Discussion Paper No. 1043

AN ENTRY GAMEWITH LEARNING
AND MARKET COMPETITION

Chia-Hui Chen
Junichiro Ishida
Arijit Mukherjee

October 2018

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

An Entry Game with Learning and Market
Competition∗
Chia-Hui Chen†, Junichiro Ishida‡ and Arijit Mukherjee§
October 26, 2018

Abstract
This paper provides a dynamic game of market entry to illustrate entry dynamics inan uncertain market environment. Our model features both private learning about themarket condition and market competition, which give rise to the first-mover and secondmover advantages in a unified framework. We characterize symmetric Markov perfectequilibria and identify a necessary and sufficient condition for the first-mover advantage
to dominate, which elucidates when and under what conditions a firm becomes a pioneer,an early follower or a late entrant. We also derive equilibrium payoff bounds to show that
pioneering entry is generally payoff-enhancing, even though it is driven by preemption
motives, and discuss efficiency properties of entry dynamics.

JEL Classification Number: D82, D83, L13
Keywords: market entry, market competition, private learning, signaling, entry timing.
∗

We thank Makoto Hanazono, Hideshi Itoh, and seminar and conference participants at Korea University,
Kyoto University, Academia Sinica, SAET 2018, Pan Pacific Conference in Economic Research, and Contract
Theory Workshop (CTW) for helpful comments and suggestions. The first author acknowledges financial
support from JSPS KAKENHI Grant-in-Aid (B) JP16H03615. The second author acknowledges financial
support from JSPS KAKENHI Grant-in-Aid (S) JP15H05728, (A) JP20245031, (B) JP16H03615, and (C)JP24530196 as well as the program of the Joint Usage/Research Center for Behavioral Economics at ISER,
Osaka University. All errors are our own.
†
Institute of Economic Reserach, Kyoto University. Email: chchen@kier.kyoto-u.ac.jp‡
Institute of Social and Economic Research, Osaka University. Email: jishida@iser.osaka-u.ac.jp§
Notthingham University Business School, University of Notthingham.
Email: arijit.mukherjee@nottingham.ac.uk

1Introduction

In the early 1990s, Boeing and Airbus independently began to investigate the potential profitability of the superjumbo market, i.e., the market for aircrafts with capacity for 500-1000
passengers. At the time, neither was sure whether the market would develop large enoughto justify the huge investment required to enter this market. Specifically at issue was theuncertainty looming over the evolution of airline networks: if hub-and-spoke networks would
remain dominant, it would favor bigger, though less cost efficient, aircrafts; if point-to-point
networks would emerge instead, it would favor smaller medium-sized aircrafts. In the face of
this uncertainty, the two companies eventually chose to follow divergent paths. In December
2000, Airbus formally announced to enter the market with commitment to developing andlaunching A380. In the meanwhile, Boeing came to believe that the fragmentation of themarket due to the rise of point-to-point networks would limit demand for a superjumbo anddecided to withdraw from the market.
This is the type of situation firms routinely encounter whenever there is a new (geographical) market, a new product, or a new technology becoming available; facing a new market
opportunity, a potential entrant needs to conduct careful marketing research to gauge its
profitability and decide whether and when to enter the market. The timing of market entry
is in fact one of the most critical strategic decisions a firm must make, as is often emphasizedin the fields of strategic management and marketing.1 Even as a single-firm decision problem, it is never straightforward to determine whether and when to enter the market when theeventual value of market entry is dissolved only gradually over time. The situation is evenmore compounded when there are multiple parties who interact and compete with each other,
especially with information and payoff spillovers among them. In the superjumbo market,
for instance, Airbus could have entered earlier, to which Boeing might have reacted differently, or Airbus’s entry could have trigerred Boeing’s subsequent entry, either immediately
or after some additional marketing research. All these decisions and anticipated reactions are
strategically linked and affect each other, possibly generating a plethora of dynamic entry
patterns.
In this paper, we construct a dynamic game of market entry to capture this type ofsituation, with focus on a tradeoff faced by firms which contemplate to enter a market of
unknown profitability. We consider an environment with two potential entrants, each ofwhich independently decides whether and when to enter a new market. The profitability ofthe market is determined by the market condition (e.g., market size, production cost) which1

For instance, Lilien and Yoon (1990) note that “the choice of market-entry time is one of the major reasonsfor new product success or failure.”1

is not known to anyone initially. Each firm privately collects information about the market
condition over time, and decides to enter when it becomes sufficiently confident about themarket. As a key departure from the existing literature, we introduce into this setting thefollowing two factors.
• Market competition: A firm’s flow profit is decreasing in the number of firms in themarket.
• Private (pre-entry) learning and signaling via market entry: Before entry, eachfirm privately collects information about the market over time. As a consequence, afirm’s market entry serves as a signal of its private information.
These two factors are often regarded as the primary sources of early-mover and latemover advantages in the literature,2 and generate the tradeoff of our focus: the first one,the presence of market competition, implies a benefit of acting first whereas the second one,
private learning and signaling via market entry, implies a benefit of waiting and learning fromthe rival’s action. Our model thus captures the first-mover and second-mover advantages
in a unified framework. The overall benefit of becoming a “market pioneer” is essentiallydetermined by the way this tradeoff resolves.
To see the workings of our model, it is important to observe that since each firm starts witha common prior and gains more information over time, the signaling effect of market entry is
generally weaker at early stages of the game. As time passes, however, it eventually reaches
a point where a firm’s entry reveals so much information that the rival firm immediately
follows suit, thereby entirely dissipating the first-mover advantage. Due to this force, thegame is generally divided into two distinct phases, called preemption and waiting for clarity,
where the game begins with the preemption phase and then later switches to the waiting
phase. The strategic nature of the problem differs significantly between the two phases. In
the preemption phase, as a firm’s entry can push back the timing of the rival firm’s entry
by cutting its residual demand, each firm prefers to enter slightly earlier than the rival. In
the waiting phase, on the other hand, there is no such gain, and each firm prefers to enter
slightly later to learn from the rival. For expositional purposes, we say that pioneering entry
occurs, or a market pioneer emerges, when a firm enters the market in the preemption phase.
2
For instance, Lieberman and Montgomery (1988) raise preemption of rivals as one of the first-moveradvantages but also emphasize that “late movers can gain an edge through resolution of market or technologicaluncertainty.” Incidentally, these two factors also played a central role in the Boeing-Airbus example: themarket for large jetliners is a typical duopoly where the two firms compete intensely for the global market
share; given the high entry cost into each submarket, pre-entry marketing research is also an integral part oftheir business operations.

2

Although we base our analysis on a relatively simple setup, a technical complicationarises from the fact that each firm privately observes signals and at the same time also
learns from the other’s action (or inaction). As a consequence, the firms’ beliefs evolvein a rather complicated manner depending on their entry strategies, which along with thepresence of market competition can easily make the analysis intractable. One of the major
contributions of our analysis is to devise an analytical framework that is tractable enough todeliver some key insights for when and how a market pioneer emerges in a market where bothlearning and market competition are crucial factors. Specifically, we characterize symmetricMarkov perfect equilibria and derive a necessary and sufficient condition for pioneering entry
to occur in this environment. When this condition is satisfied, the firms enter the market
at some positive rate in the preemption phase until it reaches a “saturation point” where
the signaling effect of market entry becomes strong enough just to offset the first-moveradvantage. Market entry then ceases to occur past this point, with neither firm taking any
action. After a while, though, an uninformed firm would eventually become confident enoughand willing to enter the market even without the chance to earn the monopoly rent. The
model thus exhibits a rich, on-and-off, dynamics of market entry where the firms graduallyenter the market at early and late stages, with the period of no entry in between.
We also identify equilibrium payoff bounds and argue that pioneering entry is generally
payoff-enhancing for the firms even though it is driven by preemption motives. To see thisresult, note that the impact of pioneering entry on each firm’s ex ante expected payoff is
not entirely positive because the presence of market competition induces the firms to enterprematurely, compared to the optimal timing, in order to preempt the rival firm. Despite
this efficiency loss, pioneering entry can still be socially beneficial as it reveals valuable
information and inevitably facilitates the rival firm’s learning. We show that due to thispositive information spillover, pioneering entry is generally payoff-enhancing in that eachfirm’s expected payoff is higher in the equilibrium with pioneering entry than the one without.
Finally, as noted earlier, the timing and order of market entry are a topic of utmost concern
in the strategic management and marketing literature.3 The literature often classifies entry
timing into three broad categories: pioneers, early followers, and late entrants. Our model
generates a range of entry patterns consistent with this classification and hence contributes tothis literature by clarifying when and under what conditions a firm becomes a pioneer, an early
follower or a late entrant. We also examine the optimal timing of entry that maximizes thejoint profit of the firms and find that the welfare impact of an entry pattern depends crucially3

Since the seminal work of Lieberman and Montgomery (1988), there is now a voluminous literature
examining, mainly empirically, the extent of the first-mover advantages which is also known as the order-ofentry effect.

3

on the timing of the first entry. Drawing on this fact, we argue, somewhat paradoxically, thatthe firms enter the market too early when there is a late entrant while they enter too latewhen there is an early follower.

