In a recurring auction early bids may reveal bidders’ types, which in turn affects bidding in later auctions. Bidders take this into account and may bid in a way that conceals their private information until the last auction is played. The present paper analyzes the equilibrium of a sequence of ?rst-price auctions assuming bidders have stable private values. We show that signal-jamming occurs and explore the dynamics of equilibrium prices.Auctions; Signaling; Price Competition

Ding, Wei

Jeitschko, Thomas D.

Wolfstetter, Elmar G.

English

Research Papers in Economics

Sonderforschungsbereich/Transregio 15  www.sfbtr15.de 
Universität Mannheim  Freie Universität Berlin  Humboldt Universität zu Berlin  Ludwig Maximilians Universität München 
Rheinische Friedrich Wilhelms Universität Bonn  Zentrum für Europäische Wirtschaftsforschung Mannheim 
 
Speaker: Prof. Dr. Urs Schweizer.  Department of Economics  University of Bonn  D 53113 Bonn, 
Phone: +49(0228)739220  Fax: +49(0228)739221 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
*Bonn Graduate School of Economics (BGSE), University of Bonn, Kaiserstr. 1, 53113 Bonn, 
Germany, Email: ding@uni-bonn.de.  
**Department of Economics, Michigan State University, East Lansing, MI 48824, USA, Email: 
jeitschk@msu.edu 
***Corresponding Author: Department of Economics, Institute of Economic Theory I, Humboldt 
University at Berlin, Spandauer Str. 1, 10178 Berlin, Germany, Email: wolfstetter@gmail.com 
 
 
 
 
June 2009 
 
 
 
Financial support from the Deutsche Forschungsgemeinschaft through SFB/TR 15 is gratefully acknowledged. 
Discussion Paper No. 261 
Signal-Jamming in a Sequential 
Auction 
 
