Sealed-bid first-price auctions with an unknown
number of bidders
Erik Ekström
Department of Mathematics, Uppsala University
Carl Lindberg
The Second Swedish National Pension Fund
e-mail: [email protected], [email protected]
April 13, 2016
Abstract
We consider sealed-bid first-price auctions under uncertainty about the
total number of bidders, and we determine the optimal trading behaviour
for an invited bidder in the case of fully observable common values. More
precisely, we find a symmetric Nash equilibrium in mixed strategies, and
we relate our equilibrium with the classical equilibrium in the setting of
independent private values. Additionally, extensions to cases with several
potential buyers or asymmetric views are discussed.
Key words: game theory; auctions
Mathematics Subject Classification (2010): 91B26, 91A05
1
Introduction
In sealed-bid auctions there is often a natural uncertainty about the number of
bidders. For example, consider a house owner who wants her house repainted.
Naturally, to get a good price, she should ask as many painters as possible.
However, if each invitation has an associated cost, then there will be a trade-off
between inviting many contractors for a competitive price and fewer ones to
keep the total invitation cost small. From the painters’ perspective, this yields
an uncertainty about the number of invited bidders.
We study sealed-bid first-price auctions with an unknown number of riskneutral bidders under the assumption that all auction participants agree about
the value of the good and that this value is observable. While a set-up with
an observable value which is common to all bidders is more simplistic than
the standard model in auction theory using bidders with independent private
values (IPV), the equilibrium we derive under uncertainty about the number of
bidders is surprisingly involved. In fact, there is no equilibrium in pure bidding
strategies, but we instead show that there is a unique symmetric equilibrium
1
in mixed (randomized) strategies. Thus a bidder who faces uncertainty about
the competition would place a random bid drawn from a certain distribution.
In some sense, and in contrast to the IPV setting in which equilibrium bids are
deterministic functions of the value, this captures a speculative element often
observed in real-world bidding.
Of course, the set-up with an observable common value may be regarded
as a special case of an auction with independent private values drawn from
a one-point probability distribution. However, that case is typically excluded
from the analysis of such auctions since strong non-degeneracy conditions on the
distribution of private values are imposed. Nevertheless, we show that our solution, based on mixed bidding strategies, is closely related to the deterministic
bidding strategies in the classical IPV setting. For example, as the distribution
of private values shrinks to a one-point distribution, the distribution of bids
in the IPV setting converges to the distribution of the mixed strategy in the
observable value setting. Moreover, the equilibrium in mixed strategies serves,
in a certain sense, as a building block for the solution in the case of auctions
with independent private values drawn from a discrete distribution.
In Section 2 we study a basic version of sealed-bid first-price auctions in
which the number of bidders is unknown but at most two. In particular, we
prove existence and uniqueness of a symmetric equilibrium in mixed strategies.
In Section 3 we discuss connections between our set-up with a known common
value with the classical set-up with independent private values. Sections 4-6
extend the basic version studied in Section 2 in various directions. In Section 4,
we show that in the presence of an additive invitation cost, there is an auction
equilibrium which includes also the seller as one of the players. In particular,
we show that there exists a probability distribution on the number of invited
bidders that in equilibrium should be used by the seller. Section 5 extends our
results to the case of an arbitrary number of potential bidders, and Section 6
contains a short study of the case with asymmetric, but still fully observable,
private values and probability distributions of the number of bidders.
2
The basic version: two potential bidders
In this section we study an auction from the perspective of an invited bidder
who does not know whether a second potential bidder was also invited or not.
Instead, the probability of being the only bidder has been estimated to be
p ∈ (0, 1), and thus the probability that the other potential bidder was also
invited is 1 − p. We assume that all bidders agree on the value of the good
and that this value is a known constant v > 0. We also assume that the game
is symmetric in the sense that also the second player, if invited, estimates the
probability of being the only bidder to be p. Then, if the invited bidder bids
x, and the second bidder (if invited) bids y, the expected payoff for the first
invited bidder is
p(v − x) + (1 − p)(v − x)1{x>y} + (1 − p)
2
v−x
1{x=y}
2
(here the last term accounts for the possibility of equal bids, in which case we
assume that the profit is split evenly between the bidders).