1.1

Related literature

The major contribution of our analysis is that it incorporates market competition and private
learning (and hence signaling via market entry) in an analytically tractable manner to deliver
a comprehensive description of when and under what conditions a market pioneer emerges
in equilibrium. Despite the fact that these two factors are often regarded as the primarysources of early-mover and late-mover advantages (Lieberman and Montgomery, 1988), they
have been investigated rather independently and separately from one another in the existing
literature: there are many related works which incorporate either one of these elements invarious ways but very few which combine them in a unified framework.
First, our paper is related to works which examine the optimal timing of entry or exit
with information externalities among the players. To name some, Chamley and Gale (1994)
consider a model of strategic investment in which there are N players, a random number
n of whom have an investment option. Assuming that the value of investment depends
positively on the random number n, there is an incentive to wait and see others’ investment
decisions. Grenadier (1999) analyzes an option exercise game in which each player is endowedwith private information about the true value of the option and hence the timing of exercise
becomes a signal for other players. Murto and Välimäki (2011) analyze an exit game withprivate learning where each player receives a signal in each period which partially reveals
his own type. A common thread of those works is that they do not consider the elementof market competition, or negative payoff externalities,4 and focus more on issues such as
investment delays and waves.
In contrast, models of market entry in industrial organization generally focus on market
competition but not on dynamic learning.5 For instance, Levin and Peck (2003) analyze aduopoly model of market entry in which each firm privately observes its entry cost at theoutset of the game. The market environment is similar to ours in that the first mover can
4

In Chamley and Gale (1994), the value of investment increases as more players invest, which is a form of
positive payoff externality. Genadier (1999) also extends the analysis to incorporate positive payoff externalities, e.g., due to network effects.
5
Profit uncertainty also plays an eminent role in the context of foreign direct investment. Horstmann andMarkusen (1996, 2015) consider a model in which a producer (the MNE) is unsure of the potential customer
size and chooses either to contract with a local sales agent or to establish an owned local sales operation.
While contracting with the local sales agent, the producer gains information about the customer size andswitches to an owned sales operation if this option is found to be profitable. In their models, however, there
is only one producer and hence no market competition.

4

earn monopoly rents until the second mover arrives. Aside from the fact that there is nolearning, a crucial difference is that the cost uncertainty in their model is individual-specificand hence a firm’s entry does not reveal any useful information to the other firm. Rasmusen
and Yoon (2012) analyze a duopoly model of market entry which incorporate both market
competition and signaling. They consider a two-period model in which one of the firms is
better informed about the market size than the other, and market entry by the informed firm
hence becomes a signal of its private information. As in Levin and Peck (2003), however,
the information structure is exogenously fixed at the outset of the game, which rules out thepossibility of learning over time.
Bloch et al. (2015) consider a similar environment to ours but with a different information
structure. They analyze a duopoly model of market entry where the firms face uncertainty
regarding the common entry cost.6 In their model, as in our setting, the firms receive private
signals about the entry cost, and the flow payoff of entry is decreasing in the number of firms
in the market. The key difference from our setting is that the learning process in their modelis “all or nothing” where a firm can perfectly identify the true state of nature with someprobability in each period or learn nothing at all. This structure makes the belief systemessentially binary: at any point in time, a firm is either fully informed about the state or
has no additional information to the prior. They then exploit this structure and identify
various kinds of equilibrium by letting the two types – informed and uninformed – coordinate
their entry timing; as such, entry by the uninformed type is generally lumpy and occurs at
some time with strictly positive probability. In contrast, the learning process in our modelis gradual and continuous where each firm builds up confidence slowly over time, so that weneed to keep track of the firms’ beliefs which evolve in a rather complicated manner dependingon their entry strategies. This difference requires us to adopt a different analytical approachand leads to different equilibrium properties.
A work that is more closely related to ours is a duopoly model of investment by Decamps
and Mariotti (2004).7 As in our model, they assume that the quality of the project, whichis common to both players, is not known ex ante and gradually revealed over time via thearrival of a bad (public) signal. In addition to this public learning, they further assume thatonce the leader has made investment, an additional signal is generated that may be used by
the follower to adjust his investment decision. Although they focus mainly on the case with
6

Bloch et al. (2014) also consider a similar setup but with firm-specific entry costs.
The optimal timing of investment under uncertainty is investigated actively in the real options literature.
Although much of the literature focuses on a single-agent decision problem, there are some attempts toincorporate strategic interactions into the real options approach (Trigeorgis, 1991; Grenadier, 1996; Weeds,
2002; Shackleton et al., 2004; Pawlina and Kort, 2006). This strand of literature generally assumes publiclearning with no information asymmetry among the competing players as in Decamps and Mariotti (2004).
7

5

no market competition, they later extend the analysis to discuss the case where the follower’s
payoff is discounted by some fraction. In their model, however, learning is public, and there
is hence no possibility of signaling.

2

Model

We consider a dynamic game of market entry where two potential entrants, labeled as firms
1 and 2, contemplate to enter the market of unknown profitability. The market conditionis either good or bad, and each firm collects information before it makes an irreversible
entry decision. There are two sources of information in this model: on one hand, each firm
may observe a signal of the market condition which arrives stochastically over time; on theother hand, the entry decision of each firm is also publicly observable and hence serves as
an additional signal. The fact that a firm can observe the rival’s entry implies a benefit of“waiting,” giving rise to the second-mover advantage. The tradeoff arises, however, as the
profitability of each firm depends negatively on the number of firms in the market, meaningthat the first one to enter the market can temporarily monopolize the market (until thesecond one arrives).
Time is continuous and extends from zero to infinity, and each firm decides whether toenter the market at discrete points in time 0, ∆, 2∆, ... by incurring the entry cost c > 0.
Throughout the analysis, we will focus on the continuous-time limit where the length of atime interval ∆ shrinks to zero. As we will see below, once a firm enters, it is weakly optimal
to stay in the market indefinitely, so that the firm has no further decision to make. We assume
that each firm can observe the other firm’s entry decisions but not the realized payoffs (or,equivalently, the market condition).8
Definition 1 A firm is called active if it has entered the market and inactive otherwise.
The market condition, which is common to both firms, is either good or bad. The priorprobability of the market being good is p0 . The market condition is initially not known toeither firm and is revealed only gradually over time. More precisely, a firm may privately
observe a signal of the market condition as long as it remains inactive. The signal is of
the bad-news type and arrives only if the market is bad: conditional on the market being
bad, for an interval [t, t + dt), an inactive firm observes a signal with probability λdt. This
specification means that the arrival of a signal indicates that the market condition is bad for
8This is essentially to assume that the profits are often realized with a substantial lag. In the superjumbo
market, for instance, it easily took a decade, if not more, to see the eventual size of the market.

6sure.9
Definition 2 An inactive firm is called informative if it has observed a signal and uninformative otherwise.
According to these definitions, we can classify each firm into three distinct categories:
active, informed, and uninformed. In particular, when we refer to a firm as either informed
or uninformed, it implies that it is inactive at the moment. As will become clear below, there
are no further decisions to make if a firm is either active or informed, so that we generally
focus on the problem of a firm that is currently uninformed.
The net profit a firm can earn is determined by the market condition and the numberof firms in the market. If the market is good, an active firm earns (π + ρ)∆ per period if
it is the only firm in the market and π∆ if both of them are in the market. We call ρ themonopoly premium, which could depend on the extent of market competition, and in generalassume ρ > 0. If the market is bad, on the other hand, an active firm invariably earns zeroprofit. The net profit for an inactive firm is also normalized at zero, implying that there isno incentive to exit from the market ex post.

3

Market entry under private learning

3.1

Belief formation

From the viewpoint of an uninformed firm, there are three possible states as we describe
below:
1. The market is good (state G);
2. The market is bad, and the other firm is uninformed (state BU );3. The market is bad, and the other firm is informed (state BI).
An uninformed firm’s belief is hence defined in two dimensions and denoted as (pt , qt )where: (i) pt is the conditional probability that the market is good (state G); (ii) qt is theconditional probability that the market is bad and the other firm is uninformed (state BU ).
By definition, 1 − pt − qt is the conditional probability that the market is bad and the otherfirm is informed (state BI). All the probabilities are conditional on the firm being inactive
and uninformed. In what follows, we focus on symmetric Markov perfect Bayesian equilibria
with the belief (pt , qt ) as the state variable.
9

We make this assumption purely for tractability. An important implication of this assumption is that afirm’s belief gradually improves over time as long as it stays uninformed, which is essential to make pre-entry
learning meaningful. Our results should hold in a qualitative sense even if we consider a more complicatedinformation process as long as this structure is retained.