Wei Ding* 
Thomas D. Jeitschko** 
Elmar Wolfstetter*** 
 Signal-Jamming in a Sequential Auction∗
Wei Ding
Bonn Graduate School of Economics
University of Bonn†
Thomas D. Jeitschko
Michigan State University‡ and
Royal Holloway, University of London
Elmar Wolfstetter
Institute of Economic Theory I
Humboldt-University at Berlin§
April 2009
Abstract
In a recurring auction early bids may reveal bidders’ types, which in turn
affects bidding in later auctions. Bidders take this into account and may
bid in a way that conceals their private information until the last auction is
played. The present paper analyzes the equilibrium of a sequence of ﬁrst-
price auctions assuming bidders have stable private values. We show that
signal-jamming occurs and explore the dynamics of equilibrium prices.
JEL classiﬁcations: D44, D02, D43
Keywords: AUCTIONS, SIGNALING, PRICE COMPETITION
∗We thank Yaron Raviv, Yosuke Yasuda, conference participants at the 2009 IIOC, and seminar
participants at Korea University. Research support by the Deutsche Forschungsgemeinschaft (DFG),
SFB Transregio 15, “Governance and Efﬁciency of Economic Systems,” is gratefully acknowledged.
†Bonn Graduate School of Economics (BGSE), University of Bonn, Kaiserstr. 1, 53113 Bonn,
Germany, Email: ding@uni-bonn.de
‡Department of Economics, Michigan State University, East Lansing, MI 48824, USA, Email:
jeitschk@msu.edu
§Corresponding Author: Department of Economics, Institute of Economic Theory I, Humboldt
University at Berlin, Spandauer Str. 1, 10178 Berlin, Germany, Email: wolfstetter@gmail.com
11 Introduction
Many market transactions have an auction structure, and many such auctions are
recurring events. For example, price competition between retailers is essentially
a (reverse) auction. And this auction is typically a recurring event in which the
relevant valuations (unit costs) are stable, at least for some time. Evidently, in
this case bidders must pay attention to the information they reveal about their val-
uations through their bids. This gives rise to a problem of strategic information
transmission.
The present paper analyzes equilibrium bidding in a sequence of ﬁrst-price auc-
tions, when bidders have stable private values. Bidders want to win each auction,
but they are also concerned with concealing their valuation in order to reduce the
intensity of price competition.
If bidders play strictly monotone strategies in the ﬁrst auction, they reveal their
private information, and the second auction is one under complete information, re-
sulting in ﬁerce competition that wipes out proﬁts. Bidders thus attempt to keep
their rival unsure about their valuation by playing non-monotone strategies, mim-
icking the bidder with a low valuation with some probability.
2 The model
Consider a sequence of two ﬁrst-price auctions for two identical objects, and two
ex ante symmetric bidders, named 1 and 2. Bidders draw their private valuation
before the ﬁrst auction and keep that valuation in second one. Before the second
object is auctioned, bidders observe both bids of the ﬁrst auction, and use this
information to update their beliefs concerning their rival’s valuation. Valuations V
are iid random variables which assume either a low value 0 (normalized) or a high
value v > 0 with the prior probability ρ := Pr{V = v} ∈ (0,1).
If bidders tie, the winner is selected by ﬂipping a fair coin, with one exception:
If bidders tie in the second auction and exactly one bid was positive in the ﬁrst
auction, the one who made a positive bid in the ﬁrst auction is selected as the
winner.
We denote bidder i’s bid in the j-th auction by b
j
i, and continuation payoffs by
π(h), where h denotes the history of the game prior to the second auction.
The possible histories of the game, h, are described by past bids observed by both
