ISSN (Print) 0473-453X

Discussion Paper No. 1132

ISSN (Online) 2435-0982

PIONEER, EARLY FOLLOWER
OR LATE ENTRANT:ENTRY DYNAMICS WITH
LEARNING AND MARKET COMPETITION

Chia-Hui Chen
Junichiro Ishida
Arijit Mukherjee

April 2021

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

Pioneer, Early Follower or Late Entrant: Entry Dynamics with
Learning and Market Competition∗
CHIA-HUI CHEN
Kyoto University

JUNICHIRO ISHIDA
Osaka University

ARIJIT MUKHERJEE
University of Nottingham
April 22, 2021

Abstract. Timing of market entry is one of the most important strategic decisions a firmmust make, but its decision process becomes convoluted with information and payoff
spillovers. The threat of competition pushes firms to enter earlier to preempt their rivals
while the possibility of learning make them cautiously wait for others to take action.
This combination amounts to a new class of timing games where first-mover advantage
first emerges as in preemption games but second-mover advantage later prevails as inwars of attrition. Our model identifies under what conditions a firm becomes a pioneer,
early follower or late entrant and shows that the timing of entry is excessively early(late) when there emerges a late entrant (early follower). We also argue that consumer
inertia is often efficiency-enhancing in this environment, highlighting an elusive link
between static market competition and dynamic entry competition.
JEL Classification Number: D82, D83, L13
Keywords: market entry, market competition, private learning, signaling, preemption, consumer
inertia.
∗This is a substantially revised version of the manuscript previously circulated under the title “An Entry Game with

Learning and Market Competition.” We thank Makoto Hanazono, Hideshi Itoh, Hülya Eraslan, Kim-Sau Chung and seminar and conference participants at Korea University, Kyoto University, Hitotsubashi University, Academia Sinica, Hong
Kong Baptist University, Chinese University of Hong Kong, 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 Numbers JP17K13699 and JP16H03615. The second author acknowledges financial
support from JSPS KAKENHI Grant Numbers JP20H05631, JP19H01468, JP18H03640, and JP16H03615 as well as the
program of the Joint Usage/Research Center for Behavioral Economics at ISER, Osaka University. All errors are our own.

1.

Introduction

It is often emphasized, mainly in the fields of strategic management and marketing, thattiming of market entry is one of the most critical strategic decisions a firm must makewhenever there is a new (geographical) market, a new product, or a new technology becoming available.1 Market entry strategies are in fact highly complicated, as the benefit ofmarket entry depends not only on the profitability of the market, which is often uncertain,
but also on potential responses of other rival firms. Should a firm take the initiative inopening up a market and be a “pioneer,” or more cautiously wait for others to take action?
In case there emerges a pioneer, should a firm follow immediately or take some time tosee how the market develops over time? Understanding this strategic decision process isof first-order importance, not only for potential entrants but also for policymakers, as itleads to immense welfare and policy implications: valuable resources are wasted if firms
are rushed to enter a failed market while potential gains must be sacrificed if they wait toolong to enter a successful one.
In a nutshell, the optimal timing of market entry comes down to the tradeoff between
becoming a leader and a follower. On this tradeoff, Lieberman and Montgomery (1988)—
one of the seminal articles on first-mover advantage—descriptively raise two strategic considerations as crucial forces in shaping market entry outcomes. On one hand, in the presence of market competition, there arises a benefit of preemption, which urges potential
entrants to enter the market before their rivals do in order to seize market power (p.4447). On the other hand, there is also a benefit of learning from rivals when there is uncertainty over potential benefits of market entry, as they argue “Late movers can gain an edge
through resolution of market or technological uncertainty” (p.47). These two considerations generate counteracting incentives and a dynamic tradeoff of our focus.
In this paper, we build on this broad yet somewhat informal insight and develop a stylized model of market entry which sheds light on determinants and welfare consequences ofentry dynamics in a tractable manner; main applications of our model include new productmarkets, technology adoption and foreign direct investment among others. We consider
an environment with two potential entrants, each of which independently decides whether
and when to enter a new market. The profitability of the market is determined by the market condition (e.g., market size, production cost, quality of labor force) which is not known
to anyone initially. Each firm thus privately investigates whether the market is profitableenough over time and enters when it becomes sufficiently confident about the market.
1

For instance, Lilien and Yoon (1990) note that “the choice of market-entry time is one of the major reasons for new

product success or failure.”

1

The strategic nature of our model is determined by two spillover effects of market entry
which stem from market competition and private learning, as summarized below.
• Payoff spillovers from market competition. The payoff from entry (in case the market
turns out to be good) is decreasing in the number of firms in the market. A firm’s
entry thus reduces the residual demand and makes the rival firm’s subsequent entry
less profitable. The payoff spillover generates a strong incentive to be the first moverto preempt the rival firm.
• Information spillovers from private learning. Since each firm privately collects information about the market condition, a firm’s entry serves as a signal of the firm’s
confidence in the market. The information spillover generates a strong incentive tobe the second mover to learn from the rival firm’s action.
These forces generate a tradeoff where the first mover can potentially capture larger
monopoly rent by entering early, but loses information that it could have obtained fromits rival, thereby capturing both first-mover advantage and second-mover advantage in a
unified framework. The overall benefit of becoming a market pioneer is essentially determined by the way this dynamic tradeoff resolves.
To see the workings of our model, it is important to observe that the signaling effectof market entry is generally weaker at early stages of the game when the structure of
information is sufficiently symmetric. As time passes, however, it eventually reaches apoint 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 effect, thegame is divided into two distinct phases: the preemption phase where the payoff spilloveris the dominant concern; and the waiting phase where each firm has an incentive to wait
for the other to make a move. Our framework provides a new class of timing games inwhich first-mover advantage first emerges as in preemption games while second-moveradvantage later prevails as in wars of attrition.
1.1.

Main results and contributions

As noted above, the timing and order of market entry are a topic of utmost concern inthe strategic management and marketing literature.2 This paper aims to contribute tothis voluminous literature and offer some important implications, both positive and normative, by generating a range of (observationally distinguishable) entry patterns. Broadly
2

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-of-entry effect.

2

speaking, the main contribution of our analysis is that it incorporates both private learning(and hence signaling via market entry) and market competition in an analytically tractablemanner. Despite the fact that these two factors are often regarded as the primary sourcesof early-mover and late-mover advantages (Lieberman and Montgomery, 1988; Mitchell
et al., 1994; Luo, 1998), they have been investigated rather independently and separatelyfrom one another in the existing theoretical literature: there are many related works which
incorporate either one of these elements in various ways but very few, if any, which combine them in a unified framework as we do here (see the literature review in section 1.2
for more on this point). More specifically, there are three findings which we would like tohighlight here.
First, we provide a novel analytical framework—a hybrid of two widely used timing
games—which allows us to pin down the temporal distribution of market entry times fora given set of parameters, thereby clarifying when and under what conditions a firm becomes a pioneer, early follower or late entrant. Although we base our analysis on a fairly
simple setup, the interaction of information and payoff externalities makes the problem
very difficult to handle technically, because the firms’ beliefs evolve in a rather complicated manner depending on their entry strategies. We provide an analytical approach toovercome this problem, while preserving the substance of the problem, which allows us toobtain a necessary and sufficient condition for the first-mover advantage to dominate inequilibrium. 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 amount of
information revealed by a market entry becomes too much. Market entry then ceases tooccur past this point, with neither firm taking any action, as the net value of entry becomes
strictly negative. After a while, though, a firm that has accumulated more favorable information becomes confident enough and willing to enter the market again, even without
the chance to earn the monopoly rent. Our model thus exhibits rich, on-and-off, dynamics
of market entry where the firms gradually enter the market at early and late stages, witha period of no entry in between.3
Second, we examine the timing of entry in comparison to the cooperative benchmark(where the firms cooperatively choose the timing of entry to maximize the joint profit,3

Our on-and-off dynamics are reminiscent of market frenzies and crashes described by Bulow and Klemperer (1994).

In an auction-like environment in which a seller has K identical units of a good, and K + L buyers each wish to purchasea single unit, they show that there emerge both market frenzies in which a single purchase at a given price causes
many others to offer the same price and market crashes in which it becomes common knowledge that no purchases will
be made for some duration of time. The key driving force of their result is the fact that the willingness-to-pay curvenecessarily becomes flatter for high-valuation buyers.

3thus eliminating the externalities created by private learning and competition) to evaluate
welfare implications of entry dynamics. An important observation is that the timing ofentry can be either too early or too late, depending crucially on the (often unobserved)
timing of first entry. Specifically, the efficiency of entry dynamics is determined by whether
the first entry occurs in the preemption phase. If it does, the timing of entry is excessively
early because: (i) in addition to the preemption motive which pushes the timing of entry
forward, the first entrant fails to take into account the positive information externality;and (ii) the second entrant fails to take into account the negative payoff externality (orthe “business-stealing effect”). If no entry occurs in the preemption phase, on the other
hand, the timing of entry tends to be excessively delayed because each firm has a strong
incentive to wait and see the rival’s action.
This observation is useful only to the extent that the timing of first entry is observable,
but the existence of a market is often not known to outside observers until its first entry
actually materializes. Fortunately, though, the time lag between early and late entrants issuggestive and provides enough information, because the temporal distribution of entry
times of the second mover is shaped entirely by the timing of first entry. A general rule ofthumb is that when market entries are clustered in a short span of time (the case of early
follower), chances are that the market was already ripe when the first entrant arrived,
implying that the game has reached the waiting phase. In contrast, when entries arespaced apart in time (the case of late entrant), the first entrant was likely to be a true
pioneer who entered when the market was still filled with uncertainty, or in the preemption
phase. Drawing on this result, we argue, somewhat paradoxically, that the timing of entry
tends to be too early when there is a late entrant while it tends to be too late when there
is an early follower.
Third, we assume for most of our analysis that there is no market friction and thefirst-mover advantage dissipates immediately after the arrival of the follower. In reality,
however, we often find instances where first-mover advantage persists over time, especiallyonce the first mover has established its presence in the market. This form of consumer inertia can indeed arise from various factors such as brand loyalty, habit formation, switchingcosts, and slow diffusion of product information.4 In an extended version of our model,
we incorporate this possibility and examine how it affects entry dynamics. We argue that
consumer inertia, which biases the allocation of surplus in favor of the first mover, is generally efficiency-enhancing in our environment because it raises the benefit of becoming
4

In fact, in the field of strategic management, the impact of the timing of entry on long-run performances has been

the major focus of the literature on first-mover advantage.

4

the first mover and facilitates information sharing between the competing firms. More
broadly, this argument points to an elusive link between static market competition anddynamic entry competition that has received little attention in the literature: market competition on equal footing may be beneficial from the ex post point of view (once all theentry decisions are made), but it may distort the timing of market entry by limiting thebenefits of becoming the first mover that are not fully internalized by potential entrants.
1.2.