7

3.2

Second mover

It is quite straightforward to derive the optimal strategy for the second mover who, with nostrategic concerns, simply enters if the current belief pt is high enough. Observe that the
belief pt increases monotonically over time as long as the firm observes no signal.
Given some belief p, the expected joint profit of entering now is given by
p

π
− c,
r

which increases over time. Alternatively, the firm may wait and enter at the next instantupon observing no signal. Since the firm observes no signal with probability 1 − (1 − p)λ∆,the expected profit of this waiting strategy is given by π(1 − r∆) p − c + (1 − p)λ∆c .
r

The cost of adopting this strategy is obviously the foregone profit. On the other hand, by
waiting, the firm can collect more information, the benefit of which is captured by (1−p)λ∆c.
It is clear that the benefit of avoiding wrong entry is decreasing in the current belief.
It follows from these that the firm enters the market now if and only if ππp − c ≥ (1 − r∆) p − (p + (1 − p)(1 − λ∆))c ,rrwhich is further simplified to πr∆ p − c ≥ (1 − r∆)(1 − p)cλ∆.
rThis condition gives the cutoff belief p which converges in the limit top=(r + λ)c,π + λcsuggesting that the second mover enters the market once and for all when the belief pt reaches
the threshold p̄. We assume that this threshold is smaller than one, so that a firm enters if
it knows that the market condition is good almost surely.
Assumption 1 p < 1 ⇔πr > c.

Now suppose that the first mover enters the market at some time t. As this necessarily
implies that the first mover has observed no signal, the second mover’s belief makes a discrete
jump after the entry. The updated belief is obtained aslim pt+∆ = φt :=

∆→0

8

pt
,
pt + q twhich indicates the amount of information revealed by a market entry. Note in particularthat no firm is informed at time 0, i.e., q0 = 1 − p0 and p0 = φ0 , meaning that a firm’s entry
reveals no additional information.
If the updated belief exceeds p, the firm will follow immediately at the next instant, sothat the first mover can appropriate almost no monopoly rent in the limit.
Lemma 1 An uninformed firm follows immediately after observing the other firm’s entry at
t if and only if φt ≥ p := (λ+r)c
π+λc .

3.3

First mover

We now turn to the first mover’s problem which is far more complicated than the second
mover’s problem described above. Throughout the analysis, we often denote by (p, q) the
current belief and by (p′ , q ′ ) the belief at the next instant when no entry will have occurred.
Let σ be the (symmetric Markov) behavior strategy where σ(p, q)∆ is the probability ofmarket entry in a time period, conditional on the firm being uninformed (and both firms
being inactive) when the current belief is (p, q). In what follows, we often write s = σ(p, q)
and s′ = σ(p′ , q ′ ) to save notation.
As mentioned, the major technical complication of our analysis arises from the fact thatthe evolution of the belief during this phase depends on the first mover’s strategy σ. Moreprecisely, in states G and BU , the rival firm enters the market with probability s∆; in stateBI, knowing that the market is bad, the rival firm never enters. Moreover, in states BU andBI, the firm observes no signal and remains uninformed with probability 1 − e−λ∆ . Finally,
the state changes from BU to BI when the rival firm observes a signal, which occurs withprobability 1 − e−λ∆ . Given the current belief (p, q) and s = σ(p, q), the next-period belief(p′ , q ′ ) is hence computed as(1 − s∆)p,(1 − s∆)(p + qe−λ∆ ) + (1 − p − q)e−λ∆(1 − s∆)e−2λ∆ q,q′ =(1 − s∆)(p + qe−λ∆ ) + (1 − p − q)e−λ∆

p′ =

(1)
(2)

with the initial prior given by q0 = 1 − p0 .
The current state of the economy is thoroughly characterized by (p, q). What is particp
ularly crucial is φ := p+q
which essentially determines the amount of information revealedby an entry. Fortunately, while (p, q) may follow a quite complicated path, it is relatively
straightforward to compute φ as it is independent of the first-mover strategy σ; from (1) and

9

(2), we obtain
φ′ :=

p′
p
=
> φ,
p′ + q ′p + qe−2λ∆which indicates that φt monotonically increases over time for any given strategy σ.
3.3.1

The waiting phase: φ ≥ p

To characterize the first mover’s optimal strategy, there are two cases we need to consider,depending on whether φ exceeds p or not. We start with the case where the current belief(p, q) is such that φ ≥ p, so that an uninformed firm immediately follows the rival firm. Notethat this is the “winner’s curse range” where the first mover can monopolize the market
only if the market condition is bad. As this is a phase where the second-mover advantage
dominates, we call it the waiting phase.
We essentially follow the same procedure to derive the continuation equilibrium in thisphase, by comparing the expected payoffs of entering now and at the next instant. Suppose
first that firm 1 chooses to enter now. Conditional on the market being good, firm 2 stays
inactive with probability 1 − s∆, which allows firm 1 to monopolize the market for a period.
As such, since firm 1 earns (π + (1 − s∆)ρ)∆ in the first period after entry and π∆ thereafter,
the expected profit of entering at t is
p

π
r


+ (1 − s∆)ρ∆ − c.

(3)

Now consider an alternative strategy in which firm 1 waits and enters at the next instant
if it is still uninformed. In the meanwhile, firm 2 may or may not enter at t. There are two
possibilities. First, firm 2 enters now and firm 1 follows at the next instant, which occurs
with probability s∆(p + q(1 − λ∆)). The expected payoff from this contingency is π
(1 − r∆)s∆ p − (p + q(1 − λ∆))c .
r
Second, firm 2 does not enter now but firm 1 enters at the next instant, which occurs withprobability (1 − s∆)(p + q(1 − λ∆)) + (1 − p − q)(1 − λ∆). The expected payoff from thiscontingency is

π

 
(1 − r∆) (1 − s∆)p
+ (1 − s′ ∆)ρ∆ − (1 − s∆)(p + q(1 − λ∆)) + (1 − p − q)(1 − λ∆) c ,rwhere s′ = σ(p′ , q ′ ). From these, the expected payoff of this waiting strategy is obtained as π
(1 − r∆) p − (p + (1 − p)(1 − λ∆))c + (1 − s∆)(1 − s′ ∆)pρ∆ ,
r
10

(4)

taking s and s′ as given.10
Comparing (3) and (4), the waiting strategy yields a lower payoff if πr p − c + (1 − s∆)s′ ∆pρ > (1 − p)λc,r(5)which shows the tradeoff between entering now and waiting for an instant. By entering now,
firm 1 may monopolize the market and obtain an additional payoff of (1 − s∆)ps′ ∆ρ for atime period. By waiting until the next instant, the firm may receive additional informationand save (1 − p)λc by avoiding wrong entry; however, a firm also suffers a loss r(p πr − c) since
the continuation payoff is realized an instant later.
The following statement establishes that the firms gradually enter at some positive rate
once the belief reaches the threshold p. It is relatively straightforward to verify that thisconstitutes an equilibrium. The proposition further suggests that this is the unique symmetriccontinuation equilibrium of the waiting phase.
Proposition 1 For φt ≥ p > pt , there exists a unique symmetric continuation equilibrium
in which:
1. Neither firm enters until pt reaches p;
2. When pt reaches p, the two firms start entering at a rate to keep pt = p;
3. Once a firm enters, the other firm immediately follows at the next instant.
Proof. See Appendix.
If p0 > p, we have a situation where φ0 > p0 > p. In this case, the firms enter with strictly
positive probability at time 0 so that p > p∆ (see Lemma 5 in Appendix). Proposition 1
thus exhausts all the possibilities and completely characterizes the continuation equilibrium
for any φt ≥ p.
3.3.2

The preemption phase: p > φ

The strategic nature of the problem flips once φ gets below this threshold p. As entry inthis phase is driven by the first-mover advantage and preemption motives, we call it the10
Note that this is the expected payoff when firm 1 enters at the next instant regardless of whether firm 2enters now or not, which may not hold if lim∆→0 s∆ > 0. For instance, if lim∆→0 s∆ = 1 and firm 2 does notenter, firm 1 knows for sure that the market is bad, making it optimal never to enter. As we will see below,
however, the firms always enter smoothly in the waiting phase, so that this condition is always valid.

11preemption phase. Let τ ∗ denote the time of the earliest possible entry in equilibrium, which
is define asτ ∗ := inf{t : σt > 0}.
For exposition, we say that pioneering entry occurs, or a market pioneer emerges, if s > 0 for
any (p, q) such that p > φ or, equivalently, p > φτ ∗ .11 We call it “pioneering” because, as we
will see later, it generates valuable information and is generally beneficial to the rival firm,
despite the fact that it is driven by preemption motives.
Given p > φ, if a firm enters now, the rival firm’s belief jumps up but is still lower than p.
The second mover thus will not enter immediately, and the first mover can monopolize themarket for some duration of time which we denote by δt . Since the second mover waits until
the belief reaches p, δt is computed as
δt =

1
pqt
ln
.
λ pt (1 − p)Note that for a given p0 , φt depends only on t, and so is δt . The expected payoff of entering
now is then given byp

π + (1 − e−rδt )ρ
− c.
r

The equilibrium allocation becomes much more complicated when pioneering entry actually occurs on the equilibrium path with positive probability. We can nonetheless establishthat any (symmetric) equilibrium must in general take the following form.
Proposition 2 Suppose that pioneering entry occurs in equilibrium with some positive probability. Then, in any symmetric equilibrium, there exist τ and τ ∈ (τ , ∞) such that(
∈ (0, ∞) for t ∈ (τ ∗ , τ ) ∪ (τ , ∞),st=0
for t ∈ (τ , τ ).
Moreover, φτ < p.
Proof. See Appendix.
In equilibrium, the firms enter smoothly over time except at the outset of the game.12
The proposition suggests that if pioneering entry ever occurs, there is an interval during
11
Note that our definition of market pioneer differs from the conventional one: in the literature, it simply
refers to the first entrant in a new market (Robinson and Fornell, 1985).
12
At time 0, the firms may enter with some strictly positive probability.