A mixed strategy consists of a distribution function F on [0, v]. For a pair
(F, G) of such distribution functions, let (X, Y ) be a pair of independent random
variables with F (x) = P(X ≤ x) and G(y) = P(Y ≤ y). The corresponding
expected payoff to the invited bidder is then
v−X
1{X=Y } .
pE[v − X] + (1 − p)E (v − X)1{X>Y } +
2
For a pair (F, F ) of distribution functions to be a symmetric Nash equilibrium,
one expects that F is continuous with F (0) = 0, F (v) = 1 and that
p(v − x) + (1 − p)(v − x)F (x)
is constant on the support of F . This leads to a candidate equilibrium (F, F )
where F is given by
px
x ∈ [0, (1 − p)v]
(1−p)(v−x)
F (x) =
(1)
1
x ∈ ((1 − p)v, v].
Theorem 1 The pair (F, F ), where F is defined in (1), is a Nash equilibrium
in mixed strategies. Moreover, it is the unique symmetric equilibrium.
Proof. First we check that (F, F ) is an equilibrium. If an invited player bids
x, then an average payoff
pv
x ≤ (1 − p)v
p(v − x) + (1 − p)(v − x)F (x) =
v − x x > (1 − p)v
is received provided the other player (if invited) uses the strategy F . Since
v − x ≤ pv for x ≤ (1 − p)v, this means that no strategy can give more than pv
on average. Since the strategy F puts all mass on [0, (1 − p)v] where the average
payoff is constant (and maximal), the strategy F is an optimal response to F ,
so (F, F ) is a Nash equilibrium.
For uniqueness, assume that (G, G) is a Nash equilibrium in mixed strategies.
First note that G has to be continuous with G(0) = 0 and G(v−) = 1. To see
this, let G(0−) := 0 and assume that G(a) − G(a−) > 0 for some a ∈ [0, v), and
define G by
G(x) if x ∈
/ [a, a + )
G (x) =
G(a) if x ∈ [a, a + ).
Then, for > 0 small enough, G is a strictly better response to G than G,
so (G, G) is not an equilibrium. Similarly, if G(v) − G(v−) = η > 0, then the
strategy given by
G(x) + η if x ∈ [0, v)
G̃(x) =
G(v)
if x = v
3
is a strictly better response to G than G, so again (G, G) cannot be an equilibrium. Consequently, a symmetric equilibrium has to have a continuous distribution function with G(0) = 0 and G(v−) = G(v) = 1.
Next consider the function
g(x) = p(v − x) + (1 − p)(v − x)G(x),
i.e. the average gains from bidding x in case the other bidder (if invited) uses
the strategy G. Then g is continuous on [0, v], and we denote its maximum g.
We claim that g ≡ g on the support of dG. To see this, assume that x ∈
supp(dG) and that g(x) < g. By continuity, g < g in some interval (x − , x + ),
and since x is in the support of dG, we have G(x + ) − G(x − ) > 0. Let x0 be
a point such that g(x0 ) = g. Then the strategy that puts all mass at x0 gives an
average payoff g, which strictly exceeds the payoff from playing G, so (G, G) is
not an equilibrium.
If y ∈ (0, v) with g(y) < g, then y ∈
/ supp(dG) so G is constant in an open
interval containing y. Therefore g is strictly decreasing, and by the continuity
of g it follows that the interval (y, v] also belongs to (supp(dG))c . This proves
that supp(dG) = [0, x] for some x ∈ [0, v]. Consequently, g(x) = g on [0, x], so
G(x) =
g
p(v−x)
−
1
p
1−p
x ∈ [0, x]
x ∈ [x, v].