players. The following histories must be distinguished; there, only the sign of
observed bids matters: 1) The history at the beginning of the game, h/ 0; 2) the
histories with both bids either zero or positive: h00 := {b1
1 = 0,b1
2 = 0} and h11 :=
{b1
1 > 0,b1
2 > 0}; 3) the histories with one positive bid and one bid equal to zero:
h10 := {b1
1 > 0,b1
2 = 0} and h01 := {b1
1 = 0,b1
2 > 0}.3 Equilibrium
A bidder with valuation V = 0 obviously bids zero with certainty in both auctions
(which will not be repeated from here onwards). This does not, however, imply that
a zero bid can only come from a bidder with valuation V = 0. Indeed, in a signal-
jamming equilibrium a bidder with V = v may also bid zero in the ﬁrst auction in
order to keep his rival in doubt about his valuation.
We now solve the equilibrium strategies of a bidder with valuation V = v in both
auctions, for all possible histories of the game, employing the equilibrium concept
of a sequential equilibrium with observable moves.
As a working hypothesis, suppose F : [0,¯ b] → [0,1] is the unique symmetric equi-
librium mixed strategy of a bidder withV =v in the ﬁrst auction (history h/ 0), where
F(0) may be positive. This allows us to characterize the equilibrium play in the
second auction, and then conﬁrm the working hypothesis concerning the equilib-
rium play in the ﬁrst auction. To avoid unnecessary duplication we state only the
equilibrium strategies and beliefs of one player, named player 1.
Equilibrium in the second auction After the ﬁrst auction, bidders observe bids,
process this information to update their beliefs about the rival’s valuation, and then
play the second auction. Updated beliefs must be consistent with the equilibrium
strategy of the ﬁrst auction and the observed bids. Hence, using Bayes’ rule when-
ever applicable, posterior beliefs are:1
Pr{V2 = v | b1
2} =
(
1 if b1
2 > 0
F(0)ρ
F(0)ρ+(1−ρ) =: q if b1
2 = 0.
(1)
Proposition 1 (Second auction). Consider the second auction. The equilibrium
strategy of player 1 (withV1 = v) depends on the history as follows: h11 ⇒ b2
1 = v,
h ∈ {h00,h01} ⇒ G(b2
1), h = h10 ⇒ H(b2
1):
G : [0,qv] → [0,1], G(b) =
b(1−q)
(v−b)q
=
b(1−ρ)
(v−b)ρF(0)
(2)
H : [0,qv] → [0,1], H(b) =
v(1−q)
v−b
=
v(1−ρ)
(v−b)(1−ρ +ρF(0))
, (3)
where H has one mass point at b = 0. The associated equilibrium continuation
payoffs are π(h11) = 0,π(h00) = π(h10) = π(h01) = v(1−q).
Proof. Equilibrium strategies and payoffs are self-evident for h11. Histories h01
and h10 result in an asymmetric, and history h00 in a symmetric one-shot auction,
solved in Jeitschko and Wolfstetter (2002, Prop. 3).
1Note, this belief system involves only a fairly innocent prescription of “off-equilibrium path”
beliefs by stipulating that Pr{V2 = v | b1
2 > 0} = 1 also for bids that are higher than “predicted”, i.e.
for b1
2 > ¯ b.
2Equilibrium in the ﬁrst auction Signal jamming pays for bidder 1 only if bidder
2 has also a high valuation. However, signal jamming is costly, and its beneﬁt
outweighs the cost only if it is sufﬁciently likely that bidder 1 meets a high value
bidder 2.
Proposition 2. Suppose ρ > 1/3. The equilibrium strategy in the ﬁrst auction (con-
ditional onV = v) is F : [0,¯ b] → [0,1]:
F(b) :=
1−ρ
ρ
b
v−b
+
v
v−b
p
3−4ρ +ρ2−2(1−ρ)
ρ
(4)
¯ b := v
￿
2−ρ −
p
3−4ρ +ρ2
￿
. (5)
F has a mass point at zero: F(0) =
p
(1/ρ−1)(3/ρ−1)−2(1/ρ −1) > 0 which
has a maximum at ρ = 3/4 and approaches zero as ρ → 1 and as ρ → 1/3.
Proof. Consider one bidder, say bidder 1 withV = v, and history h/ 0 (ﬁrst auction).
To conﬁrm the asserted equilibrium mixed strategy F, stated in (4), we must show
that this bidder is indifferent between all bids from the support of F, which is [0,¯ b]
where ¯ b is stated in (5).