12

which the firms enter at some positive rate, followed by an interval of no entry (τ , τ ). This isbecause, although the firms have less private information and face more uncertainty early on,
the fact that they have less information means that a firm’s entry reveals less information,
thereby making the preemption effect stronger. When this first-mover advantage dominates
the cost of entering prematurely with insufficient information, the firms enter with somepositive probability in the preemption phase.

4

Equilibrium

4.1

The constrained problem

To analyze the possibility of pioneering entry, we consider a hypothetical situation in whicha firm, say firm 2, can enter only after the other firm, i.e., firm 1, enters. As it turns out,
this constrained version of the problem, which excludes the possibility of entry competition,
provides enough information to see when pioneering entry occurs in the original problem.
Under the restriction that firm 2 must be the second mover, the problem faced by firm 1
is substantially simpler: firm 1 simply decides when to enter conditional on having observed
no signal. Provided that firm 2 never enters, firm 1’s belief pair at t is given bypt =(1 − p0 )e−2λt
p0
,
q
=
,
t
p0 + (1 − p0 )e−λt
p0 + (1 − p0 )e−λt(6)which depends only on t. As a consequence, the expected payoff of entering at t can also be
written as a function of t. Let Π̂s (t) denote the expected payoff of entering at t, evaluated attime s, under the restriction that firm 2 must be the second mover. If firm 1 enters at timet, firm 2 will wait until pt reaches p. Evaluated at time 0, we obtain
 π + (1 − e−rδt )ρ
 
Π̂0 (t) = e−rt p0− p0 + (1 − p0 )e−λt c ,r(7)

If φt ≥ p, then δt = ∆ which converges to zero in the limit. In contrast, if p > φt , firm 2
must wait to collect more information, giving firm 1 some time to monopolize the market.
The incentive for pioneering entry thus hinges crucially on δt .
Suppose first that there is no pioneering entry and δt = 0. Along with the fact that φt
is monotonically increasing, the continuation equilibrium characterized in Lemma 1 suggeststhat the earliest possible entry occurs when the belief reaches p. Define τ NP such that
e−λτ

NP

=

p0 (π − rc)
p0 (1 − p)=,p(1 − p0 )(1 − p0 )(λ + r)cso that given p0 , the earliest possible entry occurs at τ NP . From (6), we obtainδt =1 p(1 − p0 )e−2λt
1
pqt
ln
= ln
= τ NP − 2t,
λ pt (1 − p)λp0 (1 − p)13

NP

meaning that δt > 0 if and only if τ 2> t. Let ΠNP := maxt∈( τ NP ,∞) Π̂0 (t) denote the2

expected payoff without pioneering entry, which can be written as
 π

NP
NP 
p0 − p0 + (1 − p0 )e−λτΠNP = Π̂0 (τ NP ) = e−rτ
c .
r

NP

Now suppose that firm 1 enters early when δt is still positive or alternatively τ 2> t.

If firm 1 enters at t, firm 2’s belief jumps up to φt , but firm 2 still needs to wait until the
belief reaches p, which allows firm 1 to monopolize the market for some time δt . Therefore,NPthe expected payoff of entering at t ∈ [0, τ 2 ] is given by−rtΠ̂0 (t) = e

 π + (1 − e−r(τ NP −2t) )ρ p0− p0 + (1 − p0 )e−λt c .
rNote that this problem is well defined only if τ NP > 0. In what follows, therefore, we assume
that this is indeed the case.
Assumption 2 τ NP > 0 ⇔ p > p0 .
The optimal timing of entry under the entry restriction can be found by maximizing Π̂0 ,which is a strictly concave function of t as we show below.
NP

Lemma 2 In the limit, there exits a unique τ P ∈ [0, τ 2 ] which maximizes Π̂0 (t). Lettingµ(t) := −p0 (π + ρ − rc) − p0 ρe−r(τNP −2t)+ (1 − p0 )(λ + r)ce−λt ,the optimal timing of pioneering entry, denoted by τ P , is given by
if 0 ≥ µ(0),0NPτ P = τ̂if µ(0) > 0 > µ( τ 2 ),NP
 τ NP
if µ( τ 2 ) ≥ 0,2
where τ̂ solves µ(τ̂ ) = 0.

Proof. See Appendix.
Let ΠP := maxt∈[0, τ NP ] Π̂0 (t) denote the expected payoff under the restriction that firm
2

2 must be the second mover, which can be written as
P

P

−rτ P

Π = Π̂0 (τ ) = e π + (1 − e−r(τ NP −2τ P ) )ρP
− p0 + (1 − p0 )e−λτ c .
p0r

As the monopoly premium becomes larger, it becomes more costly to wait and collect more
information. As a consequence, the optimal timing of pioneering entry moves forward withan increase in the monopoly premium.
14

4.2

The emergence of a market pioneer

Under the restriction that firm 2 must be the second mover, it is optimal for firm 1 to enter
once and for all at τ P if ΠP > ΠNP . Clearly, though, this does not constitute an equilibrium
when firm 2 is also an active player who can enter the market at any point in time: if firm
2 knows that firm 1 enters at τ P with certainty, there is always an incentive for firm 2 toenter slightly earlier at τ P − ∆ which allows it to monopolize the market for a while. As
this argument and Proposition 2 indicate, the firms must adopt mixed strategies when they
compete to be the first mover. Even then, these payoffs are still useful as they provide anecessary and sufficient condition for a market pioneer to emerge in equilibrium.
Proposition 3 Pioneering entry occurs in equilibrium if and only if ΠP > ΠNP .
Proof. See Appendix.
The intuition behind Proposition 3 is quite simple. To see this, observe that in theequilibrium with no pioneering entry, each firm waits until time τ NP and then gradually enterspast that point; the expected payoff in this equilibrium is hence ΠNP as in the constrained
problem. Given this, the sufficiency is obvious, because if ΠP > ΠNP , a firm must havean incentive to deviate and enter with some positive probability in the preemption phase.
On the other hand, the necessity comes from the fact that the entry competition can only
lower the benefit of pioneering entry while raising the benefit of waiting due to the signaling
effect (see the next section for more detail). As such, if there is no incentive to enter in thepreemption phase in the constrained problem, then there is certainly no incentive to do so in
the original problem.
Proposition 3 implies that as ΠP −ΠNP becomes larger, the first-mover advantage becomes
more salient, rendering pioneering entry more likely. It is hence important to clarify factors
which affect ΠP − ΠNP . The following proposition clarifies under what conditions a market
pioneer is more likely to emerge, which offers crucial welfare and policy implications.
Proposition 4 There exist ρ̂ and p̂0 ∈ (0, p) such that pioneering entry occurs if and only ifρ ∈ (ρ̂, ∞) or p0 ∈ (p̂0 , p).
Proof. See Appendix.
An important determinant of the value of pioneering entry is obviously the monopoly
premium ρ, where an increase in ρ facilitates pioneering entry. This is illustrated in figures 1
and 2 which depict Π̂0 (t) for different values of π and ρ (with π + ρ fixed at one).
15

[Figures 1 and 2 about here]

4.3

Equilibrium payoff bounds

Let Π∗ denote the expected equilibrium payoff for each firm whereΠ∗ = e−rτ∗ π + (1 − e−rδτ ∗ )ρ∗
− p0 + (1 − p0 )e−λτ c .
p0r

If ΠNP > ΠP , the earliest possible entry occurs at time τ NP , and Π∗ = ΠNP as we haveseen. If ΠP > ΠNP , on the other hand, there is a clear gain from becoming the first mover.
We in general have Π∗ < ΠP because the entry competition shifts the timing of entry forward, inducing the firms to start entering before the optimal time τ P . The following result
characterizes the timing of pioneering entry, provided that ΠP > ΠNP .
NP

Proposition 5 Suppose that τ P > 0. If pioneering entry occurs in equilibrium, then τ 2 >τ P > τ ∗.
Proof. See Appendix.
The entry competition is self-defeating in that it induces the firms to enter before theycollect enough information, thereby reducing their payoffs compared to the case where firm 2
can only follow firm 1. The question is then whether this competition drives the value of thefirst-mover advantage down to zero. As it turns out, the firms’ expected payoffs can actually
improve when pioneering entry occurs with positive probability, because the second movercan benefit from the information revealed by the first mover’s entry. The following result
characterizes the equilibrium payoff bounds when ΠP > ΠNP .
Proposition 6 Suppose that ΠP > ΠNP so that pioneering entry occurs with positive probability. Then, each firm’s expected payoff falls in (ΠNP , ΠP ], i.e.,Π∗ = e−rτ∗ π + (1 − e−rδτ ∗ )ρ∗
− p0 + (1 − p0 )e−λτ c ∈ (ΠNP , ΠP ].
p0r

Proof. See Appendix.