Related literature

Our paper is most closely related to canonical preemption games such as Reinganum(1981) and Fudenberg and Tirole (1985).5 They consider an environment where firms
independently determine the timing of technology adoption with payoff spillovers. The
payoff to a firm depends on and is decreasing in the number of firms adopting the technology while the cost of doing so varies over time. Their models are pure preemption games
in which there are no strategic incentives to delay adoption.6 We extend this standard
setting by incorporating state uncertainty and private learning, which create information
externalities and are often considered important for market entry. This gives rise to a countervailing incentive of waiting for the rival’s move (and information). This new elementof private learning qualitatively changes the strategic nature of the problem and provides
a new angle to address the question of when first-mover advantage prevails.
Our model also exhibits a phase which is effectively a war of attrition, and in this sense
related to Chen and Ishida (2021) who analyze a war of attrition, as formulated by Fudenberg and Tirole (1986), with experimentation about the unobserved state of nature via
exponential bandits.7 In their model, each player may receive a signal indicating that theunderlying state is good for sure, in which case it is optimal to stay in the game indefinitely.
A crucial difference is that the state is individual-specific and independent across players,thus eliminating any possibility of learning from others.
Decamps and Mariotti (2004) consider a model of investment with a similar learning
5

See Hopenhayn and Squintani (2016) and Bobtcheff et al. (2017) for some recent examples of preemption games.

In this class of models, the game typically ends once a player takes an action, so that there is no observational learning.
6
In canonical preemption games, the only reason to delay adoption is because the cost of doing so decrease over time;otherwise, there would only be a trivial equilibrium in which all players adopt immediately. The benefit of adoption delayis thus exogenous and non-strategic in this framework.
7
Due to its tractability, the approach to model learning by exponential bandits, pioneered by Keller et al (2005), has
becomes a workhorse specification in the literature and offered many applications such as Strulovici (2010), Bonattiand Hörner (2011), Keller and Rady (2010, 2015), Guo (2016), Che and Hörner (2018), Chen and Ishida (2018), andMargaria (2020), just to name a few.

5

process to ours in a model of strategic investment.8 As in our model, they assume thatthe quality of the project, which is common to both players, is not known ex ante andgradually revealed over time via the arrival of a bad (public) signal. Although they focus
mainly on the case with no market competition, they later extend the analysis to discuss
the case where the follower’s payoff is discounted by some fraction. Their model is one of
public learning where all the information is publicly observed; as a consequence, there isagain no possibility of learning from others.9
Several works examine the optimal timing of entry or exit with information externalities
among players. To name some, Chamley and Gale (1994) consider a model in which there
are N players, a random number n of whom have an investment option. Assuming that thevalue 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 endowed with private information about the true value ofthe option and hence the timing of exercise becomes a signal for other players. Murto and
Välimäki (2011) analyze an exit game with private learning where each player receives asignal in each period which partially reveals his own type. Kirpalani and Madsen (2020)
analyze investment decisions with social learning where each player decides whether andwhen to invest in acquiring information. A common thread of those works is that they
do not consider the element of market competition, or negative payoff externalities, andfocus more on issues such as investment delays and waves.
In clear contrast, models of market entry in industrial organization generally focus onmarket competition but often assume away any dynamic learning.10 For instance, Levinand Peck (2003) analyze a duopoly model of market entry in which each firm privately
8

As we will detail below, we consider “no news is good news” whereby a firm observes a bad signal at some random

time if the market condition is bad. Bloch et al. (2020) also incorporate private learning into a model of market entry but
consider a different learning process where a firm can perfectly identify the true state of nature with some probability
in each period or learn nothing at all (“no news is no news”). See Bloch et al. (2015) for a similar setup but with
firm-specific entry costs.
9
The optimal timing of investment under uncertainty is investigated actively in the real options literature. Althoughmuch of the literature focuses on a single-agent decision problem, there are some attempts to incorporate strategicinteractions 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 public learning with no information asymmetry
among the competing players as in Decamps and Mariotti (2004).
10
Profit uncertainty also plays an eminent role in the context of foreign direct investment. Horstmann and Markusen(1996, 2018) 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 and switches to an owned sales operation if thisoption is found to be profitable. In their models, however, there is only one producer and hence no market competition.

6

observes its entry cost at the outset of the game. The market environment is similar toours in that the first mover can earn monopoly rents until the second mover arrives. Asidefrom the fact that there is no learning, a crucial difference is that the cost uncertainty intheir model is firm-specific and 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 the possibility of learning over time.
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 condition, which is common to both firms, is either good or bad, and each firm can “test thewaters” before it makes an irreversible entry decision. There are two sources of information in this model: on one hand, each firm may privately observe a signal of the market
condition which arrives stochastically over time; on the other hand, the entry decision ofeach firm is publicly observable and hence serves as an additional signal. The fact that afirm can observe the rival’s entry implies a benefit of “waiting,” giving rise to a secondmover advantage. The tradeoff arises, however, as the profitability of each firm depends
negatively on the number of firms in the market, meaning that the first one to enter cantemporarily monopolize the market (until the second 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 lengthof a time interval ∆ shrinks to zero and only state results in the limit without further
notice.11 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 decisions 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).12
11

We consider discrete time purely for technical reasons: in our model, there arises a situation where the best responseof a firm is to enter the market immediately after the rival’s entry, which is not well defined in continuous time.
12
One way to justify this assumption is that the profits are often realized with some time lag. In most cases of ourinterest, what is immediately inferrable at time of entry is not the eventual gain of market entry but the expectation heldby the entrant.

7

Definition 1. A firm is called active if it has entered the market and inactive otherwise.
Both firms start with a common prior that the market condition is good with probability
p0 and gradually acquire information via the arrival of a signal. More precisely, conditionalon the market being bad, an inactive firm privately observes a signal with probability
λdt for an interval of time [t, t + dt).13 Note that a signal arrives only if the market isbad and hence that the arrival of a signal indicates that the market is bad for sure. This
information structure, which is often called “exponential bandits,” is commonly assumed in
the literature on strategic experimentation (Keller et al., 2005; Bonatti and Hörner, 2011;
Che and Hörner, 2018) and is assumed here to highlight the fundamental forces with asmuch clarity. Later in section 6.2, we extend the analysis to incorporate “Poisson bandits”—
a setup in which a signal can come from either state and no one signal is conclusive—andshow that the extended version retains many key properties of this baseline model andyields analogous insights.
Definition 2. An inactive firm is called informed if it has observed a signal and uninformed
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 + m)∆ per period
if it is the only firm in the market and π∆ if both of them are in the market. We call m
the monopoly premium, which could depend on the extent of market competition, and ingeneral assume m > 0. If the market is bad, on the other hand, an active firm invariably
earns zero profit. The net profit for an inactive firm is also normalized at zero, implyingthat there is no incentive to exit from the market ex post even when the market condition
turns out to be bad.
3.

Equilibrium characterization

This section provides an equilibrium characterization of the model described above. Since
the problem for the second mover (after one of the firms has entered) is rather straight13

An active firm may or may not observe a signal, but this is irrelevant because there is no possibility of exit in thecurrent setup.

8

forward, our analytical focus is more on the first mover who must take into account how
the rival firm would react to its market entry. As we will detail below, the game has two
distinct phases, called waiting and preemption, depending on the way the second moverreacts.
3.1.

Belief formation

From the viewpoint of an uninformed firm, there are two unknowns: the market condition(good or bad) and the rival firm’s state of knowledge (informed or uninformed). Since therival firm is by construction uninformed if the market is good, we have three possible statesof the economy as described below:
1. The market is good (state G);
2. The market is bad, and the rival firm is uninformed (state BU);
3. The market is bad, and the rival firm is informed (state BI).
An uninformed firm’s belief is hence defined in two dimensions and denoted as (p t , q t )where: (i) p t is the conditional probability that the market is good (state G); (ii) q t isthe conditional probability that the market is bad and the rival firm is uninformed (state
BU).14 By definition, 1 − p t − q t is the conditional probability that the market is bad andthe rival firm is informed (state BI). All the probabilities are conditional on the firm being
inactive and uninformed.
In what follows, we focus on symmetric perfect Bayesian equilibrium in Markov strategies (hereafter, simply equilibrium) with belief (p t , q t ) as the state variable.15 The use of
belief (p t , q t ) as a state variable is justified in the following way. Note first that once oneof the firms has entered the market, the only relevant part of the belief is p t ,16 and theremaining firm’s strategy should be conditioned only on p t . When both firms are inactive,on the other hand, each firm needs to infer the rival firm’s state. In this case, the rival firm
is uninformed with probability p t + q t , in which case the firm makes its decisions basedon (p t , q t ) which is publicly known. With the remaining probability, the firm is informed(p t = 0), but this case is irrelevant because we know that the firm never enters. As such,as long as the firms adopt Markov strategies, the belief pair (p t , q t ) provides a sufficient
statistic which summarizes all the relevant information of the game.
14
15

The derivation of the belief is shown later in (1) and (2), after we introduce the strategies of the model.
Our model does not exclude the possibility of asymmetric equilibria. We focus on symmetric equilibrium because

asymmetric equilibrium necessarily requires coordination between the firms which is extremely difficult, if not impossible, to attain in many situations of our interest.
16
If the first entry occurs at some t 0 , then q t = 1 − p t for all t > t 0 , so that the belief is summarized by p t .

9

3.2.

Second mover

It is straightforward to derive the optimal strategy for the second mover who, with nostrategic concerns, simply enters if the current belief p t is high enough. Observe that asignal arrives only if the state is bad, and hence by Bayes’ rule, the belief p t increases
monotonically over time as long as the firm observes no signal (i.e., “no news is goodnews”).
Given current belief p, the expected profit of entering now is given byp

π − rcπ−c = p− (1 − p)c,rrwhich is an increasing function of p. Alternatively, the firm may wait until the next instant.17 Since the firm observes no signal with probability p + (1 − p)e−λ∆ ,18 the expected
profit of this waiting strategy is given byh πih π − rcie−r∆ p − [p + (1 − p)e−λ∆ ]c = e−r∆ p− (1 − p)ce−λ∆ .
rr
The cost of adopting this strategy is the foregone profit due to time discounting. On theother hand, by waiting, the firm can collect more information, the benefit of which is
captured by (1 − p)ce−λ∆ . Since a firm is less likely to observe a signal when the current
belief is high, the benefit of waiting is decreasing in the belief.
It follows from these that the firm enters the market now only ifh π − rci
π − rc− (1 − p)c ≥ e−r∆ p− (1 − p)ce−λ∆ .
prrThis condition gives the cutoff belief p which converges top=(r + λ)c,π + λcas ∆ → 0, suggesting that the second mover enters the market once and for all when
the belief p t reaches the threshold p. To focus on relevant cases, we assume that thisthreshold is smaller than one, so that a firm enters if it knows that the market condition isgood almost surely.
Assumption 1. p < 1 ⇔ πr > c.
17

Throughout the analysis, we often say “wait for an instant (or until time t)” to refer to the following entry strategy:(i) the firm does not enter now (until t); (ii) at the next instant, the firm enters if it is still uniformed at the time.
18
For an interval [t, t + δ), an uninformed firm observes a signal with probability 1 − e−λ∆ if the state is bad. The
probability of a signal arriving in [t, t + δ) is hence (1 − p)(1 − e−λ∆ ).