(2)
The only continuous function G with G(0) = 0 of the form (2) is G = F specified
in (1), which completes the proof.
Remark 2 It is easy to check that the game degenerates in the excluded cases
p = 0 and p = 1. In fact, if p = 1, i.e. if there is only one bidder, then there
is no competition, and hence the optimal strategy for the bidder is to bid 0. On
the other hand, if p = 0, then each bidder knows that there is exactly one more
bidder, and a situation with pre-emption appears. The symmetric equilibrium
in this case consists of bids of size v, thus generating no profit at all.
Remark 3 Any bid x ∈ supp(F ) gives an average payoff
p(v − x) + (1 − p)(v − x)F (x) = pv.
Consequently, the expected profit for an invited bidder is pv.
3
Connections with auctions with independent
private values
In this section we discuss the equilibrium in mixed strategies found above in
connection with the classical set-up of an auction with independent private values. In particular, we show that the equilibrium in mixed strategies for the case
of an unknown number of bidders serves as a building block for the equilibrium
4
in the case of independent private values with a discontinuous distribution. We
also show that the mixed strategy may be viewed as the limit of bids in a setting with independent private values as the distribution of values shrinks to a
one-point distribution.
3.1
Sealed-bid first-price auctions with independent private values
In this subsection we describe the classical case of an auction with independent
private values drawn from a distribution with a continuous density. Further
details in the case with a known number of bidders can be found in the classical
reference [6]; for the case of an unknown number of bidders, see [2], [4] and [5].
The results also appear in most text books on auction theory, see for example
the nice monograph [3].
In a classical set-up of an auction in which the bidders have independent
private values drawn from a distribution with a continuous density, the symmetric equilibrium strategy is not a mixed strategy, but instead an equilibrium
is obtained if each invited player bids a certain deterministic function of his/her
private value. Assume that the private values are drawn from a distribution
with distribution function H with support [a, b] ⊂ [0, ∞), and that the first
bidder uses an increasing function α : [a, b] → [a, b] to map his/her private value
into a bid. Then the expected payoff for the second bidder who bids y is
(x − y)H(α−1 (y)),
where x is the observed value. Optimizing over the bid y and using α(x) = y for
a symmetric equilibrium (for further details, see [3, Chapter 2.3]), one obtains
Rx
xH(x) − a H(z) dz
.
(3)
α(x) =
H(x)
We now keep the assumption of independent private values with distribution function H, but add uncertainty about the number of bidders. If an invited
bidder estimates the probability of being the only bidder to be p, and the probability that there is exactly one more bidder to 1 − p, then the expected payoff
for an invited bidder who bids y is
p(x − y) + (1 − p)(x − y)H(β −1 (y)).
Here β : [a, b] → [a, b] is an increasing function that is used by the second bidder
(if invited) to map his/her private value into a bid, and x is the private value
of the invited bidder. Again, optimizing over the bid y and using β(x) = y for
a symmetric equilibrium (for details, see [2] or [3, Chapter 3.2]), one obtains
Rx
(1 − p)(xH(x) − a H(z) dz)
β(x) =
.
(4)
p + (1 − p)H(x)
5
3.2
Independent private values with discrete distributions
In the classical set-up with two bidders with independent private values, a standard assumption for the derivation of the symmetric equilibrium described by
the bidding function α in (3) is the existence of a continuous and strictly positive density of the distribution function H. In the absence of such an assumption, however, typically one needs to enlarge the set of possible strategies to
include mixed strategies in order to find Nash equilibria. The equilibrium in
mixed strategies is closely related to the equilibrium under uncertainty about
the number of players.
To see this in a particularly simple case, consider two bidders with independent private values drawn from a two-point distribution H such that
p x ∈ [0, v)
H(x) =
1 x≥v
for some v > 0. Thus each bidder has value 0 with probability p and value v
with probability 1 − p. It is easy to see that the strategy of bidding 0 in case
the value is 0 and according to the mixed strategy F defined in (1) if the value
is v forms a symmetric equilibrium.