To understand this result, especially why the expected payoff is not driven down to ΠNP ,it is important to understand the roles of the two types of externality that are present in thissetting. On one hand, there is a negative payoff externality via market competition which is

16

captured by ρ. The payoff externality is clearly the source of the entry competition. This ismost clearly seen by supposing ρ = 0, in which case π Π̂0 (t) = e−rt p0 − p0 + (1 − p0 )e−λt c ,rfor all t ∈ [0, ∞), suggesting that there is no preemption phase.

In contrast, as ρ increases, the first-mover advantage becomes more salient, giving eachfirm an incentive to become a market pioneer. This entry competition forces the firms toenter earlier than the optimal timing τ P , which necessarily lowers the expected payoff ofbecoming a market pioneer. In equilibrium, this expected payoff must be driven down tothe expected payoff of becoming a follower which is strictly larger than ΠNP because of theinformation externality: with the first-mover’s entry providing additional information, thefollower can enter earlier than τ NP and hence on average achieve a higher payoff.

5

Efficiency

5.1

The social planner’s problem

To derive efficiency properties of the model, we now consider a social planner who attempts
to maximize the joint profit of the firms. Since the first-best allocation is rather trivial inthis setting,13 here we focus on the second-best allocation where the social planner is subject
to the same informational constraints as the firms. Specifically, we consider an environment
in which the social planner specifies the entry timing (τ1 , τ2 ) such that firm i can enter themarket only at time τi .14 Without loss of generality, we assume τ1 ≤ τ2 .
For a given pair (τ1 , τ2 ), the joint profit is obtained as
 π+ρ
 π−ρ
 
 
Π(τ1 , τ2 ) = e−rτ1 p0− p0 + (1 − p0 )e−λτ1 c + e−rτ2 p0− p0 + (1 − p0 )e−λ(τ1 +τ2 ) c .
rr

Two remarks are in order regarding the two types of externality in this setting. First, firm 2’s

p0, because of the positive informabelief at τ2 is p +(1−p p)e0−λ(τ1 +τ2 ) , rather than p +(1−p)e−λτ200

00tion externality of the first entry. Second, the second entry contributes only π − ρ to the joint
profit (while its private gain is π) due to the negative payoff externality, which corresponds towhat is often referred to as the “business-stealing effect” in standard static oligopoly models.
13

The first-best allocation may be defined as the one that maximizes the joint profit when the market
condition is known to the social planner. Then, the problem is clearly trivial: if the market is good, the firms
should enter immediately at time 0; if not, they should never enter.
14
Given that each firm has only one chance to enter the market, it chooses to enter if and only if it isprofitable to do so at the time.

17

It is also important to note that because of the payoff externality, there may arise a casewhere it is socially optimal to have only one firm in the market. This is the case if
c≥

π−ρ
,
r

in which case the social planner would allow only one firm to enter the market (t2 = ∞).
Since this case is relatively straightforward, we restrict our attention to the case where it issocially optimal to have two firms whenever the market condition is good.
Assumption 3 π − ρ > rc.

5.2

The socially optimal timing of entry

We begin with the optimal timing of the second entry. Given τ1 , the social planner’s problem
is defined as π−ρ− p0 + (1 − p0 )e−λ(τ1 +τ2 ) .
max e−rτ2 p0
τ2r

The first-order condition is then given by

p0 (π − ρ − rc) = (1 − p0 )(λ + r)ce−λ(τ1 +τ2 ) .

(8)The left-hand side indicates the foregone benefit of delaying the second entry while the righthand side is the benefit of collecting more information. Let τ2∗∗ (τ1 ) denote the socially optimal
timing of the second entry as a function of τ1 . If there exists an interior solution τ2∗∗ (τ1 ) > τ1 ,it must solvepτ2∗∗ =(λ + r)cp0> p.
∗∗ (τ )) =
−λ(τ
+τ
1
1
2
π − ρ + λc
p0 + (1 − p0 )e(9)Define T ∗∗ = τ1 + τ2∗∗ (τ1 ) for τ2∗∗ (τ1 ) > τ1 . From (9), we obtain
e−λT

∗∗

=

p0 (π − ρ − rc),(1 − p0 )(λ + r)c

which depends only on the primitives of the model. This means that if firm 1 enters at time∗∗t ∈ [0, T2 ), firm 2 waits until time T ∗∗ − t and enters if it is still uninformed at the time.
∗∗

Letting Π∗∗ (τ1 ) := Π(τ1 , τ2∗∗ (τ1 )), the joint profit for τ1 ∈ [0, T2 ) can be written as
 π+ρ
 π−ρ

 
∗∗ 
∗∗
Π∗∗ (τ1 ) = e−rτ1 p0− p0 + (1 − p0 )e−λτ1 c + e−r(T −τ1 ) p0
− p0 + (1 − p0 )e−λT c .
rr∗∗

If firm 1 enters after time T2 , on the other hand, firm 2 follows immediately after the first∗∗entry. In the limit, the joint profit for τ1 ∈ [ T2 , ∞) converges to
 π−ρ
 π+ρ
 
 
− p0 + (1 − p0 )e−λτ1 c + e−rτ1 p0− p0 + (1 − p0 )e−2λτ1 c .
Π∗∗ (τ1 ) = e−rτ1 p0
r
r
18

∗∗

For τ1 ∈ [0, T2 ), the first-order condition is obtained as∗∗

p0 (π + ρ − rc) = (1 − p0 )(λ + r)ce−λτ1 +

λ∗∗∗∗p0 (π − ρ − rc)e−r(T −2τ1 ) .
λ+r

(10)

Note that the second term of the right-hand side indicates the benefit of the information∗∗

externality which is increasing in τ1 .15 For τ1 ∈ [ T2 , ∞), the first-order condition is obtainedas∗∗

2p0 (π − rc) = (1 − p0 ) (λ + r) + (2λ + r)e−λτ1∗∗ce−λτ1 .

(11)

As above, let τ1∗∗ denote the socially optimal timing of the first entry.
Proposition 7 (i) T ∗∗ > τ NP . (ii) τ1∗∗ > τ P if τ P > 0. (iii) τ1∗∗ may be larger or smaller
than τ NP where τ NP > τ1∗∗ if
p0 (π − rc)λ+r>.
2λ + r(1 − p0 )(λ + r)c(12)

Proof. See Appendix.

5.3

Pioneers, early followers, and late entrants

Since the seminal work of Lieberman and Montgomery (1988), there is now a voluminous
literature, mainly in the fields of marketing and strategic management, which examines theeffects of entry timing and order.16 The literature often classifies entry timing into three broad
categories: pioneers, early followers, and late entrants (Robinson and Fornell, 1985, Lambkin,1988). Below, we first interpret our equilibrium outcomes according to this classification anddiscuss efficiency properties for each possible class of outcomes, with focus on the two typesof externality.
Observe first that for a given set of parameters, we can pin down an equilibrium with anassociated distribution of entry time. Each equilibrium outcome is generally characterized
by a pair of realized entry times (t1 , t2 ) where t1 (t2 ) denotes the timing of the first (second)
entry (ti = ∞ in the case of no entry). Our model then generates four classes of entry
dynamics that are observationally distinguishable.
1. No entry (t1 = t2 = ∞): Neither firm chooses to enter, and the market never materializes.
∗

15

Due to this effect, the joint profit may not be concave for τ1 ∈ [0, T2 ).
The literature generally focuses on whether there is any enduring advantage for early entrants. Although
we do not consider this possibility, our model can be easily extended to incorporate this type of first-moveradvantage.
16

19

2. Only one entry (t1 < t2 = ∞): Only one firm enters while the other firm chooses not
to follow. This is the case of premature entry (as in the Boeing-Airbus case).
3. Late entrant (t1 < t2 < ∞): A firm enters in the preemption phase and is followedby the other firm with some time lag. The two entries are spaced apart in time.
4. Early follower (t1 ≈ t2 < ∞): A firm enters in the waiting phase and is immediately
followed by the other firm. The two entries are clustered together.
The first two cases occur only when the market condition is bad, reflecting the obvious
fact that no successful market can be monopolized forever. Welfare implications in thesecases are rather straightforward, even without going through any detailed analysis. Given
that the market condition is bad, the first case is (first-best) efficient as the firms correctly
decide not to enter the market. In the second case, one of the firms fails to observe a signal
in time and erroneously enters the market which is not sufficiently profitable. This case isclearly inefficient regardless of the exact timing of the sole entry.
The latter two cases admit two entrants and are often the focus of attention in the existing
literature. Parts (i) and (ii) of Proposition 7 concern the case with a late entrant and suggestthat the firms enter the market too early compared to the social optimum.
• Part (i) states that the equilibrium timing of the second entry is earlier than the socially
optimal timing.17 This is due to the negative payoff externality: when the second
entry occurs, the firm’s average net payoff when the market is good is π − rc while its
contribution to the joint profit is only π − ρ − rc due to the business-stealing effect.18
In the optimal allocation, therefore, the entry threshold is higher and the firm should
wait longer to collect more information.
• Part (ii) states that the equilibrium timing of the first entry is also earlier than thesocially optimal timing. This is due to the positive information externality: if the firstentry occurs later, it reveals more information and benefits the other firm. The first
entrant not only ignores this benefit of signaling (τ1∗∗ > τ P ), but in equilibrium enterseven earlier so as to reveal less information to the rival firm and delay its subsequent
entry (τ P > τ ∗ ).
17

Given that the first entry occurs at some t, the second entry occurs at τ NP − t whereas the socially optimal
timing is given by T ∗∗ − t.
18
The celebrated excess entry theorem (Mankiw and Winston, 1986; Suzumura and Kiyono, 1987; Lahiri
and Ono, 1988) generally builds on this effect and demonstrates that the number of firms in a market can be
too many in static oligopoly models. Our analysis complements this literature by extending this argument toa dynamic context, showing that market entry is too early with this same effect.