If bidder 1 withV = v makes a bid b ∈ (0,¯ b] his payoff is equal to
(v−b+π(h11))ρ(F(b)−F(0))+(v−b+π(h10))((1−ρ)+ρF(0)). (6)
Whereas if he bids zero, his payoff is
￿v
2
+π(h00)
￿
(ρF(0)+(1−ρ))+π(h01)ρ(1−F(0)). (7)
And the assertion follows immediately.
We mention that no signal jamming occurs if ρ ≤ 1/3; in that case, a high value
bidder plays the myopic strategy K : [0,ρv] → [0,1], K(b) :=
1−ρ
ρ
b
v−b in the ﬁrst
auction, and in the second auction bids v if he has observed a positive ﬁrst-auction
bid from his rival, and otherwise bid zero. While this is the unique Bayesian Nash
equilibrium, that equilibrium is not a sequential equilibrium.2
4 Signal-jamming
Signal-jamming occurs if a bidder with a high valuation bids zero in the ﬁrst auc-
tion with positive probability, F(0) > 0, and thus sometimes mimics a bidder with
a low valuation in order to keep the rival uninformed.
2For, suppose V1 = v and consider history h00; consistency of beliefs requires that each bidder
believes that his rival has a zero valuation; but then bidding zero is not the best response of bidder 1
(in fact, no best response exists in that case).
3Altogether, bidder 1 beneﬁts from signal jamming if and only if bidder 2 also has
a high valuation. Signal jamming leads bidder 2 to update his belief from ρ to
Pr{V1 = v | b1
1 = 0} = q < ρ instead of revealing bidder 1’s type, which in turn
induces him to bid stochastically lower, no matter how he bid in the ﬁrst auction.
If bidder 2 made a positive bid in the ﬁrst auction, he plays the stochastically lower
mixed strategy H(b)>G(b),∀b∈[0,qv], and if he also engaged in signal jamming,
both bidders play the mixed strategy G which preserves a positive expected proﬁt
in the second auction.
However, signal jamming is also costly since it entails the risk of losing the ﬁrst
auction. It follows that it pays to “invest” in signal-jamming only if it is sufﬁ-
ciently likely that the rival has a high valuation. Interestingly, this relationship is
not monotone, and F(0) has a global maximum at ρ = 3/4.
We mention that signal-jamming induces pointwise less aggressive bidding in the
ﬁrst auction, in the sense that the myopic strategy K ﬁrst-order stochastically dom-
inates the (continuously extended) strategy ¯ F(b) := max{F(b),1}, i.e., ¯ F(b) ≥
K(b),∀b ∈ [0,ρv].
5 Dynamics of equilibrium prices
The study of price sequences in sequential auctions has received much attention
in the literature (see, for example, McAfee and Vincent, 1993, Gale and Hausch,
1994, Jeitschko, 1999). In the present context, one might expect that prices be
stochastically increasing since signal jamming involves bidding low in the ﬁrst
auction.
In the following assume ρ > 1/3 (because otherwise no signal jamming occurs and
equilibrium prices are stationary), and denote the continuously extended strategies
for the enlarged domain [0,v] by ¯ G(b) := min{G(b),1}, ¯ H(b) := min{H(b),1}.
Lemma 1. The probability distribution of the equilibrium price in the ﬁrst auction,
FP1 : [0,v] → [0,1], is
FP1(p) : = Pr{P1 ≤ p}
= Pr{˜ b1
1 ≤ p and ˜ b1
2 ≤ p}
=
(
(1−ρ +ρF(p))
2 if p ≤ ¯ b
1 if p ≥ ¯ b.
(8)
FP1 has exactly one mass point, FP1(0) =
￿
(1−ρ)−
p
3−4ρ +ρ2
￿2
> 0, which
is strictly decreasing in ρ with limρ→1FP1(0) = 0.
4Lemma 2. The probability distribution of the equilibrium price in the second auc-
tion, FP2 : [0,v] → [0,1], is
FP2(p) = Pr{P2 ≤ p}
= ρ2￿
(1−F(0))2·0+F(0)2 ¯ G(p)2+2F(0)(1−F(0)) ¯ G(p) ¯ H(p)
￿
+2ρ(1−ρ)
￿
F(0) ¯ G(p)+(1−F(0)) ¯ H(p)
￿
+(1−ρ)2.
(9)
FP2 has mass points at p = 0 and p = v:
FP2(0) = (1−ρ)
p
3−4ρ +ρ2 > 0 (10)
Pr{P2 = v} =
￿
2−ρ −
p
3−4ρ +ρ2
￿2
> 0. (11)
Pr{P2 =v} is strictly increasing in ρ, and FP2(0) decreasing with limρ→1FP2(0)=
0.
Proposition3. Equilibriumpricesareincreasinginthesenseofﬁrst-orderstochas-