More generally, assume that the independent private values are drawn from
an n-point distribution with mass qk in vk , where 0 ≤ v1 ≤ ... ≤ vn . Consider
the strategy of a bidder with private value vk who uses a mixed strategy Fk ,
where
0 x < v1
F1 (x) =
1 x ≥ v1
and
Fk (x) =







0
P
(x−vk−1 ) k−1
i=1 qi
qk (vk −x)
1
x ≤ vk−1
Pk−1
qi +vk qk
Pi=1
k
i=1 qi
Pk−1
vk−1 i=1 qi +vk qk
Pk
i=1 qi
vk−1 < x <
x≥
vk−1
for k = 2, 3, ..., n. It is straightforward to check that this strategy leads to a
symmetric Nash equilibrium. We omit the details, but point out the resemblance of this strategy with the equilibrium strategy under uncertainty about
the number of bidders.
3.3
A common known value as the limit of independent
private values
Our setting, where all participants agree and know about the value of the good,
can be obtained as the limit of the classical set-up as the support of the distribution of private values shrinks.
To see this in a particular case, assume that private values are distributed
uniformly on [1 − , 1]. Consider an invited bidder who is the sole bidder with
probability p and who has competition by one other invited bidder with probability 1 − p. By (4), the symmetric equilibrium strategy consists of bidding β(x)
6
if the private value is x ∈ [1 − , 1], where
β(x) =
(1 − p)(x2 − (1 − )2 )
.
2( − (1 − p)(1 − x))
Take y < 1 − p, and let X be a random variable that represents the private
value of an invited bidder. Then straightforward calculations show that
r
1−p−
P(β(X) ≤ y) = −1 y − 1 + + y 2 − 2
y + (1 − )2
1−p
py
→
(1 − p)(1 − y)
as → 0. Thus, as the support of the distribution of private values collapses
to {1}, the bid converges in distribution to the corresponding Nash equilibrium
determined in Theorem 1.
4
Two potential bidders and one seller
In this section we view the whole auction as a game, thus including also the
seller as a player. The seller’s strategy amounts to inviting a certain number
of bidders, but with the assumption that each invitation is made at a cost
c ∈ (0, v). Accordingly, the seller wants to invite many bidders to ensure that
the winning bid is sufficiently high, but not too many in order to control the
total cost.
A strategy is now a triplet (H1 , H2 , q), where H1 and H2 are distribution
functions on [0, v] and q denotes the probability that the seller invites only one
bidder, and 1 − q is the probability that the seller invites two bidders.
First note that if the seller uses a strategy q, then an invited bidder is alone
(from the bidder’s perspective) with probability p = q/(2 − q), and there is one
more invited bidder with probability 2(1 − q)/(2 − q). Consequently, the bidder
will use the strategy F specified in (1) with p = q/(2 − q). Let X and Y be
two independent random variables with distribution function F . The expected
benefit for the seller if inviting only one bidder is then
p
q
q
E[X] − c = v 1 +
ln p − c = v 1 +
ln
− c. (5)
1−p
2(1 − q) 2 − q
Similarly, the expected benefit for the seller if inviting two bidders is
1 − 3p
p2
E[X ∨ Y ] − 2c = v
−2
ln
p
− 2c
1−p
(1 − p)2
1 − 2q
q2
2−q
= v
+
ln
− 2c.
1−q
2(1 − q)2
q
(6)
Consequently, the seller is indifferent between inviting one and inviting two
bidders precisely when q satisfies
c
−q
q
2−q
=
+
ln
.
v
1 − q 2(1 − q)2
q
7
(7)
Defining
f (q) :=
−q
2−q
q
ln
+
,
1 − q 2(1 − q)2
q
one can check that f > 0 on (0, 1) with f (0+) = 0 and f (1−) = 0. By continuity,
there exists a maximum q ∈ (0, 1), and we denote by f = f (q) the maximal value
of f .