20

In contrast, Part (iii) of the proposition concerns the case with an early follower. Althoughτ1∗∗ can be larger or smaller than τ NP , we argue that (12) is more likely to hold under plausible
conditions, meaning that the firms enter too late in this contingency.19 The intuition behind
this case is relatively clear. Once the game reaches the waiting phase, the clear winner is theone that becomes the follower as it can minimize the risk of wrong entry while losing almost
no monopoly rent. This incentive to wait for the rival’s action is often excessively strong,preventing the firms from entering the market at an opportune time.
Note that Proposition 7 implies a paradoxical fact which is worth emphasizing: the firms
enter too early when there is a later entrant, and too late when there is an early follower. Thisis because the efficiency properties of entry dynamics are determined largely by the timing of
the first entry which shapes the temporal distribution of market entry. The problem is thatthe timing of the first entry is hard to observe empirically in many cases, as the existenceof a market is often not known to outside observers until its first entry actually occurs.
Fortunately, though, the time lag between early and late entrants, is suggestive and provides
enough information about the timing of the first entry. A general rule of thumb is that whenmarket entry occurs successively in a short span of time (the case of early followers), chances
are that the market was already ripe when the first entrant arrived, implying that the market
opened up too late. In contrast, when market entry occurs sparsely over time (the case oflate entrants), the first entrant is likely to be a true pioneer who entered when the market
was still filled with uncertainty, even though the market opened up too early with substantial
risk of premature entry.

6

Conclusion

In the existing literature, the roles of pre-entry learning and subsequent market competition
have been investigated extensively but almost independently, and there are very few works
which combine them in a unified framework. To fill this gap and provide a more comprehensive description of the tradeoff faced by potential market entrants, this paper constructs adynamic model of market entry which features these two elements simultaneously. Primary
applications of our analysis include new product markets, technology adoption and foreign
direct investment among others.
As final note, we would like to make a brief remark about the possibility of asymmetricequilibria. Although we restrict our attention to symmetric equilibria throughout the analysis,
our model also admits some asymmetric equilibria. We can indeed find at least one form
19
Proposition 4 shows that if p0 is small, (i) pioneering entry is less likely to occur, and (ii) (12) is more
likely to hold. Therefore, if no market pioneer emerges in equilibrium, (12) is more likely to hold.

21

of such equilibrium, as the strategies considered in the constrained problem constitute an(asymmetric) equilibrium when ΠNP − ΠP is sufficiently large: if a firm can somehow commit
to being the second mover, the optimal choice for the other firm is to enter once and for all attime τ NP .20 We did not pursue this possibility in our analysis because asymmetric equilibria
typically require a level of coordination that is somewhat unrealistic in the context of entry
games.21 It is nonetheless of some interest, at least theoretically, to explore this possibility
in future to obtain a more detailed characterization of asymmetric equilibria.

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20

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21
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23

Appendix
Proof of Proposition 1. To prove the proposition, we first establish the following facts.
Lemma 3 lim∆→0 s∆ < 1 for any φ ≥ p.
Proof. Suppose that lim∆→0 s∆ = 1 for some (p, q). This means that each firm earns p ρπr −cby entering now. Now suppose that a firm deviates and delays entry until the next instant.
In this case, if the other firm does not enter now, the firm knows for sure that the market isbad, and the belief drops to zero, making it optimal never to enter. Taking this into account,the expected profit is π(1 − r∆) p − (p + q(1 − λ∆))c ,rwhich is larger than p πr − c if ∆ is sufficiently small, which is a contradiction.
Lemma 4 s = 0 if φ ≥ p > p.
Proof. Suppose otherwise, i.e., s > 0. This is the case if π(1 − s∆)s′ ∆pρ > (1 − p)λc − r p − c .
rNote that the right-hand side is strictly positive by definition when p > p, which implies
that lim∆→0 s′ ∆ > 0 and p > p′ . Given this, we can apply the same argument to show thatlim∆→0 st ∆ > 0 for all future periods. Given p > p, we can take an arbitrarily small ε anddefineNε (∆) := max{k ∈ N | ε > k∆},N (∆)ε(1 − st+k∆ ∆) = 0. This means that if thefor some ε. Then, we must have lim∆→0 Πk=0

other firm does not enter by t + Nε (∆)∆, the firm’s belief will drop almost to zero. By thesame argument as in Lemma 3, it is strictly better to wait until t + Nε (∆)∆, and hence s = 0
which is a contradiction.
Lemma 5 For φ ≥ p,
1. s > 0 if p ≥ p;
2. lim∆→0 s∆ > 0 such that p > lim∆→0 p′ if p > p, where p′ indicates the belief at thenext instant (in case of no entry).
24

Proof. We first show that s > 0. Suppose that s = 0. Then p′ > p, and the continuation
payoff is weakly lower than π+ ρ∆ − (p + (1 − p)(1 − λ∆))c ,(1 − r∆) prwhich is in turn lower than p( πr + ρ∆) − c, the expected payoff of entering now, if p ≥ p.
To prove the second statement, note that π
(1 − p)λc − r p − c < 0,r
so that (5) must be satisfied. As such, it is strictly better to enter now than at the next instant.
By Lemma 3, however, a firm, say firm 1, waits for an instant with positive probability. Thisimplies that firm 1’s entry decision at the next instant must depend on what firm 2 does now,
for otherwise there would be no incentive for firm 1 to wait. The next-period belief musthence be low enough to satisfy p > p′ so that firm 1 enters at the next instant if and only iffirm 2 enters now.
Lemma 6 For φ ≥ p, s ∈ (0, ∞) such that p′ = p if p = p.
Proof. From Lemmas 3 and 4, we know that a firm must play a mixed strategy if p ≥ p.
Suppose first that s is small enough and p′ > p = p. Then, s′ ∆ > 0 by Lemma 5, which
implies that (5) strictly holds given p = p. This implies, however, that it is strictly betterto enter now than at the next instant, which is a contradiction. Now suppose that s is large
enough and p′ < p. Then, s′ = 0 by Lemma 4, so that π(1 − s∆)s′ ∆pρ = (1 − p)λc − r p − c = 0.
rThis means that the expected payoff of entering now is equal to that of entering at the next
instant regardless of the other firm’s action. Given p′ < p, however, the optimal strategy at
the next instant is to enter if and only if the other firm enters now. Therefore, a firm canearn a higher payoff by waiting for an instant, which is a contradiction. This proves that the
belief must be kept at p once it reaches this level.
These results show that starting from some pt < p, no firm enters until pt reaches p. Oncethe belief reaches p, then the firms start randomizing so as to keep pt = p as suggested by
Lemmas 5 and 6.
Proof of Proposition 2. We first establish some important properties which must hold in
the preemption game.
25

Lemma 7 σt ∆ < 1 for any φt < p.
Proof. If σt ∆ = 1, neither firm can earn monopoly profit. Then, one of the firms can achievea higher payoff by waiting until the belief reaches p.
Lemma 8 There exists no t > 0 such that φt < p and σt = ∞ for all s ∈ (0, t).
Proof. Note that if pt is close enough to zero, σt = 0. Suppose that there exists some t suchthat σs = ∞ for all s ∈ (0, t). However, if σs = ∞ for all s ∈ (0, 2t ], then p t → 0 and hence2

σ t → 0 as ∆ → 0, which is a contradiction.
2

Lemma 9 lim∆→0 σt < ∞ for any φt < p and t > 0.
Proof. Given some equilibrium strategy σ, suppose that σt = ∞ for some φt < p and t > 0.
By Lemma 8, there exists some s ∈ (0, t) such that σs < ∞. Without loss of generality,
consider pt and qt such that σt−∆ < ∞ with pt−∆ and qt−∆ satisfying (1) and (2). Then, if
a firm enters at t − ∆, the expected payoff ispt−∆