10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 adiscrete jump after the entry. In the limit as ∆ → 0, the updated belief becomes
lim p t+∆ = φ t :=

∆→0

pt
,
pt + qt

which indicates the amount of information revealed by a market entry; throughout theanalysis, we often refer to φ t as the post-entry belief for expositional clarity. Note in particular that since no firm is informed at time 0, i.e., q0 = 1 − p0 and φ0 = p0 , a firm’s
immediate entry reveals no additional information.
If the post-entry belief exceeds p, the firm will follow immediately at the next instant,so that 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(λ+r)ct if and only if the post-entry belief exceeds the threshold, i.e., φ t ≥ p := 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 (p0 , q0 ) the belief at the next instant when no entry will have occurred.
Let σ be the (symmetric Markov) behavior strategy where 1 − e−σ(p,q)∆ is the probability ofmarket entry in a time period, conditional on the firm being uninformed (and both firms
being inactive). In what follows, we often write s = σ(p, q) and s0 = σ(p0 , q0 ) to savenotation.
As mentioned, a 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 1 − e−s∆ ; instate BI, knowing that the market is bad, the rival firm never enters. Moreover, the firm
observes a signal with probability 1 − e−λ∆ if the market is bad (in states BU and BI).19
Finally, the state changes from BU to BI when the rival firm observes a signal, which occurs
also with probability 1 − e−λ∆ .
Given current belief (p, q) and strategy s = σ(p, q), the next-period belief (p0 , q0 ) is19

Note that state BI means that the rival firm is informed. The firm in question is uninformed by construction, because

the problem it faces would be trivial otherwise.

11

inactive and uninformed

state Gsactivesstate BIλstate BU

λλ

informed

Figure 1. State transition. There are three possible states {G, BI, BU} when a firm is inactive and uninformed.
Thick arrows indicate transition by the arrival of information; thin arrows indicate transition by the firm’s
strategic choice.

computed aspe−s∆,(p + qe−λ∆ )e−s∆ + (1 − p − q)e−λ∆qe−(s+2λ)∆0q =,(p + qe−λ∆ )e−s∆ + (1 − p − q)e−λ∆

p0 =

(1)
(2)

with the initial prior given by q0 = 1−p0 . Figure 1, which graphically summarizes the statetransition process, helps illustrating how we obtain these equations: recall that p and q
are the probabilities of state G and BU respectively, both conditional on being inactive
and uninformed (i.e., inside the dotted square). Taking the limit gives the laws of motion
which clarify how the belief evolves over time:ṗ = p [(1 − p)λ − (1 − p − q)s] ,q̇ = −q [(1 + p)λ + (1 − p − q)s] .
The current state of the economy is thoroughly characterized by (p, q). What is particp
ularly crucial is the post-entry belief φ := p+q which essentially determines the amount of
information revealed by an entry. Fortunately, while (p, q) may follow a quite complicatedpath, it is relatively straightforward to compute φ as it is independent of the first-mover12

strategy σ; from (1) and (2), we obtainφ 0 :=

p0
p
=
> φ,
0
0
p +q
p + qe−2λ∆which indicates that φ t monotonically increases over time for any given strategy σ. Thisis an essential technical property of our model which enables us to simply the analysis
substantially while preserving the substance of the issue at hand.
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 withthe case where the current belief (p, q) is such that φ ≥ p, so that an uninformed firm
immediately follows the rival firm. Note that this is the “winner’s curse range” where thefirst mover can monopolize the market only if the market condition is bad. As this is aphase 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 e−s∆ , which allows firm 1 to monopolize the market for a period.
As such, since firm 1 earns (π + e−s∆ m)∆ in the first period after entry and π∆ thereafter,
the expected payoff of entering at t isπ − rcp+ e−s∆ m∆ − (1 − p)c.
(3)r
Now consider an alternative strategy in which firm 1 waits for an instant. In the meanwhile, firm 2 may or may not enter, but the conditional probability that the project is good,from the viewpoint of today, is still p because the expectation of posteriors coincides with0
the prior. Since firm 1 can monopolize the market with probability e−(s+s )∆ , the expected
payoff of this waiting strategy is obtained asih π − rc−(s+s0 )∆−λ∆−r∆m∆ − (1 − p)ce
,
(4)
+e
e
p
r
taking s and s0 as given.20
Comparing (3) and (4), in the limit, it is better to enter now if π
r p − c + (1 − s∆)s0 ∆pm > (1 − p)λc.
r
20

(5)

Note that this is the expected payoff when firm 1 enters at the next instant regardless of whether firm 2 enters now

or not, which may not hold if lim∆→0 s∆ > 0. For instance, if lim∆→0 s∆ = 1 and firm 2 does not enter, firm 1 knowsfor sure that the market is bad, making it optimal never to enter. As we will see below, however, the firms always entersmoothly in the waiting phase, so that this condition is always valid.

13

This condition 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∆)s0 ∆pm for an instant. By waiting until the next instant, the firm may receive additional
information and save (1 − p)λc by avoiding wrong entry; however, a firm also suffers aloss r(p πr − c) since the continuation payoff is realized an instant later.
The following statement establishes that equilibrium in the waiting phase is generally
in mixed strategies. To see this, suppose that a firm adopts a pure strategy and enters at
some t with probability one. In this case, if the firm does not enter at t, it means thatthe firm is informed and hence the market condition is bad for sure. Since this createsa discrete jump in the belief, the best response for the rival firm is to wait until t + ∆ togain this extra information (when ∆ is sufficiently small). Clearly, this does not constitutean equilibrium because the firm also has an incentive to deviate and wait until t + 2∆ to
see what the rival firm does at t + ∆. As such, in equilibrium, the firms gradually enter atsome positive rate to keep the belief constant at p. The proposition further suggests that
this is the unique symmetric continuation equilibrium in this phase.
Lemma 2 (Continuation equilibrium in the waiting phase). For φ t ≥ p > p t (i.e., when
the game is in the waiting phase), there exists a unique symmetric continuation equilibrium
in which:
1. Neither firm enters until the belief p t reaches the threshold p;
2. When p t reaches p, the two firms start entering at a rate to keep p t = p;
3. Once a firm enters, the other firm immediately follows at the next instant.
Proof. See Appendix A.
If p0 > p to begin with, 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 3 in Appendix).
Lemma 2 thus exhausts all the possibilities and completely characterizes the continuation
equilibrium for any φ t ≥ p.
It is important to emphasize that the game in this phase is not a preemption gamewhere each firm has an incentive to deviate slightly earlier than the rival.21 This is a departure from the canonical preemption game such as Fudenberg and Tirole (1985) where21

In a preemption game, if the other firm adopts a pure strategy to enter at some time t, then the best response isto enter slightly earlier at t − ∆. In the waiting phase of our model, on the other hand, the situation is the opposite as
noted above: if the rival firm is to enter at t, the best response is to enter slightly later at t + ∆.

14

there is no strategic benefit of becoming the second mover. To illustrate the key difference,
it is worth emphasizing that the equilibrium identified in Lemma 2 is not a “joint-adoption
equilibrium” of Fudenberg and Tirole (1985): the entry times of the two firms are arbitrarily close but not simultaneous—the type of equilibrium that does not exist in theirframework (or more generally in preemption games).22 In this phase, neither firm in fact
wants to move first, because the firm must bear the risk of being the first mover to earn abenefit that is vanishingly small. The game in this phase thus resembles a war of attrition
where each firm waits for the rival to move first.
Preemption phase: p > φ. The strategic nature of the problem flips if φ starts below thisthreshold p. As entry in this phase is driven by the first-mover advantage and preemption
motives, we call it the preemption phase. Let τ∗ denote the time of the earliest possible
entry in equilibrium, which is defined asτ∗ := inf{t : σ t > 0},
where we assume τ∗ > 0 for now; a sufficient condition for this will be provided laterin section 4.1. For exposition, we say that there is pioneering entry in equilibrium (or an
equilibrium with pioneering entry) if s > 0 for any (p, q) such that p > φ or, equivalently,
p > φτ∗ .23 We call it “pioneering” rather than “preemptive” because, as we will see later,
it generates valuable information to the other firm and is generally socially beneficial,despite the fact that it is driven by preemption motives.
Observe that p > φ means that if a firm enters now, the other firm’s belief jumps up
but is still lower than p. The second mover thus will not enter immediately, and the firstmover can monopolize the market for some duration of time which we denote by δ t . Since
the second mover waits until the belief reaches p, δ t is computed as
δt =

pq t
1
ln
.
λ p t (1 − p)Note that for a given p0 , φ t depends only on t, and so is δ t . The expected payoff ofentering now is then given byp22

π + (1 − e−rδt )m − r c− (1 − p)c.
r

Fudenberg and Tirole (1985) show that the joint-adoption equilibrium exists in continuous time but not in discrete

time. This is because in discrete time, there is always an incentive to deviate and enter slightly earlier, which goes
unpunished for a unit of time if the other firm cannot react immediately, generating a discrepancy between continuoustime and discrete-time equilibria. This argument of course does not apply to our setting where there is no incentive toenter earlier, so that the limit of our discrete-time equilibrium is well defined.
23
Note that our definition of market pioneer differs from the conventional one: in the literature, it simply refers to thefirst entrant in a new market (Robinson and Fornell, 1985).

15

The equilibrium allocation becomes much more complicated when pioneering entry
actually occurs on the equilibrium path with positive probability. We can nonetheless establish that any (symmetric) equilibrium must in general take the following form.
Proposition 1 (Characterization). In any symmetric equilibrium,(a) If there is pioneering entry in equilibrium, there exist τ∗ , τ and τ such that the firms
enter the market gradually (i.e., at some finite rate) in (τ∗ , τ) but with zero probabilityin (τ, τ), where the belief p t reaches p at τ; after τ, the firms again enter gradually as
described in Lemma 2.
(b) If there is no pioneering entry in equilibrium, the firms wait until the belief p t reaches
p, and then enter gradually as described in Lemma 2.
Proof. See Appendix A.
Proposition 1 fully characterizes symmetric equilibrium of this model. If there is no
pioneering entry in equilibrium, i.e., φτ∗ ≥ p, the firms wait until the belief p t reaches p
and start entering at some rate past that point as described in Lemma 2. Otherwise, entry
occurs in two disjoint intervals. The firms start entering at some rate from τ∗ , which isby definition the earliest time of entry, but stops at τ. This is followed by an interval ofno entry (τ, τ) where neither firm takes any action. After a while, though, the belief p t
eventually reaches p at τ, at which point the firms start entering again as in the case withno pioneering entry.
The reason why we have this period of no entry in between pertains to the amount of
information revealed by an entry which increases over time. Although the firms have lessprivate information and face more uncertainty early on, the fact that they have less private
information means that there is less to learn from the rival’s action, thereby making the preemption effect stronger. When this first-mover advantage dominates the cost of entering
prematurely with insufficient information, the firms enter with some positive probability
in the preemption phase. As each firm accumulates more information over time, however,the signaling effect of entry becomes stronger and the game reaches a point where thenet payoff of becoming the first mover is negative. In any equilibrium with pioneering
entry, therefore, there must be an in-between phase where market entry ceases to occur,
generating on-and-off dynamics of market entry.

16

4.

Emergence of a market pioneer

The previous section provides an equilibrium characterization and establishes that there
are two forms of equilibrium, depending on whether pioneering entry occurs on the equilibrium path. Given this, we now explore under what conditions we have pioneering entry
in equilibrium.
4.1.