Theorem 4 For c ∈ (0, f ], denote by q ∗ a solution to (7) in (0, 1), and let F ∗
be given by (1) with p = q ∗ /(2 − q ∗ ). Then (F ∗ , F ∗ , q ∗ ), is a Nash equilibrium.
Proof. First assume that all invited bidders use the mixed strategy F ∗ . By
construction, the seller is then indifferent between inviting one or two bidders,
so there is no strategy for the seller that strictly dominates the strategy q ∗ .
Next, assume that Player 1 is invited to the auction, that the seller uses the
strategy q ∗ and that Player 2 uses the mixed strategy F ∗ if invited. By the same
argument as in the proof of Theorem 1, F ∗ is an optimal strategy for Player 1.
From the above, and by symmetry, (F ∗ , F ∗ , q ∗ ) is a Nash equilibrium.
Since f > 0 on (0, 1) with f (0+) = 0 and f (1−) = 0, the equation f (q) = c/v
may have multiple solutions. Note that the expected benefit for the seller if inviting exactly i bidders, i = 1, 2 (see (5) and (6), respectively) is decreasing in the
invited bidder’s estimate p for the probability of being the only bidder (equivalently, in the probability q for the seller to invite only one bidder). Therefore,
the seller prefers the equilibrium (F ∗ , F ∗ , q ∗ ) where q ∗ is the smallest solution
of (7) to any other equilibrium on this form. Also, if q̂ ∈ (q ∗ , 1) solves (7) and
F̂ is as in (1) with p = q̂/(2 − q̂), then the strategy q ∗ is an optimal strategy for
the seller also if all invited bidders use F̂ , so the equilibrium (F̂ , F̂ , q̂) will not
be used.
Remark 5 We end this section with pointing out the existence of a degenerate Nash equilibrium, which exists for any invitation cost c ∈ (0, v). Denote
by G(x) = 1 for x ≥ 0 the degenerate strategy of always bidding zero. Then
(G, G, 1) is a Nash equilibrium. Indeed, first assume that all invited bidders bid
0. Then the pay-off for the seller is −cq−2c(1−q), which is maximized for q = 1,
so q = 1 is optimal for the seller. Similarly, if q = 1, then the optimal response
for an invited bidder is to bid 0, so (G, G, 1) is a Nash equilibrium. However,
this equilibrium is of minor practical imortance since it yields a negative value
for the seller.
5
Several potential bidders
In this section we extend the set-up of Section 2 to include an arbitrary number
of potential bidders. Again we take the viewpoint of an invited bidder, but now
there are n − 1 additional potential bidders. The invited bidder estimates the
probability that exactly
k bidders have been invited to be pk , k = 1, ..., n, where
Pn
0 < p1 < 1 and k=1 pk = 1. The case studied in Section 2 thus corresponds
to the set of parameters n = 2, p1 = p, and p2 = 1 − p.
8
Assuming that each invited player uses the same mixed strategy represented
by a continuous distribution function F , the expected payoff if the invited player
bids x is
n
X
(v − x)
pk F k−1 (x).
k=1
Assuming the expected payoff is constant on the support of F , and imposing
boundary conditions F (0) = 0 and F (1) = 1, yields a symmetric candidate
equilibrium given by F , where F (x) solves
(v − x)
n
X
pk F k−1 (x) = p1 x
(8)
k=2
for x ∈ [0, v(1 − p1 )] and F (x) = 1 for x ∈ (v(1 − p1 ), v]. Note that for a fixed
x ∈ [0, v(1 − p1 )] there is a unique solution F (x) ∈ [0, 1] to (8). Moreover, the
unique solution F to (8) is continuous and increasing in x and satisfies F (0) = 0,
so F is a distribution function.