π + (1 − σt−∆ ∆)(1 − e−rδt−∆ )ρ
− c.
r

If it waits and enters at t, the expected payoff is(1 − r∆)pt−∆  π + (1 − σt ∆)(1 − e−rδt )ρ
−c ,
pt
pt
r

which is strictly lower than the payoff of entering at t − ∆. We should thus have σt−∆ ∆ = 1,
which is a contradiction.
We can now prove the proposition. Since φt is strictly increasing, it will reach p sooner
or later, and the firms enter at some positive rate as shown in Proposition 1. This meansthat there must be a period in which σt = 0, followed by a period in which σt ∈ (0, ∞). To
show that σt ∈ (0, ∞) for t ∈ (τ ∗ , τ ), note from Lemma 9 that σt < ∞ for t ∈ (τ ∗ , τ ). Thismeans that we only need to show that σt > 0 for all t ∈ (τ ∗ , τ ). Suppose that there exists aninterval (a, a) such that σt = 0 for t ∈ (a, a) but σt > 0 for t ∈ (a − ε, a) ∪ (a, a + ε) where εis some positive number. This implies that a firm obtains a higher payoff by entering at a orat a than at any time in (a, a). Given that σt = 0 for t ∈ (a, a), by entering at t + dt insteadof entering at t, the firm’s expected payoff increases by(r + λ)c − pt π + (1 + e−rδt )ρ + λc ,26

which is decreasing in t. This means that the expected payoff is concave in (a, a), and hence
we cannot have the payoff maximized at a and a in this interval. Therefore, σt ∈ (0, ∞) fort ∈ (τ ∗ , τ ).
Finally, note that when φt = p, it is strictly optimal for a firm to wait until pt reaches p.
Therefore, shortly before φt reaches p, it is still optimal to wait, meaning that φτ < p.
Proof of Lemma 2. Observe that τ P is the solution tomaxtp0 (π + ρ − rc) −rt p0 ρ −r(τ NP −t)e−e− (1 − p0 )ce−(λ+r)t .
r
r

The first-order condition is then given byµ(t) := −p0 (π + ρ − rc) − p0 ρe−r(τNP −2t)+ (1 − p0 )(λ + r)ce−λt .
NP

It is straightforward to verify that µ is strictly decreasing in t for t ∈ [0, τ 2 ], meaning that
there exists at most one τ̂ such that µ(τ̂ ) = 0. In the limit, the optimal timing of pioneering
entry, denoted by τ P , is given by0Pτ = τ̂ τ NP2

where τ̂ solves µ(τ̂ ) = 0.

if 0 ≥ µ(0),NP
if µ(0) > 0 > µ( τ 2 ),NP
if µ( τ 2 ) ≥ 0,

Proof of Proposition 3. The sufficiency is obvious. If ΠP > ΠNP , there is an incentive fora firm to enter the market when p > φt . There must be an equilibrium with pioneering entry.
To show the necessity, we define ΠNP as a function of the initial prior p0 . We can thenestablish the following fact.
Lemma 10 If pioneering entry occurs,pτ ∗π + (1 − e−rδτ ∗ )ρ− c > ΠNP (pτ ∗ ).
r

Proof. Suppose that pioneering entry occurs. Then, by Proposition 2, σt < ∞ for t > 0,
and the expected payoff of entering at t ispt

π + (1 − e−rδt )ρ
− c.
r

As discussed in the proof of Proposition 2, a firm must be indifferent between entering at τ
and at τ , at which the belief reaches p, i.e.,pτ

π + (1 − e−rδτ )ρ
− c = ΠNP (pτ ).
r
27

Moreover, at any t ∈ (τ ∗ , τ ), a firm must be indifferent between entering and waiting. Thismeans that a firm’s payoff of entering at τ ∗ must equal to that of becoming the second mover(observes the other firm’s entry and then waits until the belief reaches p), i.e., in either
contingency, the firm’s payoff is
π + (1 − e−rδτ ∗ )ρ
− c.
r

pτ ∗

Since ΠNP is the payoff when a firm does not gain any information from the rival, it isnecessarily lower than the payoff of becoming the second mover, which proves the lemma.
∗

Given this, since ΠNP (p0 ) = e−rτ pp0∗ ΠNP (pτ ∗ ) andτΠP (p0 ) ≥ e−rτ∗p0  π + (1 − e−rδτ ∗ )ρ
pτ ∗
−c ,
pτ ∗
r

we can show that ΠP (p0 ) > ΠNP (p0 ).
Proof of Proposition 4. Note that ΠNP and ΠP are given by
 π

NP 
p0 − p0 + (1 − p0 )e−λτ
c ,
r
NP
P


π + (1 − e−r(τ −2τ ) )ρPP
− p0 + (1 − p0 )e−λτ c .
ΠP (p0 ) = e−rτ p0rΠNP (p0 ) = e−rτ

NP

(13)
(14)

It is clear that (14) can only increase with ρ while (13) is independent of it. Therefore, there
must be a threshold ρ̂ such that the condition for pioneering entry is satisfied if and only if
ρ > ρ̂.
p′

For the effect of p0 , define p′′0 := p′ +(1−p0′ )e−λ∆ for a given p′0 < p where ∆ is assumed to00

be arbitrarily small. Also, we write τ NP and τ P both as functions of p0 . SinceΠNP (p0 ) = e−rτNP (p )0


p0  πp −c ,
pr

we haveNPΠNP (p′′0 ) =′′′′e−rτ (p0 ) p′′0 NP ′r∆ p0 NP ′
Π (p0 ),
Π
(p
)
=
e
0
NP ′
p′0
e−rτ (p0 ) p′0

for any p > p′′0 > p′0 .
As for ΠP , observe first that by Lemma 2,
lim µ(0) = −p0 (π + ρ − rc) − p0 ρe−rτ

NP

p0 ↑p

=−

+ (1 − p0 )(λ + r)c(λ + r)c(λ + r)c −rτ NP π − rc+
(π + ρ − rc) −ρe(λ + r)c < 0,π + λcπ + λcπ + λc28(15)

suggesting that there exists some threshold p̃ < p such that τ P = 0 if and only if p0 ∈ [p̃, p).
Also, define p+ :=p′0
p′0 +(1−p′0 )e−λτ

P (p′ )
0.

First consider the case where p0 < p̃, or alternatively p+ ≥ p′′0 and τ P (p′0 ) ≥ ∆. We then
have
P

Π

 π + (1 − e−r(τ NP (p′0 )−2τ P (p′0 )) )ρ

P ′ 
p′0
− p′0 + (1 − p′0 )e−λτ (p0 ) cr
NP (p′ )−2τ P (p′ ))

′ 
−r(τ
0
0 )ρ
P(p′0 ) p0
+ π + (1 − e= e−rτ
−
c
.
p
p+
r

P ′
(p′0 ) = e−rτ (p0 )Similarly, π + (1 − e−r(τ NP (p′′0 )−2τ P (p′′0 )) )ρ
P ′′ 
P ′′− p′′0 + (1 − p′′0 )e−λτ (p0 ) c
ΠP (p′′0 ) = e−rτ (p0 ) p′′0
r
P ′


−r(τ NP (p′′′′0 )−2(τ (p0 )−∆) )ρ
−r(τ P (p′0 )−∆) p0+ π + (1 − e
>e
−
c
p
p+
r
NP (p′ )−2τ P (p′ ))


′′
−r(τ
0
0 )ρ
P ′
p0 + π + (1 − e> e−r(τ (p0 )−∆) +
−c
p
p
r
′′
p
= er∆ 0′ ΠP (p′0 ).
p0Here, the second line shows the payoff when the firm enters at τ P (p′0 ) − ∆ at which point the
belief reaches p+ . From the second line to the third, we use the fact that
τ NP (p′′ ) + 2∆ > τ NP (p′′ ) + ∆ = τ NP (p′0 ).
Next, consider the case where p′0 ≥ p̃, or alternatively p′′0 > p+ and τ P (p′0 ) = 0. We then
have
P

Π

−rτ NP (p′0 ) )ρp′′
′ π + (1 − e(p0 ) = p0
− c > e−r∆ ′′0 ΠP (p′′0 ),
r
p0

and
P

Π

−rτ NP (p′′0 ) )ρ′′′′ π + (1 − e− c.
(p0 ) = p0

r

It follows from above that if π NP (p′0 ) = π P (p′0 ), then π NP (p′′0 ) < π P (p′′0 ) for p̃ > p′0 and
π NP (p′′0 ) > π P (p′′0 ) for p0 ≥ p̃. This suggests that ΠNP and ΠP intersect at most twice forp0 ∈ (0, p]. Note also that p > p̃ and ΠNP (p) = ΠP (p). This proves that there is a thresholdp̂ ∈ (0, p) such that the condition for pioneering entry is satisfied if and only if p0 ∈ (p̂, p).
Proof of Proposition 5. For the first inequality, observe that limt↓ τ NP Π̂′0 (t) > 0 by
2NP

definition. This implies that if τ P = τ 2 , then ΠNP > ΠP .
For the second inequality, we make use of the following results.
29

Lemma 11 If pioneering entry occurs in equilibrium, then τ P ≥ τ ∗ .
Proof. If τ > τ̂ , a firm can deviate by entering at t ∈ [τ̂ , τ ) and obtain a higher payoff.
Lemma 12 If pioneering entry occurs in equilibrium,p0 (π + (1 + e−r(τNP −2τ P )P)ρ − rc) ≥ (1 − p0 )e−λτ (r + λ)c.