tic dominance, FP2(p)≤FP1(p) (with strict inequality except for p=v), if and only
if ρ is sufﬁciently large, i.e. ρ > ρ∗ := 2−3/5
√
5.
This strong stochastic order does not apply to ρ ∈ (1/3,ρ∗). However, in either
case E[P2] > E[P1].
Proof. 1) Suppose ρ > ρ∗. Let p ∈ [0,v) and deﬁne (where the functions G and H
are applied to the enlarged domain [0,v)):
˜ FP2(p) : = ρ2￿
(1−F(0))2·0+F(0)2G(p)2+2F(0)(1−F(0))G(p)H(p)
￿
+2ρ(1−ρ)(F(0)G(p)+(1−F(0))H(p))+(1−ρ)2.
By deﬁnition, G(p) ≥ ¯ G(p),H(p) ≥ ¯ H(p), with equality for all p ∈ [0,qv] and
strict inequality for all p ∈ (qv,v). Therefore, it follows immediately that ˜ FP2(p)
is a pointwise upper bound of FP2(p), i.e., ˜ FP2(p) ≥ FP2(p), for all p ∈ [0,v), and
hence, in particular, for all p ∈ [0,¯ b].
As one can easily conﬁrm, FP1 > ˜ FP2(p),∀p ∈ [0,¯ b] ⇐⇒ ρ > ρ∗. Since FP1(p) =
1,∀b ≥ ¯ b and FP2 < 1,∀b < v (since it has a mass point at p = v), we conclude
that ρ > ρ∗ ⇒ FP2(p) ≤ FP1(p), with strict inequality everywhere except at p =
v, as illustrated in the right-hand Figure 1. Of course, the established ﬁrst-order
stochastic dominance relationship implies E[P2] > E[P1].
2) Suppose ρ ∈ (1/3,ρ∗). Then, as one can easily conﬁrm, FP2(0) > FP1(0). More-
over, FP1(p) = 1 > FP2(p),∀p ∈ [¯ b,v) (since FP1(¯ b) = 1 and FP2 has a mass point
at p = v). Therefore, FP2(p) and FP1(p) must intersect at least once; hence no ﬁrst-
order stochastic dominance relationship applies to P2 and P1, as illustrated in the
left-hand Figure 1.
51.0
0.8
0.6
0.4
0.2
0.0  0.2  0.4  0.6  0.8  1.0
FP1, FP2
FP1
FP2
p
1.0
0.8
0.6
0.4
0.2
0.0  0.2  0.4  0.6  0.8  1.0
FP1, FP2
FP1
FP2
p
Figure 1: Comparison between FP1 (dashed) and FP2 (solid) for v=1 and ρ = 1/2<
ρ∗ (left) resp. ρ = 3/4 > ρ∗ (right)
3) Computing expected equilibrium prices, one ﬁnds, using (8) and then (4), (5)),
E[P1] =
Z v
0
pdFP1(p) =
Z ¯ b
0
2p(1−ρ +ρF(p))ρdF(p)
= v
￿
7−2ρ(4−ρ)−2(2−ρ)
p
3−4ρ +ρ2
￿
.
Similarly, the expected price in the second auction auction is, using equations (9)
and (11),
E[P2] =
Z v
0
pdFP2(p) =
Z qv
0
pdFP2(p)+vPr{P2 = v}
=
v
￿
14−5ρ3+25ρ2−34ρ −(8+5ρ2−14ρ)
p
3−4ρ +ρ2
￿
p
3−4ρ +ρ2−(1−ρ)
Therefore, after some rearrangements, for all ρ > 1/3:
E[P1]−E[P2] = v
p
3−4ρ +ρ2
￿
5ρ −7+4
p
3−4ρ +ρ2
￿
< 0,
as asserted.
6 Discussion
In the present paper we assumed that bidders have stable valuations. An alterna-
tive framework would be to assume that valuations are subject to stochastic scale
effects, as in Jeitschko and Wolfstetter (2002).
We also assumed that bidders observe all ﬁrst auction bids before they bid in the
second auction. If instead bidders could only learn whether they either won or
lost the ﬁrst auction, in some subgames bidders would know the rank order of
valuations, as in Landsberger et al. (2001).
6Our analysis also assumed a passive auctioneer. Therefore, signal jamming served
exclusively the purpose of misleading the rival bidder. The incentive for signal
jamming is further increased if the auctioneer is able adjust reserve prices, taking
advantage of information acquired during the ﬁrst auction.
References
Gale, D., Hausch, D., 1994. Bottom–ﬁshing and declining prices in sequential auc-
tions. Games and Economic Behavior 7, 318–331.
Jeitschko, T., 1999. Equilibrium price paths in sequential auctions with stochastic
supply. Economics Letters 64, 67–72.
Jeitschko, T., Wolfstetter, E., 2002. Scale economies and the dynamics of recurring
auctions. Economic Inquiry 40, 403–414.
Landsberger, M., Rubinstein, J., Wolfstetter, E., Zamir, S., 2001. First–price auc-
tions when the ranking of valuations is common knowledge. Review of Eco-
nomic Design 6, 461–480.
McAfee, R. P., Vincent, D., 1993. The declining price anomaly. Journal of Eco-
nomic Theory 60, 191–212.
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