12] ≥ 0
s
as Qzji is increasing in its arguments.
Substituting the utility function and β(xi ) to [11] and [12], LHS[12] − LHS[11] is
s
s
j
j
[Q(1,1,0)
(bhi1 , bhi2 ) − Q(1,1,0)
(bli1 , bli2 )] · [xhi1 xhi2 − xli1 xli2 ]
[13]
s
s
s
s
j
j
+ [P(1,0,1)
(bhi ) − P(1,0,1)
(bli )] · [xhi1 xhi3 − xli1 xli3 ]
j
j
+ [P(0,1,1)
(bhi ) − P(0,1,1)
(bli )] · [xhi2 xhi3 − xli2 xli3 ].
s
Note that ∀k, k ′ ∈ K, xhik xhik′ − xlik xlik′ ≥ 0, and Qzji is increasing in its arguments.
s
s
However, unlike Qzji , a value of Pzij depends on all of the three objects’ bids bi and is not
s
s
monotone in its arguments. Also, a sign of [Pzij (bhi ) − Pzij (bli )] varies in sj . Therefore, a
sign of [13] cannot be generally determined, while (13) can in the proof of Lemma 1.
4.1.2
Quasisupermodularity of Vi (bi , xi , sj )
As noted in section 2.3.2, we want to show that LHS[20] ≥ LHS[19] and RHS[19] ≥
RHS[20]. Since LHS[20] − LHS[19] = 0 in this example, we only need to check
RHS[19] − RHS[20] as follows.
[21]
j
j
j
j
[Q(1,1,0)
(b̌i1 , b̌i2 ) + Q(1,1,0)
(b̂i1 , b̂i2 ) − Q(1,1,0)
(bi1 , bi2 ) − Q(1,1,0)
(b′i1 , b′i2 )] · xi1 xi2
s
s
s
s
j
j
j
j
+ [P(1,0,1)
(b̌i1 , b̌i3 ) + P(1,0,1)
(b̂i1 , b̂i3 ) − P(1,0,1)
(bi1 , bi3 ) − P(1,0,1)
(b′i1 , b′i3 )] · xi1 xi3
s
s
s
s
j
j
j
j
(b′i2 , b′i3 )] · xi2 xi3 .
(bi2 , bi3 ) − P(0,1,1)
+ [P(0,1,1)
(b̌i2 , b̌i3 ) + P(0,1,1)
(b̂i2 , b̂i3 ) − P(0,1,1)
s
s
s
s
s
s
s
s
j
j
j
j
(b′i1 , b′i2 )] ≥ 0 by
(bi1 , bi2 ) − Q(1,1,0)
While [Q(1,1,0)
(b̌i1 , b̌i2 ) + Q(1,1,0)
(b̂i1 , b̂i2 ) − Q(1,1,0)
s
Lemma 2, supermodularity of Pzij for zi ∈ {(1, 0, 1), (0, 1, 1)} in [21] cannot be guaranteed because of its dependence on bi , sj , and its non-monotonicity. Therefore, a sign of
[21] cannot be generally determined, while (21) can in Lemma 2.
4.2
Symmetric complementarity
Suppose that complementarity only depends on the number of goods that each bidder
wins. We specifically consider a utility function below.
• u((1, 0, 0), xi ) = xi1 , u((0, 1, 0), xi ) = xi2 , u((0, 0, 1), xi ) = xi3 .
18
• u((1, 1, 0), xi ) = xi1 + xi2 + η,
u((1, 0, 1), xi ) = xi1 + xi3 + η,
u((0, 1, 1), xi ) = xi2 + xi3 + η.
• u((1, 1, 1), xi ) = xi1 + xi2 + xi3 + 2η.
This utility function represents a situation where bidders’ utility level is calculated
by adding up the private values of the obtained objects and additional η(> 0) for an
increase of the number of winning.6
Then, β(xi ) = −η, and the expected utility function is
Vi (bi , xi , sj ) =
∑
s
Qejk (bik ) · [xik − bik ]
k=K
s
j
+ Q(1,1,0)
(bi1 , bi2 ) · η
s
j
+ Q(1,0,1)
(bi1 , bi3 ) · η
s
j
+ Q(0,1,1)
(bi2 , bi3 ) · η
s
j
+ Q(1,1,1)
(bi1 , bi2 , bi3 ) · (−η).
Unlike the example in section 4.1, Vi (bi , xi , sj ) in this example satisfies single crossing
s
differences. Specifically, RHS[11] − [12] ≥ 0 as Qzji is increasing in its arguments,
and LHS[12] − LHS[11] = 0 as the complementary value is a constant. However,
quasisupermodularity of Vi (bi , xi , sj ) cannot be guaranteed due to the same reason of
the previous example. Specifically, [21] can be rewritten for the current example as
follows.
[21′ ]
j
j
j
j
[Q(1,1,0)
(b̌i1 , b̌i2 ) + Q(1,1,0)
(b̂i1 , b̂i2 ) − Q(1,1,0)
(bi1 , bi2 ) − Q(1,1,0)
(b′i1 , b′i2 )] · η
s
s
s
s
j
j
j
j
(b′i1 , b′i3 )] · η
(bi1 , bi3 ) − Q(1,0,1)
(b̂i1 , b̂i3 ) − Q(1,0,1)
(b̌i1 , b̌i3 ) + Q(1,0,1)
+ [Q(1,0,1)
s
s
s
s
j
j
j
j
(bi2 , bi3 ) − P(0,1,1)
(b′i2 , b′i3 )] · η.