Proof. By the definition of τ P , ifP

(1 − p0 )e−λτ (r + λ)c > p0 (π + (1 + e−r(τNP −2τ P ))ρ − rc),NPthen τ P = τ 2 , so that the first mover cannot earn any monopoly profit. This means,however, that ΠNP > ΠP , i.e.,
 π
 π


NP
NP
P
P
p0 − (p0 + (1 − p0 )e−λτ )c > e−rτ p0 − (p0 + (1 − p0 )e−λτ )c ,e−rτrr
which suggests no pioneering entry in equilibrium, a contradiction.
We are now ready to prove the proposition. If a firm, say firm 1, enters at τ ∗ , the expected
payoff isΠτ ∗ := pτ ∗π + (1 − στ ∗ ∆)(1 − e−rδτ ∗ )ρ
− c.
r

(16)

If firm 1 waits for a period, on the other hand, firm 2 enters with probability στ ∗ ∆, inwhich case firm 1 waits until τ ∗ + δτ ∗ when pt reaches p; with the remaining probability,
firm 2 does not enter at τ ∗ , and firm 1 enters at τ ∗ + ∆ if it observes no signal. Letting
xt := (1 − σt ∆)(pt + qt e−λ∆ ) + (1 − pt − qt )e−λ∆ , the expected payoff of waiting for a period
is given byστ ∗ ∆e−rδτ ∗pτ ∗  π
π + (1 − στ ∗ +∆ ∆)(1 − e−r(δτ ∗ −2∆) )ρ
p − c + (1 − r∆) (1 − στ ∗ ∆)pτ
− xτ ∗ c .
p
r
r
(17)

Now suppose that τ ∗ = τ P > 0. Since lim∆→0 στ ∗ < ∞ by Lemma 9 and lim∆→0 στ ∗ +∆ =στ ∗ by the continuity of σt , (17) is reduced to(1 − r∆)Πτ ∗ − στ ∗ ∆Λ − pτ ∗2re−rδτ ∗ ρ∆ − (1 − pτ ∗ )λc∆,r

whereΛ := Πτ ∗ + (1 − pτ ∗ − qτ ∗ )c − e−rδτ ∗30pτ ∗  π
p −c .
pr(18)

From (16) and (18), the firm is indifferent ifστ ∗ Λ + pτ ∗ π + (1 + e−rδτ ∗ )ρ + λc = (r + λ)c.

(19)

Given this, we next show that Λ > 0 if τ ∗ = τ P . Note that πNP∗ pτ ∗∗ pτ ∗∗ pτ ∗
ΠP , e−r(τ −τ )ΠNP .
Πτ ∗ = erτ
p − c = erτ
p0
p
r
p0

It follows from these thatΛ = erτ∗NP∗pτ ∗  πpτ ∗ P(Π − ΠNP ) + (1 − pτ ∗ − qτ ∗ )c − (e−rδτ ∗ − e−r(τ −τ ) )p −c .
p0
pr

Since ΠP > ΠNP when pioneering entry occurs, we obtain(1 − pτ ∗ − qτ ∗ )c > (e−rδτ ∗ − e−r(τNP −τ ∗ )∗)pτ ∗  πp − c ⇒ Λ > 0.
pr

Given that qτ ∗ = (1 − pτ ∗ )e−λτ and δt = τ NP − 2t, this condition can be written as
 π

NP
∗
∗ pτ ∗∗(1 − pτ ∗ )(1 − e−λτ )c > e−r(τ −2τ ) (1 − e−rτ )p −c .
pr
To show that (20) holds, by the definition of p, we have πpτ ∗  π(1 − p)λc = r p − c ⇔ (1 − pτ ∗ )e−λδτ ∗ λc = r
p −c .
rpr

Observe that for t ≤ τ ∗ < δτ ∗ ,(1 − pτ ∗ )e−λt λc > (1 − pτ ∗ )e−λδτ ∗ λc,andr
We hence obtainZ τpτ ∗  πpτ ∗  πNP∗
p − c > e−r(τ −2τ ) r
p − c e−rt .
p
r
p
r
−λt

(1 − pτ ∗ )e

λcdt >Z τe−r(τNP −2τ ∗ )00which can be written as∗(1 − pτ ∗ )(1 − e−λτ )c > e−r(τNP −2τ ∗ )This proves Λ > 0.

rpτ ∗  πp − c e−rt dt,pr∗(1 − e−rτ )pτ ∗  π
p −c .
pr

Now suppose that στ ∗ = 0. Then, (19) holds ifpτ ∗ π + (1 + e−rδτ ∗ )ρ + λc = (r + λ)c.
31

(20)

If this is the case, since σt is continuous, there exist ε > 0 such that for t ∈ (τ, τ + ε), Λt > 0
and σt is so close to zero thatpt π + (1 + e−rδt )ρ + λc > (r + λ)c.

We then haveσt Λ + pt π + (1 + e−rδt )ρ + λc > (r + λ)c,

meaning that it is strictly better to enter at t than at t + ∆, which is a contradiction.
Therefore, στ ∗ > 0.
Finally, if τ ∗ = τ P ,pτ ∗ π + (1 + e−rδτ ∗ )ρ + λc ≥ (r + λ)c,by Lemma 12. Moreover, since Λ > 0 and στ ∗ > 0, we haveστ ∗ Λ + pτ ∗ π + (1 + e−rδτ ∗ )ρ + λc > (r + λ)c.

This implies τ ∗ 6= τ P by (19) and Lemma 11 further implies τ P > τ ∗ .
Proof of Proposition 6. By Lemma 10,
e−rτ

∗


p0  π + (1 − e−rδτ ∗ )ρpτ ∗− c > ΠNP .
pτ ∗r

By Lemma 11, τ ∗ = τ P if τ P = 0. The equilibrium payoff in this case isπ + (1 − lim∆→0 σ0 ∆)(1 − e−rτ
p0
r

NP

)ρ

π + (1 − e−rτ− c ≤ Π = p0
r
P

NP

)ρ

− c.

If τ P > 0, on the other hand, it is clear that the equilibrium payoff is strictly lower than ΠP .
Proof of Proposition 7. (i) Note that τ NP must solve
p0 (π − rc) = (1 − p0 )(λ + r)ce−λτNP.

The only difference from (8) is that the left-hand side is p0 (π − rc) instead of p0 (π − ρ − rc),
which implies that T ∗∗ > τ NP .
∗∗

NP

∗∗

(ii) Since T2 > τ 2 ≥ τ P , τ1∗∗ > τ P holds if τ1∗∗ ≥ T2 . This means that we can focus on∗∗

τ1∗∗ < T2 . In this case, τ1∗∗ must solve (10), which implies that∗∗e−λτ1 <p0 (π + ρ − rc).
(1 − p0 )(λ + r)c32

On the other hand, if τ P > 0, it must satisfy
P

p0 (π + ρ − rc) = (1 − p0 )(λ + r)ce−λτ − p0 ρe−r(τNP −2τ P ).

As this implies thatPe−λτ >p0 (π + ρ − rc).
(1 − p0 )(λ + r)c

we have τ1∗∗ > τ P .
(iii) Plugging τ1 = τ NP into (11), the right-hand side becomes
π − rc  −λτ NP
p0ce.
(1 − p0 ) (λ + r) + (2λ + r)
1 − p0 (λ + r)cp0 (π−rc)λ+r< 2λ+rIf (1−p, the right-hand side is smaller than 2(1−p0 )(λ+r)ce−λτ0 )(λ+r)cNP= 2p0 (π−rc),

and thus
2p0 (π − rc) > (1 − p0 ) (λ + r) + (2λ + r)e−λτ1 ce−λτ1 ,for all τ1 ≥ τ NP . Therefore, τ1∗∗ < τ NP .

p0 (π−rc)λ+r> 2λ+rWe next show that τ1∗∗ can be larger than τ NP if (1−p. Suppose that
0 )(λ+r)cρ = 0. Then T ∗∗ = τ NP , so the right-hand side of (10) is larger than the left-hand side forp0 (π−rc)λ+r,> 2λ+rτ1 ∈ [0, T2 ). Given that (1−p0 )(λ+r)c∗∗∗∗
2p0 (π − rc) < (1 − p0 ) (λ + r) + (2λ + r)e−λτ1 ce−λτ1 ,for τ1∗∗ ∈ [ T2 , τ NP ]. Therefore, τ1∗∗ > τ NP . The same argument applies as long as ρ is closeenough to 0.

33

ΠPΠ̂0 (t)
ΠNP
0

τ NP
2

τP

τ NP

t

Figure 1: The emergence of a market pioneer(λ = 0.1, r = 0.1, p0 = 0.3, c = 3, π = 0.5, ρ = 0.5)

ΠNPΠPΠ̂0 (t)
0

τP

τ NP
2

τ NP

Figure 2: No market pioneer(λ = 0.1, r = 0.1, p0 = 0.3, c = 3, π = 0.6, ρ = 0.4)34t