The constrained problem

To derive a precise condition for pioneering entry to occur in equilibrium, we first consider
a hypothetical situation in which a 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 (unconstrained) 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 any t is givenbyp0(1 − p0 )e−2λtpt =, qt =,
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 canalso be written as a function of t. Let Π̂(t) denote the expected payoff of entering at t,
evaluated at time 0, under the restriction that firm 2 must be the second mover. If firm 1
enters at time t, firm 2 will wait until p t reaches p. We thus obtain
π + (1 − e−rδ t )m − r c−r t−λtΠ̂(t) = ep0
− (1 − p0 )ce.
(7)r
If φ t ≥ p, then δ t = ∆ which converges to zero in the limit. In contrast, if p > φ t , 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 δ t .
Suppose first that there is no pioneering entry and δ t = 0. In this case, the earliest
possible entry occurs when the belief p t reaches p. Define τNP as the time at which p tequals p in the equilibrium with no pioneering entry, which must solveNPe−λτ =

p0 (1 − p)p0 (π − r c)=.
p(1 − p0 ) (1 − p0 )(λ + r)c17

From (6), we obtain
δt =

pq t
1
1 p(1 − p0 )e−2λtln= ln= τNP − 2t,
λ p t (1 − p) λp0 (1 − p)NPmeaning that δ t > 0 if and only if τ2 > t. Let ΠNP := max t∈( τNP ,∞) Π̂(t) denote the2
expected payoff without pioneering entry, which can be written as
Π

NP

NP

= Π̂(τ ) = e

−rτNP

h

i
π − rc
NP p0 λ(π − r c)
−λτNP
− (1 − p0 )ce= e−rτ.
p0
r
r(λ + r)Now suppose that firm 1 enters in the preemption phase. If firm 1 enters at t, firm
2’s belief jumps up to φ t , but firm 2 still needs to wait until the belief reaches p, whichallows firm 1 to monopolize the market for some time δ t . Therefore, the expected payoff
NPof entering at t ∈ [0, τ2 ] is given by
π + (1 − e−rδ t )m − r c−r t−λtΠ̂(t) = ep0
− (1 − p0 )ce.
rNote that this problem is well defined only if τNP > 0. In what follows, we assume that
this is indeed the case.
Assumption 2. τNP > 0 ⇔ p > p0 .
Let τP denote the optimal timing of entry in the preemption phase, which can be foundby maximizing Π̂. The first-order condition is then obtained asµ̂(t) := −p0 (π + m − r c) − p0 me−rδt + (1 − p0 )(λ + r)ce−λt = 0.
NP

Lemma 3. In the limit, there exists a unique τP ∈ [0, τ2 ] which maximizes Π̂(t). The
optimal timing of pioneering entry, denoted by τP , is given by
if 0 ≥ µ̂(0),

0
τP =

τ̂


 τNP
2

NP

if µ̂(0) > 0 > µ̂( τ2 ),NP

if µ̂( τ2 ) ≥ 0,

where τ̂ solves µ̂(τ̂) = 0.
Proof. See Appendix A.

18

Let ΠP := max t∈(0, τNP ) Π̂(t) denote the expected payoff under the restriction that firm
2
2 must be the second mover, which can be written as
π + (1 − e−rδτP )m − r c
P
P
−rτP
−λτP
Π = Π̂(τ ) = ep0− (1 − p0 )ce.
r
As the monopoly premium m becomes larger, it becomes more costly to wait and collect
more information. As a consequence, the optimal timing of pioneering entry moves forward with an increase in the monopoly premium.
Finally, we have thus far assumed, somewhat loosely, that leaning is essential and τ∗ >0. Assumption 2 is obviously necessary but not sufficient for τ∗ > 0. To ensure this, theexpected payoff of entering immediately at time 0 must be smaller than ΠNP , i.e.,ΠNP≥ Π̂(0) ⇔ e−rτNP p0 λ(π − r c)r(λ + r)NP

π + (1 − e−rτ )m − r c> p0− (1 − p0 )c.
r

Note that as p0 → 0, the left-hand side converges to zero while the right-hand side dips
below zero, so that this condition, as well as Assumption 2, holds if p0 is sufficiently small.
Assumption 3. ΠNP > Π̂(0).
4.2.

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 τP 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. As
Proposition 1 indicates, the firms must adopt mixed strategies when they compete to be
the first mover. Even then, these payoffs are still useful as they provide a necessary andsufficient condition for a market pioneer to emerge in equilibrium.
Define Π(t) as the equilibrium payoff of entering at t evaluated at time 0 (note thedifference from Π̂(t) which is the payoff when the other firm never enters first). In equilibrium, Π(t) must be constant for t ∈ (τ∗ , τ) ∪ (τ, ∞). Two initial points, τ∗ and τ, areparticularly crucial. If a firm waits from τ∗ to τ, the other firm enters with some probability. Suppose that the other firm enters at some τ, in which case it is optimal to wait untilτNP − τ. The expected payoff of becoming the second mover, when the other firm entersat τ, is henceh π − rciNPNP
e−r(τ −τ) p0− (1 − p0 )ce−λτ = e rτ ΠNP .
r
19

Given this, the expected payoff of waiting until τ (in the continuous-time limit) is obtained
as


Zτ
Zτ
Π(τ) =

NPτ∗e rτ ΠNP dF (τ) + e−r(τ−τ

)

1−

τ∗

dF (τ) ΠNP ,(8)

where F denotes the unconditional distribution of entry times. In equilibrium, this must
be equal to the expected payoff of becoming the first mover at τ∗ , which can be written as
π + (1 − e−rδτ∗ )m − r c
∗
∗
−rτ∗
−λτ∗
Π(τ ) = Π̂(τ ) = ep0− (1 − p0 )ce.
(9)r
Proposition 2 (Existence). There always exists a symmetric equilibrium. The equilibrium
entails pioneering entry if and only if ΠP > ΠNP .
Proof. See Appendix A.
To see the intuition behind Proposition 2, observe that in the equilibrium with no pioneering entry, each firm waits until time τNP and then gradually enters past that point; theexpected payoff in this equilibrium is hence ΠNP as in the constrained problem. Given this,sufficiency is obvious, because if ΠP > ΠNP , a firm must have an incentive to deviate andenter with some positive probability in the preemption phase. On the other hand, necessity
comes from the fact that the entry competition can only lower the benefit of pioneering
entry while it raises 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 the preemption phase in theconstrained problem, then there is certainly no incentive to do so in the original problem.
Proposition 2 implies that as ΠP − ΠNP becomes larger, the first-mover advantage becomes more salient, rendering pioneering entry more likely. The following proposition
clarifies under what conditions a market pioneer is more likely to emerge, which offers
crucial welfare and policy implications.
Proposition 3 (Determinants of pioneering entry). There exist m̂ and p̂ ∈ (0, p) such that
there is an equilibrium with pioneering entry if and only if m > m̂ or p > p0 > p̂.
Proof. See Appendix A.
Two factors are particularly crucial as determinants of entry dynamics. First, it is clearthat the monopoly premium m, which measures the extent of market competition, has a20ΠPΠ̂0 (t)
ΠNP
0

τNP
2

τP

τNP

t

Figure 1: The emergence of a market pioneer

Figure 2. The emergencemarketpioneer(λ c==0.1,
0.1,= 0.3, c = 3, π = 0.5, m = 0.5)
(λof
= a0.1,r = 0.1,
p0 = 0.3,
3, πr==0.5,
ρp=0 0.5)decisive impact on the timing of entry. Since an increase in m only raises the value of
preemption, it generally favors pioneering entry. See figures 2 and 3 which depict Π̂(t)for different values ofNPπ and m (with π + m fixed). Second, the prior belief p0 , whichΠP uncertainty faced by the firms, also plays an important role inmeasures the extent Πofshaping entry dynamics. Although the effectΠ̂0 (t) of p0 is less clear, as an increase in p0 canPNPraise both Π and Π 0 , a high p0 tends to favor pioneering entry. To see why, observethat the expected payoff upon success is larger for the first mover; a firm can thus enterwith more confidence earlier while revealing less information. From these findings, we
can conclude that pioneering entry is more likely when: (i) market competition is intense;
and/or (ii) there is less uncertainty regarding the eventual likelihood of success.
4.3.

τP

τNP
2

Pioneer, early follower and late entrant

τNPt

Figure 2: No market pioneer

Since the seminal work of Lieberman and Montgomery (1988), there is now a voluminous
39 and strategic management, which examines
literature, mainly in the fields of marketingthe effects of the timing and order of market entry. The literature often classifies timing ofentry into three broad categories: pioneer, early follower, and late entrant (Robinson andFornell, 1985, Lambkin, 1988).
Our analysis allows us to pin down the temporal distribution of entry times for a given
set of parameters and hence offers some insight for this classification. To see this moreclearly, note that each realized equilibrium allocation is characterized by a pair of entry
times (t 1 , t 2 ) where t 1 (t 2 ) denotes the timing of first (second) entry (t i = ∞ in the case
21

τNP
2

τP

τNP

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

τNP
2

τNP

t

Figure 2: No market pioneer

Figure 3. No market pioneer (λ = 0.1, r = 0.1, p0 = 0.3, c = 3, π = 0.6, m = 0.4)
39

of no entry). Our model then generates four classes of entry dynamics that are observationally distinguishable.
1. No entry (t 1 = t 2 = ∞): Neither firm chooses to enter, and the market nevermaterializes.
2. Only one entry (t 1 < t 2 = ∞): Only one firm enters while the other firm chooses
not to follow. This is the case of premature entry.
3. Late entrant (t 1 < t 2 < ∞): 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 (t 1 ≈ t 2 < ∞): A firm enters in the waiting phase and is immediately
followed by the other firm. The two entries are clustered together in time.
The first two cases occur only when the market condition is bad, reflecting the obvious
fact that no successful market can be monopolized forever. The latter two cases admit
two entrants and are often the focus of attention in the existing literature. Note that we
observe a late entrant only if there is pioneering entry. This fact suggests that even though
the timing of first entry is often not observable, the time lag between entries can tell us a
lot about efficiency properties of an observed entry pattern, which we will discuss next.

22

5.

Welfare implications

5.1.

Equilibrium payoff bounds

Let Π∗ denote the expected equilibrium payoff for each firm where
π + (1 − e−rδτ∗ )m − r c
∗
−rτ∗
−λτ∗
Π =e
p0
− (1 − p0 )ce.
r
If ΠNP > ΠP , the earliest possible entry occurs at time τNP , and Π∗ = ΠNP as we have seen.
If ΠP > ΠNP , on the other hand, there is a clear gain from becoming the first mover. We
in general have Π∗ < ΠP (if ΠNP < ΠP ) because the entry competition is self-defeating andshifts the timing of entry forward, inducing the firms to start entering before the optimal
time τP . The question is then whether this competition drives the value of the first-moveradvantage down to zero, i.e., Π∗ → ΠNP . As it turns out, this is not the case because thesecond mover can benefit from the information revealed by the first mover’s entry. The
following result characterizes the equilibrium payoff bounds when ΠP > ΠNP .
Proposition 4 (Equilibrium payoff bounds). Suppose that ΠP > ΠNP so that pioneering
entry occurs with positive probability. Then, each firm’s expected payoff is between ΠNP andΠP , i.e.,
π + (1 − e−rδτ∗ )m − r c
∗
−rτ∗
−λτ∗
Π =e
p0
− (1 − p0 )ce∈ (ΠNP , ΠP ).
r
Proof. See Appendix A.
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 whichis captured by m. The payoff externality is clearly the source of the entry competition.
This is most clearly seen by supposing m = 0, in which caseh π − rci
−r t−λt
Π̂(t) = ep0
− (1 − p0 )ce,r
for all t ∈ [0, ∞), suggesting that there is no preemption phase.
In contrast, as m increases, the first-mover advantage becomes more salient, giving
each firm an incentive to become a market pioneer. This entry competition forces the firms
to enter earlier than the optimal timing τP , which necessarily lowers the expected payoff
23

of becoming a market pioneer. In equilibrium, this expected payoff must be driven down
to the expected payoff of becoming a follower which is strictly larger than ΠNP because of
the information externality: with the first-mover’s entry providing additional information,
the follower can enter earlier than τNP and hence on average achieve a higher payoff. The
presence of pioneering entry thus accelerates the learning process at the industry level.
5.2.