Theorem 6 The n-tuple (F, ..., F ) defined above is a Nash equilibrium in mixed
strategies for the game with n potential bidders. Moreover, it is the unique
symmetric equilibrium.
Proof. If an invited player bids x and all other invited players use mixed
strategies represented by the distribution function F , then the average payoff is
(v − x)
n
X
pk F
k−1
(x) =
k=1
p1 v
v−x
x ≤ (1 − p1 )v
x > (1 − p1 )v.
Thus the maximal reward is p1 v, and since the strategy F puts all mass on
[0, (1 − p1 )v], there is no strategy giving a higher average reward than F .
The proof of uniqueness follows along the same lines as in Theorem 1 and is
therefore omitted.
6
Asymmetries among bidders
In this section we give a generalization to the case when bidders have asymmetric
values and asymmetric estimates for the probability of competition. For this,
assume in a two-player setting that Bidder 1 has value v1 and estimates the
probability of being alone in the auction to be p1 , whereas Bidder 2 has value v2
and estimates the probability of being alone in the auction to be p2 . However,
we still assume complete information in the sense that all parameters pi , vi ,
i = 1, 2 are known to both players.
By heuristic reasoning as in the various cases treated above, one arrives at a
candidate equilibrium. We omit the details for this, but instead state and prove
our main result directly.
9
Theorem 7 Assume that (1 − p1 )v1 ≤ (1 − p2 )v2 . Define the distribution
functions F1 and F2 with support on [0, (1 − p1 )v1 ] by
(
(1−p2 )v2 −(1−p1 )v1 +p2 x
x ∈ [0, (1 − p1 )v1 )
(1−p2 )(v2 −x)
F1 (x) =
1
x ≥ (1 − p1 )v1
and
F2 (x) =
p1 x
(1−p1 )(v1 −x)
1
x ∈ [0, (1 − p1 )v1 )
x ≥ (1 − p1 )v1 .
Then the pair (F1 , F2 ) of mixed strategies forms a Nash equilibrium in the case
of asymmetry among bidders.
Remark 8 Note that in the special case that v1 = v2 and p1 = p2 , the equilib1 )v1
rium in Theorem 1 is recovered. Also note that F1 (0) = 1 − (1−p
(1−p2 )v2 > 0, so
Bidder 1 bids 0 with positive probability.
Proof. Assume that Bidder 2 uses the mixed strategy F2 if invited. Then any
bid x ∈ [0, (1 − p1 )v1 ] gives Bidder 1 the constant average payoff
p1 (v1 − x) + (1 − p1 )(v1 − x)F2 (x) = p1 v1 ,
and any bid x ≥ (1 − p1 )v1 gives a payoff
v1 − x ≤ p 1 v1 .
Consequently, there is no strategy for Bidder 1 that is strictly better than the
mixed strategy F1 .
Similarly, if Bidder 1 uses F1 , then any bid x ∈ (0, (1 − p1 )v1 ] gives Bidder 2
an average payoff
p2 (v2 − x) + (1 − p2 )(v2 − x)F1 (x) = v2 − (1 − p1 )v1 ,
whereas a bid x = 0 gives
p2 v2 +
(1 − p2 )v2 F1 (0)
< v2 − (1 − p1 )v1
2
and a bid x ≥ (1 − p1 )v1 gives
(v2 − x) ≤ v2 − (1 − p1 )v1 .
Thus the mixed strategy F2 is optimal for Bidder 2, and hence (F1 , F2 ) is an
equilibrium.
Remark 9 It is straightforward to check that if p1 = p2 and v1 ≤ v2 , then
F1 ≥ F2 . Consequently, in such a setting the bidder with the smaller value will
bid more agressively in the sense that the probability for a bid smaller than any
given threshold is bigger than for the other bidder. A similar feature has been
observed in the case of asymmetry among bidders in the independent private
value setting, e.g. see [3, Proposition 4.4].
10
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11