(b̂i2 , b̂i3 ) − P(0,1,1)
+ [P(0,1,1)
(b̌i2 , b̌i3 ) + P(0,1,1)
s
s
s
s
∑3
While supermodularity of Qzji for zi ∈ {zi |
2, supermodularity of
s
Pzij
s
k=1 zik
= 2} is satisfied by Lemma
for zi = (0, 1, 1) in [21′ ] can’t be guaranteed because of its
dependence on bi , sj and its non-monotonicity. Therefore, a sign of [21′ ] cannot be generally determined, while (21) can in Lemma 2.
5
Conclusion
This paper has shown the existence of monotone pure-strategy Bayesian Nash equilibria in two simultaneous first-price auctions, where the objects have complementarity
6
It also satisfies superadditivity (A2) and monotone complementarity (A3).
19
in bidders’ utility. Under the complementarity assumptions, single crossing differences
and quasisupermodularity of interim utility are satisfied, so McAdams (2003)’s existence
theorem is applicable to this environment. The monotone pure-strategy Bayesian Nash
equilibrium also exists in the general N number of simultaneous first-price auctions if
bidders obtain complementarity solely by acquiring all the N objects on the auctions.
However, it is also confirmed that such extension cannot be compatible with some other
types of complementarity. Therefore, as future research, it is natural to study other
necessary and sufficient conditions for the extension.
Also, it is meaningful to think about the simultaneous auctions environment in a
continuum action space setting. As Athey (2001) argues in the paper, it would support
a differential equation approach to verify existence conditions. She provides the extension with a continuum action space by showing ties do not occur in the game. Reny
(2011) also presents the extension through a better-reply secure game and a limit of a
pure-strategy of ϵ equilibria.
Appendix
A.1 Example of the two objects and two agents case
Consider an environment of simultaneous first-price auctions with two objects and two
agents. For each agent i, the private valuation of each good xik is either 0 or 2 with
probability 21 , and each bid bik for k ∈ {1, 2} is submitted from A = {0, 1} simultaneously. Thus, there exist four types of xi ∈ {(0, 0), (0, 2), (2, 0), (2, 2)}, and 28 types of
pure-strategies. u(xik ) = xik and w(xi1 , xi2 ) = xi1 + xi2 + α, where α = 2. That is, if
the bidder i wins both items, complementarity value α = 2 is created.
In this environment, there exist five different pure-strategy Bayesian Nash equilibria
, which are all monotone. Four equilibria are symmetric (Table 2), and one equilibrium
is asymmetric (Table 3). In Table 2(a)-(d) and Table 3, the first three columns present
each equilibrium strategy si (xi ). For example, in Table 2(a), the cell (0, 1) corresponds
to si (0, 2). The next three columns specify expected utility V , probability of winning
both goods Q3 , and expected payment E(pay) of a bidder for each equilibrium. In the
result of this example, there are three notable points.
First, we can observe over-bidding behavior in equilibrium strategies due to the complementarity. In the equilibrium described in Table 2(b), si (2, 0) = (1, 1). Therefore,
under this equilibrium, when bidders’ type is (2, 0), bidders bid 1 for the second object,
even though the single valuation for the second object is 0. Such over-bidding is also
20
observed in the equilibrium of Table 2(c), 2(d), and 3(a).
Second, the asymmetric equilibrium is consist of an aggressive (high) bidding strategy
(Table 3(a)) and an conservative (low) bidding strategy (Table 3(b)). Such equilibrium
can be justified in the following way. Given the fact that the opponent plays an conservative bidding without any over-bidding, the bidder i has high chances to win both items
and to enjoy the complementary value if he bids aggressively for both goods. Similarly, if
the opponent plays an aggressive bidding with over-bidding, the bidder i has low chances
to win both items, so bidding based on single good valuation can be rational in such case.
Finally, average probability of winning both goods is maximized in a symmetric
equilibrium (Table 2(d)) where both players employ an aggressive bidding strategy.
Note that utility functions are linear and the utilities are transferable between sellers
and bidders. Therefore, a social planner who maximizes the sum of utilities of two agents
and two sellers would be interested in increasing a probability that the complementary
value is created. Such probability is maximized when both bidders bid aggressively as
Q3 = 0.3438 in Table 2(d).
21
Table 2: Symmetric Equilibria
(a)
xi1 \xi2
0
2
V
Q3
E(pay)
0
(0,0)
(0,1)
1.25
0.25
0.75
2
(1,0)
(1,1)
(b)
xi1 \xi2
0
2
V
Q3
E(pay)
0
(0,0)
(0,1)
1.1562
0.3125
0.8438
2
(1,1)
(1,1)
(c)
xi1 \xi2
0
2
V
Q3
E(pay)
0
(0,0)
(1,1)
1.1562
0.3125
0.8438
2
(1,0)
(1,1)
(d)
xi1 \xi2
0
2
V
Q3
E(pay)
0
(0,0)
(1,1)
1
0.3438
0.9375
2
(1,1)
(1,1)
22
Table 3: Asymmetric equilibrium
(a) s1
xi1 \xi2
0
2
V
Q3
E(pay)
0
(0,0)
(1,1)
1.25
0.4375
1.125
2
(1,1)
(1,1)
(b) s2
xi1 \xi2
0
2
V
Q3
E(pay)
0
(0,0)
(0,1)
1
0.1875
0.6250
2
(1,0)
(1,1)
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