Efficient timing of entry

To derive efficiency properties of the model, we now consider a social planner who attempts to maximize the joint payoff of the firms. Since the efficient allocation under complete information is rather trivial in this setting,24 here we focus on the situation where thesocial planner is subject to the same informational constraints as the firms. Specifically,
we consider an environment in which the social planner specifies the entry times (τ1 , τ2 )such that firm i can enter the market at time τi if it is uninformed at the time.
Let W (τ1 , τ2 ) denote the joint payoff for a given pair (τ1 , τ2 ). Without loss of generality, we assume τ1 < τ2 .25 The social planner’s problem is defined asmax(τ1 ,τ2 )

W (τ1 , τ2 ),subject to τ1 < τ2 , whereh π + m − rc
h π − m − rcii
W (τ1 , τ2 ) = e−rτ1 p0− (1 − p0 )ce−λτ1 + e−rτ2 p0− (1 − p0 )ce−λ(τ1 +τ2 ) .
rr
Two remarks are in order regarding the two types of externality in this setting. First,ppfirm 2’s belief at τ2 is p +(1−p )e0 −λ(τ1 +τ2 ) , rather than p +(1−p0 )e−λτ2 , because of the positive00
00
information externality of the first entry. Second, the second entry contributes only π − mto the joint profit (while its private gain is π) due to the negative payoff externality, whichcorresponds to what is often referred to as the “business-stealing effect” in standard staticoligopoly models.
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≥
24

π−m
,
r

The problem is clearly trivial when the market condition is known to the social planner: if the market is good, thefirms should enter immediately at time 0; if not, they should never enter.
25
It is clearly without loss to assume τ1 ≤ τ2 . We can also show that it is never optimal to choose τ1 = τ2 in the limit
because W (τ, τ + ∆) > W (τ, τ) for any τ. This is because firm 2 can learn from firm 1’s decision at τ1 while the social
cost of slightly delaying the second entry is negligible as ∆ → 0.

24

in which case the social planner would allow only one firm to enter the market (t 2 = ∞).
Since this case is relatively straightforward, we restrict our attention to the case where it
is socially optimal to have two firms whenever the market condition is good.
Assumption 4. π − m > r c.
Taking derivative of the joint payoff W , the first-order conditions are obtained as


∂W
(10)
= −p0 (π + m − r c)e−rτ1 + (1 − p0 ) (λ + r)e−rτ1 + λe−(λ+r)τ2 ce−λτ1 = 0,
∂ τ1
∂W
= −p0 (π − m − r c)e−rτ2 + (1 − p0 )(λ + r)ce−rτ2 −λ(τ1 +τ2 ) = 0.
(11)
∂ τ2
Let (τ∗∗, τ∗∗) denote the socially optimal timing of entry and T ∗∗ = τ∗∗+ τ∗∗. From (11),12
1
2
we obtain
p0 (π − m − r c)∗∗,e−λT =(1 − p0 )(λ + r)c
which depends only on the primitives of the model. Since τ2 ≥ τ1 , for τ1 ∈ [0, T2 ), (10)can be written as∗∗
p0 (π + m − r c) = (1 − p0 ) (λ + r)e−λτ1 + λe−λT −r(τ2 −τ1 ) c∗∗= (1 − p0 )(λ + r)ce−λτ1 + p0λ(π − m − r c) −r(τ2 −τ1 )
e
.
λ+r

(12)

If there is no τ1 < T2 that can satisfy (12), the social optimal timing is “almost simultaneous” where τ1 = τ and τ2 = τ + ∆. In the limit, the first-order condition in this case isreduced to
1 − p0 
λ + r + (2λ + r)e−λτ ce−λτ ,(13)p0 (π − r c) =2which is independent of m. The following proposition yields some important implications
regarding the timing of entry which we will discuss in depth below.
∗∗

Proposition 5 (Efficient timing of entry).
(a) T ∗∗ > τNP .
(b) τ∗∗> τP if τP > 0.
1(c) τNP > τ∗∗
if p0 is sufficiently small.
1
Proof. See Appendix A.
Parts (a) and (b) of Proposition 5 concern the case with a late entrant and suggest thatthe firms enter the market too early compared to the social optimum.
25

• Part (a) states that the equilibrium timing of second entry is earlier than the socially
optimal timing.26 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 − r c while
its contribution to the joint profit is only π − m − r c due to the business-stealing
effect.27 In the optimal allocation, therefore, the entry threshold is higher and thefirm should wait longer to collect more information.
• Part (b) states that the equilibrium timing of first entry is also earlier than the socially
optimal timing. This is mainly 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 (τ∗∗> τP ), but in equilibrium enters
1even earlier so as to reveal less information to the rival firm and delay its subsequent
entry (τP > τ∗ ).
In contrast, Part (c) of the proposition concerns the case with an early follower. Thiscase emerges when p0 is relatively small, in which case the firms tend to enter too late.
The intuition behind this is relatively clear. Once the game reaches the waiting phase, theclear winner is the one 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 isoften excessively strong, preventing the firms from entering the market at an opportune
time.
Note that the timing of first entry is often not observable to the econometrician who
typically lacks exact knowledge of calender time (or of time 0). Even in this case, we canmake inference about whether pioneering entry occurs or not from the temporal distribution of entry times. Note that we have a late entrant only when there is pioneering
entry; the fact that entries are spaced apart in time hence suggests that the first entry indeed occurs in the preemption phase. Combined with the earlier discussion in section 4.3,
Proposition 5 implies a paradoxical fact which is worth emphasizing: the firms enter too
early when there is a late entrant, and too late when there is an early follower.
26

Given that the first entry occurs at some t, the second entry occurs at τNP − t whereas the socially optimal timing isgiven by T ∗∗ − t.
27
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 staticoligopoly models. Our analysis complements this literature by extending this argument to a dynamic context, showingthat market entry is too early with this same effect.

26

6.

Extensions

6.1.

Benefits of consumer inertia

As we have seen, efficiency properties of the model depend crucially on the monopoly
premium m which measures the extent of the payoff externality. More precisely, the possibility of business stealing generates two important forces which generally induce the firms
to enter too early: first, it gives the first mover an additional strategic incentive to enter
early so as to reveal less information to the other firm; second, it also induces the second
mover to enter early as it fails to internalize the loss to the first mover.
In reality, however, first movers can benefit from establishing their presence early on
due to consumer inertia, which could arise from various factors such as brand loyalty, habit
formation, switching costs, patent protection, and slow diffusion of product information.
It is hence more realistic to assume that the monopoly premium m decays only slowly overtime. To capture this aspect in a simple way, we now suppose that the payoff to the firstmover is π + ηm and to the second mover is π − ηm, instead of both receiving π, whereη ∈ [0, 1] measures the extent of consumer inertia.28
Since detailed analysis of this extended case is out of the scope of this paper, we briefly
describe important forces that are generated by consumer inertia; throughout this subsection, we restrict our attention to the equilibrium with pioneering entry. Since the expected
flow payoff upon successful entry is now π − ηm, the second mover waits until the belief(λ+r)creaches pF := π−ηm+λc , which is larger than p for any η > 0. Define τF such thatFp0 (π − ηm − r c) = (1 − p0 )(λ + r)ce−λτ .
If the other firm enters at τ, the expected payoff is
h π − ηm − r ciλ(π − ηm − r c)FFFΠF := e−r(τ −τ) p0− (1 − p0 )ce−λτ = e−r(τ −τ) p0.
rr(λ + r)Given this, (8) and (9) are modified, respectively, asZτZτΠ(τ) =

NPτ∗e rτ ΠF dF (τ) + e−r(τ−τΠ(τ ) = Π̂(τ ) = e
∗

28

∗

−rτ∗



)

1−

τ∗



dF (τ) ΠNP ,
F∗
π + m − (1 − η)me−r(τ −2τ ) − r c−λτ∗p0− (1 − p0 )ce.
r

Since only the discounted sum of payoffs matters after both firms enter, η constitutes a sufficient statistic for our
0

purpose. For instance, suppose that the payoff for the first mover when the second mover enters at some t 0 is π+me−ξ(t−t )rfor t > t 0 , in which case η = ξ+r. Our baseline model then corresponds to the limit case of this specification whereξ → ∞.

27

Note that with the firms’ strategies fixed, an increase in η (more consumer inertia) unambiguously raises Π(τ∗ ) but lowers Π(τ). On one hand, more consumer inertia benefits
the first mover because: (i) the duopoly payoff increases from π to π + ηm; and (ii) it
reduces the payoff for the second mover and hence delays its arrival. On the other hand,more consumer inertia is clearly detrimental to the second mover. This negative effect on
the second mover implies more intense entry competition and hence brings τ∗ forward.
The overall impact on the equilibrium payoff depends on this tradeoff. We argue, however,that the positive impact on the first mover tends to dominate the negative effect for a rangeof parameters values, meaning that consumer inertia is often efficiency-enhancing. Thiscan be seen from the fact that the positive effect necessarily dominates if the probabilityRτof entry in the preemption phase, τ∗ dF (τ), is sufficiently small. Observe that this is thecase when ΠP is sufficiently close to ΠNP .
This argument suggests that there are possible efficiency gains from consumer inertia.
As discussed, although pioneering entry is socially beneficial, the firms fail to internalize
this benefit. Biasing the allocation of surplus in favor of the first mover alleviates this inefficiency by making pioneering entry more attractive. From a broader perspective, ourargument offers crucial policy implications by highlighting how the extent of market competition shapes entry dynamics. Consider a regulatory authority who has policy tools tomanipulate η in some ways. If the authority is concerned only about ex post static gains,
it may be tempted to reduce consumer inertia (lower η) as much as possible, in order
to intensify market competition among incumbent firms. Although these ex post gains,which are assumed away in our analysis, are certainly important, the extent of market
competition can have huge impacts on the way firms enter a new market or adopt a newtechnology, thereby providing a relevant point of view that has not been discussed muchin the literature.
6.2.

A more general specification: Poisson bandits

Our baseline model assumes a particular information structure, often called exponential
bandits in the literature, where the arrival of one signal fully resolves any underlying
uncertainty. We adopt this approach, pioneered by Keller et al. (2005), because modelinglearning with general jump processes can typically be quite complicated, thereby making
the main messages of the paper somewhat obscure. Clearly, though, this assumption is notvery realistic, nor is it meant to be so: in reality, information rarely arrives in such a starkmanner. It is important to verify the key tradeoff of our focus remains in a more realistic
setting.

28

One way to generalize our learning process is to adopt Poisson bandits, in the spiritof Keller and Rady (2010), where no one signal is conclusive about the underlying state.
Formally, suppose that an inactive firm observes a signal with probability λθ dt for aninterval [t, t + dt) where θ = 1 if the state is good and θ = 0 if bad. Assume λ0 > λ1 > 0,so that a signal can come from either state. For clarity, let λ0 = λ and λ1 = νλ whereν ∈ (0, 1).29 The baseline model corresponds to the case with ν = 0.
In the baseline model, there are only two types—either informed or uninformed—
and the belief is conditional on the firm being uninformed (zero observed signals). In
the Poisson-bandits model, on the other hand, a firm’s type is defined by the number ofsignals it has observed in the past; we say that a firm is of type j if it has observed jsignals. Given this, the belief space is now extended to the infinite dimension and written
j,k
j,k
j,k
j,k
as (p t , q t )∞where p t (q t ) is the probability that the market is good (bad), and the
k=0
P∞ j,kjrival firm has observed k signals, conditional on the firm’s type j.30 Let p t = k=0 p t andP∞ j,kjjjjq t = k=0 q t where we must have p t + q t = 1 for all j, and s t be the entry probability ofj,ktype j. Observe that in the baseline model, (i) p t is relevant and defined only for j = 0;(ii) p0,k = 0 for all k ≥ 1; and (iii) q tk = 0 for all k ≥ 2. As such, the belief space is reducedto (p0,0, q0,0, q0,1) where p0,0
+ q0,0+ q0,1= 1 with p0,0
= p t and q0,0
= qt .
t
t
t
t
t
t
t
t
Although the Poisson-bandits model is obviously very complicated, as we need to keep
track of the infinite-dimension belief, and full characterization of the model is beyond thescope of this paper, there are still some properties which continue to hold in this setup;the details are relegated to Appendix B. First, given ν ∈ (0, 1), the belief jumps downward
when a signal arrives while it gradually improves otherwise. A firm thus becomes morejj+1
pessimistic as it observes more signals where p t > p t holds for all j and t regardlessof the entry strategies. Second, by the same argument as in the baseline model, there isno equilibrium in which a firm enters with strictly positive probability because that would
necessarily create a payoff discontinuity and an incentive to deviate. These observations
imply that: (i) market entry occurs smoothly over time; and (ii) only the most optimistic
type can enter the market at any point in time.
A crucial element of our information structure is that different firms have potentiallydifferent beliefs due to private learning, so that a firm can learn from the action of its rival.
This feature is clearly preserved in the Poisson-bandits model. The impact of this signaling
29

The fact that ν < 1 implies that the underlying information structure is still “no-news-is-good-news,” where thebelief gradually improves as long as the firm observes no signal.
30
Note that in our model, learning is private and each firm does not know how many signals the rival firm has observed.
This is a crucial difference from Keller and Rady (2010) where actions and outcomes are observable, so that all players
hold a common posterior belief.

29jeffect is again captured by the post-entry belief: let φ t be the belief for type j immediately
after observing the rival firm’s entry at t. Even though the belief evolves in a much more
complicated way, the post-entry belief still has a simple, closed-form, representation owing
to the fact that only the most optimistic type can enter at any point:
φ tj =

p0 ν j+J> pt ,
p0 ν j+J + (1 − p0 )e−2(1−ν)λtwhere J is the most optimistic type at t and p0 is the prior belief of the state being good.
Note that the post-entry belief is independent of the entry strategy and monotonicallyincreasing in t as in the baseline model. Moreover, we have φ00 = p0 so that an immediate
entry at time 0 reveals no additional information to the rival firm,31 while the probability
that the rival firm follows immediately, conditional on the state being good, converges to
1 as t gets larger. We can conclude from this that the first-mover advantage is more salientat early stages of the game while the second-mover advantage becomes more dominant atlater stages.
In addition, it is straightforward to verify that there is a sequence of equilibria thatconverge to that of the baseline model as ν → 0. This can be seen from the fact that for
any finite t, limν→0 p tk = 0 for all k ≥ 1, so that the belief space is almost reduced to theone in the baseline model. Our baseline model can thus closely approximate the moregeneral Poisson-bandits variant when ν is relatively small.
7.

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, if any, which combine them in a unified framework. To fill this gap and provide amore comprehensive description of the tradeoff faced by potential market entrants, thispaper constructs a dynamic model of market entry which features these two elements simultaneously. We fully characterize symmetric equilibrium of this game and identify anecessary and sufficient condition for the first-mover advantage to dominate. The condition clarifies when and under what conditions a firm becomes a pioneer, early follower orlate entrant. We also argue that consumer inertia is generally efficiency-enhancing, whichhighlights an elusive link between static market competition and dynamic entry competition and suggests some policy implications.
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31

Note that all firms are of type 0 at the outset of the game.

30

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Appendix A: Proofs
Proof of Lemma 2. To prove the proposition, we first establish the following facts.
Fact 1. 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 mπr −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
is bad, and the belief drops to zero, making it optimal never to enter. Taking this into
account, the expected profit ish π − rcie−r∆ p− qce−λ∆ ,rcwhich is larger than p π−rr − (1 − p)c if ∆ is sufficiently small, a contradiction.

Fact 2. s = 0 if φ ≥ p > p.
Proof. Suppose otherwise, i.e., s > 0. This is the case if
 π − rc
(1 − s∆)s0 ∆pm > (1 − p)λc − r p− (1 − p)c .
r
Note that the right-hand side is strictly positive by definition when p > p, which implies
that lim∆→0 s0 ∆ > 0 and p > p0 . Given this, we can apply the same argument to show thatlim∆→0 s t ∆ > 0 for all future periods. Given p > p, we can take an arbitrarily small " anddefineN" (∆) := max{k ∈ N | " > k∆},N (∆)"
for some ". Then, we must have lim∆→0 Πk=0
(1 − s t+k∆ ∆) = 0. This means that if theother firm does not enter by t + N" (∆)∆, the firm’s belief will drop almost to zero. By thesame argument as in Fact 1, it is strictly better to wait until t + N" (∆)∆, and hence s = 0
which is a contradiction.

Fact 3. For φ ≥ p,
1. s > 0 if p ≥ p;34

2. lim∆→0 s∆ > 0 such that p > lim∆→0 p0 if p > p, where p0 indicates the belief at thenext instant (in case of no entry).
Proof. We first show that s > 0. Suppose that s = 0. Then p0 > p, and the continuation
payoff is weakly lower thanh π − rc
i

−r∆
−λ∆
p
,
e
+ m∆ − (1 − p)cercwhich is in turn lower than p( π−rr + m∆) − (1 − p)c, the expected payoff of entering now,
if p ≥ p.

To prove the second statement, note thath π − rci
(1 − p)λc − r p− (1 − 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 Fact 1, however, a firm, say firm 1, waits for an instant with positive probability.
This implies 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 must hence be low enough to satisfy p > p0 so that firm 1 enters at the next instant
if and only if firm 2 enters now.
Fact 4. For φ ≥ p, s ∈ (0, ∞) such that p0 = p if p = p.
Proof. From Facts 1 and 2, we know that a firm must play a mixed strategy if p ≥ p.
Suppose first that s is small enough and p0 > p = p. Then, s0 ∆ > 0 by Fact 3, which
implies that (5) strictly holds given p = p. This implies, however, that it is strictly better toenter now than at the next instant, which is a contradiction. Now suppose that s is large
enough and p0 < p. Then, s0 = 0 by Fact 2, so thath π − rci0e−s∆ (1 − e−s ∆ )pm = (1 − p)λc − r p− (1 − 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 p0 < p, however, the optimal strategy
at the next instant is to enter if and only if the other firm enters now. Therefore, a firm
can earn a higher payoff by waiting for an instant, which is a contradiction. This provesthat the belief must be kept at p once it reaches this level.

35

These results show that starting from some p t < p, no firm enters until p t reaches p.
Once the belief reaches p, then the firms start randomizing so as to keep p t = p as suggested by Lemmas 3 and 4.
Proof of Proposition 1. We start with part (a) of the proposition. Let s t = σ(p t , q t ). Since
φ t is strictly increasing, it will reach p sooner or later, and the firms enter at some positive
rate as shown in Lemma 2. This means that there must be a period in which s t = 0,followed by a period in which s t ∈ (0, ∞). This means that we just need to show thats t ∈ (0, ∞) for t ∈ (τ∗ , τ).
Suppose first that the firms enter with strictly positive probability, i.e., s t = ∞, at somet ∈ (τ∗ , τ). This means that the expected payoff of entering at t drops discretely, because
the firms may enter at the same time with positive probability. Then, we must also haves t−∆ = ∞, for otherwise it is not optimal to have s t > 0. This process continues until
t = 0, but then it violates the assumption that learning is essential and τ∗ > 0.
Given this, we now show that s t > 0 for all t ∈ (τ∗ , τ). Suppose that there exists aninterval (a, a) such that s t = 0 for t ∈ (a, a) but s t > 0 for t ∈ (a − ", a) ∪ (a, a + ") where" is some positive number. This implies that a firm obtains a higher payoff by entering at
a or at a than at any time in (a, a). Given that s t = 0 for t ∈ (a, a), by entering at t + d t
instead of entering at t, the firm’s expected payoff increases by(r + λ)c − p t π + (1 + e−rδt )m + λc ,which is decreasing in t. This means that the expected payoff is concave in (a, a), andhence we cannot have the payoff maximized at a and a in this interval. Therefore, s t ∈(0, ∞) for t ∈ (τ∗ , τ).
Finally, if φ t = p > p t , it is strictly optimal for a firm to wait until p t reaches p. Therefore, shortly before φ t reaches p, it is still optimal to wait, meaning that φτ < p. From
Lemma 2, the firms wait until the belief p t reaches p at τ so that s t = 0 for t ∈ (τ, τ).
Part (b) of the proposition follows immediately from Lemma 2. Once the belief p t
reaches p, it is optimal to enter even if it is immediately followed by the rival firm. The
firms thus enter at some positive rate to keep p t at p.
Proof of Lemma 3. It is straightforward to verify that µ̂ is strictly decreasing in t forNPt ∈ [0, τ2 ], meaning that there exists at most one τ̂ such that µ̂(τ̂) = 0. In the limit, the36

optimal timing of pioneering entry, denoted by τP , is given by
if 0 ≥ µ̂(0),

0
τP =

τ̂


 τNP
2

NP

if µ̂(0) > 0 > µ̂( τ2 ),NP

if µ̂( τ2 ) ≥ 0,

where τ̂ solves µ̂(τ̂) = 0.
Proof of Proposition 2. We first show that pioneering entry occurs if and only if ΠP > ΠNP .
We then construct an equilibrium to show its existence.
Necessary and sufficient condition: The sufficiency is obvious. If ΠP > ΠNP , there is an
incentive for a firm to enter when p > φ t . Pioneering entry must occur with some probability.
To establish the necessity, suppose that pioneering entry occurs. Then, if a firm entersat τ∗ , the expected payoff ispτ∗

π + (1 − e−rδτ∗ )m − r c− (1 − pτ∗ )c,rwhich equals to the payoff of waiting until any t ∈ (τ∗ , τ) ∪ (τ, ∞). Now suppose that thefirm instead waits until τNP − τ∗ regardless of what the other firm does, in which case theexpected payoff isih π − rc
NP∗
NP∗
− (1 − pτ∗ )ce−λ(τ −τ ) .
e−r(τ −τ ) pτ∗r
Since this is the payoff when a firm does not utilize any information from the other firm,
we must haveh π − rci
π + (1 − e−rδτ∗ )m − r c
NP
∗
NP
∗
− (1 − pτ∗ )c > e−r(τ −τ ) pτ∗− (1 − pτ∗ )ce−λ(τ −τ ) .
pτ∗rr
Note that the right-hand side can be written as
h π − rci
∗ p ∗−r(τNP −τ∗ )−λ(τNP −τ∗ )epτ∗− (1 − pτ∗ )ce= e rτ τ ΠNP .
r
p0
Since
P

Π ≥e

−rτ∗ p0pτ∗
π + (1 − e−rδτ∗ )m − r cpτ∗− (1 − pτ∗ )c ,r
37

it follows that ΠP > ΠNP .
Equilibrium existence: If a firm enters now, the expected payoff isp

π + e−s∆ (1 − e rδt )m − r c− (1 − p)c.
r

If the firm waits for an instant, on the other hand, the other firm may enter with someprobability, in which case it follows at τNP − t if it is uninformed; otherwise, the firm
enters at the next instant if it is uninformed. The expected payoff of this waiting strategy
is given by


−s0 ∆
(1 − e−rδt+∆ )m − r c−r∆−s∆ π + e
−s∆
−λ∆
e
pe
− [1 − p − q(1 − e )]cer
 π − rc+ (1 − e−s∆ )e−rδt p− qce−λδ t .
rAs ∆ → 0, the indifference condition is obtained as π − rc
π + (1 − e−rδt )m − r c
−rδ t
−λδ t
− qc − ep− qce= µ(p, t),
s p
r
r

(14)

where
µ(p, t) := −p(π + m − r c) − pme−rδt + (1 − p)(λ + r)c.
Since
δt =

pq
1
ln
,
λ p(1 − p)this gives us a Markov strategy for any given (p, q).
Observe that at τ∗ , (14) is reduced to∗ 
se rτ Π̂(τ∗ ) − ΠNP = µ̂(τ∗ ).
For any τ∗ < τP , µ̂(τ∗ ) > 0. This means that the firms enter at positive rate only if theleft-hand side is positive. By definition, this is the case for any τ∗ ∈ Γ . For any arbitrarychoice of τ∗ < τP , we can drive a sequence of strategies {s t } and the consequent belief
path {p t }. At τ, we must haveµ(pτ , τ) = 0.

38

To see this, if µ(pτ , τ) < 0, it is strictly better to enter now. If µ(pτ , τ) > 0, on the other
hand, the expected payoff increases for t > τ, so it is strictly better to wait. Therefore, we
can pin down τ and τ as a function of τ∗ .
Note that τ(τP ) = τP by definition. On the other hand, we havelim Π(τ∗ ) = ∗lim Π̂(τ∗ ) < ΠNP .

τ∗ →inf Γτ →inf Γ

Since Π(t) is continuous in t, there must be some τ∗ ∈ Γ such that Π(τ∗ ) = Π(τ(τ∗ )) =Π(τ(τ∗ )).
Proof of Proposition 3. Note that ΠNP and ΠP as functions of p0 are given byh π − rc
i
NP
−rτNP
−λτNP
Π (p0 ) = ep0− (1 − p0 )ce,r π + (1 − e−rδτP )m − r cPPΠP (p0 ) = e−rτ p0− (1 − p0 )ce−λτ .
r

(15)
(16)

It is clear that (16) can only increase with m by the envelope theorem while (15) is independent of it. Therefore, there must be a threshold m̂ such that the condition for pioneering
entry is satisfied if and only if m > m̂.
p0

For the effect of p0 , define p000 := p0 +(1−p00 )e−λ∆ for a given p00 < p where ∆ is assumed to00
be arbitrarily small. Also, we write τNP and τP both as functions of p0 . Since π
NP−rτNP (p0 ) p0
Π (p0 ) = e
p −c ,
pr
we haveNPΠNP(p000 ) 00
p0 NP 0p00 NP 0r∆ 0Π (p0 ),
Π
(p
)
=
e
NP 0
0
p00
e−rτ (p0 ) p00

e−rτ(p000 ) =(17)

for any p > p000 > p00 .
As for ΠP , observe first that by Lemma 3,
NP

lim µ̂(0) = −p0 (π + m − r c) − p0 me−rτ + (1 − p0 )(λ + r)c
p0 ↑p

=−

(λ + r)c(λ + r)c −rτNP π − r c(π + m − r c) −me+
(λ + r)c < 0,π + λcπ + λcπ + λcsuggesting that there exists some threshold p̃ < p such that τP = 0 if and only if p0 ∈ [p̃, p).
p00
Also, define p+ := 00 .
0 −λτP (p )p0 +(1−p0 )e039Observe that p0 < p̃, or alternatively p+ ≥ p000 and τP (p00 ) ≥ ∆ by assumption. We thenhave−r(τNP (p00 )−2τP (p00 ))
π
+
(1
−
e
)m
−
r
c
P 0
P 0
− (1 − p00 )ce−λτ (p0 )ΠP (p00 ) = e−rτ (p0 ) p00r
0 −r(τNP (p00 )−2τP (p00 ))P(p0 ) p
π
+
(1
−
e
)m
−
r
c
0
0
= e−rτ
− (1 − p+ )c .
p+p+r
Similarly,
P 00
Π (p000 ) = e−rτ (p0 )Pπ + (1 − ep000−r[τNP (p000 )−2τP (p000 )]r)m − r c

P 00− (1 − p000 )ce−λτ (p0 )





NP 00
P 0
p00 + π + (1 − e−r[τ (p0 )−2(τ (p0 )−∆)] )m − r c−r(τP (p00 )−∆) 0+>ep− (1 − p )cp+r>e−r(τP (p00 )−∆)

= e r∆

p000
p00

p000
p+



p

+ π + (1 − e−r[τNP (p00 )−2τP (p00 )]r)m − r c− (1 − p )c+

ΠP (p00 ).

Here, the second line shows the payoff when the firm enters at τP (p00 ) − ∆ at which point
the belief reaches p+ . From the second line to the third, we use the fact that
τNP (p00 ) + 2∆ > τNP (p00 ) + ∆ = τNP (p00 ).
It follows from above that if ΠNP (p00 ) = ΠP (p00 ), then ΠNP (p000 ) < ΠP (p000 ). This suggests that ΠNP and ΠP intersect at most twice for p0 ∈ (0, p]. Note also that p > p̃ andΠNP (p) = ΠP (p). This proves that there is a threshold p̂ ∈ (0, p) such that the condition
for pioneering entry is satisfied if and only if p0 ∈ (p̂, p).
Proof of Proposition 4. In the proof of Proposition 2, we observe that
pτ∗
Since

π + (1 − e−rδτ∗ )m − r c∗ p ∗− (1 − pτ∗ )c > e rτ τ ΠNP .
rp0

1−p0 −λτ∗
1−pτ∗, we obtainpτ∗ = p0 ee−rτ∗ p0pτ∗
π + (1 − e−rδτ∗ )m − r cpτ∗− (1 − pτ∗ )c = Π∗ > ΠNP .
r40

We can also show that the firms start entering earlier than τP , i.e., τP > τ∗ , if ΠP > ΠNP .
To see this, suppose on the contrary that τ∗ ≥ τP . Let (p, q) be the belief at τ∗ − ∆. If afirm, say firm 1, deviates and enters unilaterally at τ∗ − ∆, the expected payoff is∗

π + (1 − e−r(δ +2∆) )m − r cp− (1 − p)c.
r

(18)

where δ∗ := τNP − 2τ∗ . Now suppose that the firm waits for a period and enters at τ∗ (ifit is still uninformed), for which the expected payoff is computed as


−s0 ∆
−rδ∗
π
+
e
(1
−
e
)m
−
r
c
e−r∆ p− (1 − p)ce−λ∆ .
(19)r
By comparing (18) and (19), it is better to deviate if


∗
0
∗
π + [1 − e−r(δ +2∆) ]m − r c
π + e−s ∆ (1 − e−rδ )m − r c
−r∆
−λ∆
p
− (1 − p)c > ep− (1 − p)ce.
rr
As ∆ → 0, this reduces to∗s0 (1 − e−rδ )m∗p> (1 − p)(λ + r)c − pme−rδ − p(π + m − r c).
r
Observe that the right-hand side is proportional to µ̂(t) and takes a non-positive value forNPt ≥ τP if τ2 > τP . This means that firm 1 always has an incentive to deviate by enteringat τ∗ − ∆, a contradiction. This means that τP > τ∗ and hence Π∗ < ΠP , giving the payoff
bounds.
Proof of Proposition 5. (i) Note that τNP must solveNPp0 (π − r c) = (1 − p0 )(λ + r)ce−λτ .
The only difference from (11) is that the left-hand side is p0 (π − r c) instead of p0 (π − m −r c), which implies T ∗∗ > τNP .
NP(ii) Since T2 > τ2 ≥ τP , τ∗∗> τP holds if τ∗∗≥ T2 . This means that we can focus on
1
1
∗∗
τ∗∗
< T2 (with an interior optimum satisfying the second-order condition). In this case,1τ∗∗must solve (12), which implies that
1
∗∗

∗∗

∗∗

e−λτ1 <p0 (π + m − r c).
(1 − p0 )(λ + r)c

On the other hand, if τP > 0, it must satisfyPNP

p0 (π + m − r c) = (1 − p0 )(λ + r)ce−λτ − p0 me−r(τ41−2τP ).

As this implies thatPe−λτ >p0 (π + m − r c),
(1 − p0 )(λ + r)c

we have τ∗∗> τP .
1(iii) Observe that τ∗∗≥ T2 if1∗∗p0 (π + m − r c) < (1 − p0 )(λ + r)ce−λτ1 + p0λ(π − m − r c) −r(T ∗∗ −2τ1 )e,
λ+r

for all τ1 ∈ [0, T2 ). This obviously holds if p0 is sufficiently small, in which case τ1 = τand τ2 = τ + ∆, where τ must solve
1 − p0 
p0 (π − r c) =λ + r + (2λ + r)e−λτ ce−λτ ,2NPas ∆ → 0. Plugging τ1 = τ into this, the right-hand side becomes
1 − p0p0 (π − r c)NP
(λ + r) + (2λ + r)
ce−λτ .
2
(1 − p0 )(λ + r)c∗∗

NPThis is smaller than (1 − p0 )(λ + r)ce−λτ = p0 (π − r c) if p0 is small enough to satisfy
p0 (π−r c)λ+r(1−p0 )(λ+r)c < 2λ+r . We then have
2p0 (π − r c) > (1 − p0 ) (λ + r) + (2λ + r)e−λτ1 ce−λτ1 ,for all τ1 ≥ τNP and hence τ∗∗< τNP .
1
Appendix B: Poisson bandits
In this appendix, we develop a generalized version of the baseline model—the Poissonbandits model—where a signal can come from either state and no one signal is conclusive.
The purpose of this appendix is to provide a heuristic argument to illustrate that this extended version exhibits many important qualitative properties and analogous insights.
B.1.

Belief updating

Suppose that the entry probability of type j is given by s j and the most optimistic type iscurrently J so that p j,k = q j,k = 0 for k < J. Given current belief (p j,k , q j,k )∞and strategy
k=0
j ∞
j,k 0
j,k 0 ∞(s ) j=0 , the next-period belief is (p , q )k=0 if the firm does not observe a signal, where
k

p

j,k 0

q

j,k 0

k−1

e−νλ∆ [p j,k e−(s +νλ)∆ + p j,k−1 e−s ∆ (1 − e−νλ∆ )]
P∞
P∞
=
,
e−νλ∆ `=J p j,` e−s` ∆ + e−λ∆ `=J q j,` e−s` ∆kk−1e−λ∆ [q j,k e−(s +λ)∆ + q j,k−1 e−s ∆ (1 − e−λ∆ )]
P∞
P∞
=
,
e−νλ∆ `=J p j,` e−s` ∆ + e−λ∆ `=J q j,` e−s` ∆
42

0

0

If the firm observes a signal, the next-period belief is (p j+1,k , q j+1,k )∞, wherek=0
p

j+1,k 0

k

k−1

k

k−1

(1 − e−νλ∆ )[p j,k e−(s +νλ)∆ + p j,k−1 (1 − e−(s +νλ)∆ )]
P∞
P∞
=
,
(1 − e−νλ∆ ) `=J p j,` e−s` ∆ + (1 − e−λ∆ ) `=J q j,` e−s` ∆(1 − e−λ∆ )[q j,k e−(s +λ)∆ + q j,k−1 (1 − e−(s +λ)∆ )]P∞P∞.
(1 − e−νλ∆ ) `=J p j,` e−s` ∆ + (1 − e−λ∆ ) `=J q j,` e−s` ∆P∞P∞
Let p j := k=J p j,k and q j := k=J q j,k where p j is the probability of the market beingjgood condition on having observed j signals. The belief p j rises to p+ if the firm observesjno signal and declines to p− if it observes a signal, where
P∞ j,k −sk ∆
−νλ∆
e
p e
j
k=J
P∞
P∞
,
(20)
p+ =
k
e−νλ∆ k=J p j,k e−s ∆ + e−λ∆ k=J q j,k e−sk ∆P∞k(1 − e−νλ∆ ) k=J p j,k e−s ∆
j
P∞
P∞
p− =
,
(21)
(1 − e−νλ∆ ) k=J p j,k e−sk ∆ + (1 − e−λ∆ ) k=J q j,k e−sk ∆0

q j+1,k =In general, as in the baseline model, market entry must occur smoothly over time in any
equilibrium because there would be a payoff discontinuity otherwise, giving an incentive
to deviate. This means that the most optimistic type must be indifferent between entering
and waiting while all other types strictly prefer to wait. As a consequence, (20) and (21)can be written as
J

j
p+ =

e−νλ∆ [p j − p j,J (1 − e−s ∆ )],e−νλ∆ [p j − p j,J (1 − e−sJ ∆ )] + e−λ∆ [q j − q j,J (1 − e−sJ ∆ )]jp− =(1 − e−νλ∆ )[p j − p j,J (1 − e−s ∆ )].
(1 − e−νλ∆ )[p j − p j,J (1 − e−sJ ∆ )] + (1 − e−λ∆ )[q j − q j,J (1 − e−sJ ∆ )]J

In particular, if there is no market entry, they are further reduced tojp+ =e−νλ∆ p j
(1 − e−νλ∆ )p j
j
, p =
.
e−νλ∆ p j + e−λ∆ (1 − p j ) − (1 − e−νλ∆ )p j + (1 − e−λ∆ )(1 − q j )jjObserve that lim∆→0 p+ = p j > lim∆→0 p− , so that there is a downward jump upon observing a signal as in the baseline model.
Let J be the most optimistic type, and a type j firm observes the rival firm’s entry. The
firm’s belief then jumps up to the post-entry belief φ j given by
φj =

p j,J
1
=
,
p j,J + q j,J
1 + L j,J43

q j,J

where L j,J := p j,J is simply the likelihood ratio. Since p j,J−1 = q j,J−1 = 0 by definition, thepost-entry belief evolves according toJe−νλ∆ p j,J e−(s +νλ)∆φ = −νλ∆ j,J −(sJ +νλ)∆ep e+ e−λ∆ q j,J e−(sJ +λ)∆
p j,J
= j,J
> φ j.
p + q j,J e−2(1−ν)λ∆j0

The post-entry belief gradually increases over time independently of the entry strategy andhence behaves in a manner quite similar to that in the baseline model.
Although the dynamics of the whole belief system is quite complicated, the post-entry
belief has a simple, closed-form, representation as in the baseline model. As we have seen
above, the post-entry belief is independent of the entry strategy, and is pinned down by the
likelihood ratio L j,J . Observe that since the number of observed signals follows a Poisson
distribution and is conditionally independent, the conditional probability that a firm has
observed j signals and the other firm has observed k signals at any t isP( j, k | θ , t) =(λθ t) j −λθ t (λθ t)k −λθ t
e
×
e
.
j!
k!

Although this probability does not take into account the possibility of market entry, thisevent happens equally across the two states (as it depends only on j) and hence is exactly
canceled out. The likelihood ratio thus equals to(λt) j(λt)k(1 − p0 ) j! e−λt × k! e−λt
(1 − p0 )e−2(1−ν)λt
q j,k
j,k
=
,
L t = j,k =(νλt) j −νλt(νλt)k −νλtpp0 ν j+k
p0
e
×
e
j!

k!

for any ( j, k), where p0 denotes the prior probability of the state being good, and thepost-entry belief is given by
φ tj =

p0 ν j+J=.
j,J
p0 ν j+J + (1 − p0 )e−2(1−ν)λt
1 + Lt
1

(22)
j

j−1

The monotone likelihood ratio property of the Poisson distribution implies φ t < φ t
all t and j ≥ 1 if ν ∈ (0, 1), which can be easily confirmed from the formula.

for

Given this, it is easy to verify that the post-entry belief is strictly higher than the current
belief p j , i.e.,p j,Jφ = j,J> p j.
j,J
p +q
j

44

Alternatively, this condition can be written as
p j,J > (p j,J + q j,J )p j .
Summing over k = J, J + 1, ..., we obtain
∞
X

p

j,k

k=J

∞
X
=
(p j,k + q j,k )p j ,
k=J

P∞

P∞
which holds because k=J p j,k = p j and k=J (p j,k + q j,k ) = 1 by definition. Suppose on
the contrary that p j ≥ φ j or, equivalently, (p j,J + q j,J )p j ≥ p j,J . Since L j,k > L j,k−1 , we thenhave (p j,k + q j,k )p j > p j,k for all k > J. This implies
∞
X
k=J

p

j,k

∞
X
<
(p j,k + q j,k )p j ,k=Jwhich is a contradiction. This shows that a firm’s entry is always a good signal in equilibrium.
B.2.

Entry dynamics

Suppose that the first mover enters at t. Then, the second mover, which must be of typejjj ≥ J, updates the belief to φ t and enters the market at τ t . As such, each market entryj+1
induces a countably infinite set of deterministic entry times {τJt , τJ+1
, ...} where τ t ≥tjjjτ t ≥ t for each j; let τ t = t if the second mover follows immediately and τ t = ∞ if itwill never enter. Notice the difference from the baseline model where there are only two
types, either j = 0 or j = 1, and τ1t = ∞ for any t. In contrast, in the Poisson-bandits
jmodel, the belief is always bounded away from 0, and hence τ t < ∞ for any finite j.
The fact that market entry must occur smoothly and only the most optimistic type
can enter at any point in time suggests that any equilibrium must have a sequence ofjintervals (T 0 , T 1 , ...), where T j := (τ j , τ ), such that only type j enters at a (weakly)
positive rate for t ∈ T j . In each interval T j , type j is the most optimistic type and bothj−1
firms must be of type k ≥ j. Notice that those intervals are disjoint, i.e., τ < τ j for allj, because different types have different beliefs that are bounded away from each other.
j−1
More precisely, suppose τ = τ j and consider τ j − " and τ j + ". We then haveφτk j −" =

1
1
k
>
φ
,
j +" =
τ
1 + L k, j−1
1 + L k, j

for any arbitrarily small " > 0 and k ≥ j (which follows from the monotone likelihood
ratio property). This means that the waiting time is strictly longer, making it strictly betterfor the first mover to enter at τ j + " than at τ j − ".
45

It is also worth mentioning that the baseline model is a special case where T 0 =0(τ∗ , ∞). Later, we will show that as ν → 0, τ → ∞ and the equilibrium converges
to that of the baseline model.
B.3.

Tradeoff between entering early and lateSince different types have different best responses, the Poisson-bandits model exhibits no
sharp distinction between the preemption phase and the waiting phase as in the baseline model. Even then, the benefit of becoming the first mover is still determined by
the expected waiting time of the second mover although it now varies across types. Letj
I t := { j : τ t = t} be the set of types that would immediately follow upon observing therival firm’s entry. The extent of the first-mover advantage can roughly be measured by thePjprobability that the rival firm follows immediately when the state is good; let Pt := j∈I t p t
be the probability of that happening. This probability is either 0 or 1 in the baseline model,
thereby giving rise to the sharp distinction between the two phases, whereas it can takeany value between 0 and 1 in the Poisson-bandits model.
To quantify the first-mover advantage, it is instructive to examine (22) and see how
the post-entry belief evolves over time. There are three observations we can make. First,
at time 0, all firms are of type 0, and φ00 = p0 , i.e., a market entry at time 0 reveals no
information. Second, the post-entry belief is monotonically increasing in t and converges
to 1 for any j and J. These are the properties that are present in the baseline model andjj−1
continue to hold in the Poisson-bandits model. Third, since φ t < φ t , there is a thresholdtype j t for each t such that j ∈ I t if and only if j ≤ j t .
These properties suggest that the amount of information revealed by a firm’s entry isarbitrarily small at early stages of the game. Since the benefit of market entry is discretely
lower for the second mover, this means that Pt = 0 if t is sufficiently small, giving riseto the first-mover advantage. As time passes, however, the game eventually reaches apoint where a market entry reveals sufficient information and is followed by the rival firmwith sufficiently high probability. This can be seen from the fact that for any j, J and ",jjthere is t such that φ j (t; ν) > 1 − " for all t > t . Moreover, as t gets arbitrarily large,
the distributions of types degenerate towards the means: νλt when the state is good andλt when the state is bad. Given these facts, therefore, Pt must approach 1 as t tends toinfinity, thereby rendering the first-mover advantage dissipate at some point. The Poissonbandits model is hence characterized by the same tradeoff as in the baseline model, where
the first-mover advantage emerges at early stages of the game while the second-moveradvantages prevails at later stages.
46

B.4.

Limit results

Now consider a limit case where ν → 0. Note that for any t and j + J ≥ 1, we havejlimν→0 φ j (t; ν) = 0. This in turn means that τ t → ∞ for any t and j + J ≥ 1. The strategic
effects of all types j ≥ 1 become negligibly small and can be excluded from considerationas ν becomes arbitrarily small. The firm’s optimization problems thus converge to those in
the baseline model, yielding arbitrarily close solutions. Note also that if the state is good, atype 0 firm remains type 0 almost surely for any t, no matter how large it is, and a strictly0
positive measure of them always stay in the game, so that τ → ∞. Our baseline model
thus closely approximates the equilibrium allocation when ν is relatively small.

47