Second Mover Advantage and Market Entry

SECOND MOVER ADVANTAGE AND ENTRY TIMING*

VINH DU TRAN†

DAVID S. SIBLEY‡

SIMON WILKIE§

We describe a model of entry timing assuming that a second mover canbenefit from observing the experience of a first mover. We focus onhow market attractiveness characteristics such as size and cost affectthe time until first entry. The effects depend on whether the number ofparticipants is exogenous or endogenous. In the former case, a moreattractive market leads to earlier entry. In the latter case, it leads tolater entry. Treating the number of firms as an integer, free entry leadsto non-monotone, but testable, effects of market attractiveness onentry timing.

I. INTRODUCTION

THE MARKETING OF A NEW PRODUCT OR ENTRY into a new market is among themost important decisions that a firm has to cope with. Sometimes, being thefirst in a new market endows the firm with a competitive advantage—whichexists in the form of a temporary monopolistic period before other firmsenter. Other times, entry to a new market involves great uncertainty aboutthe new market environment, uncertainty that can only be resolved byactual entry. However, when entry occurs, it may be impossible for the firstentrant to prevent potential rivals from using information generated by itsentry.

As an example, it has been widely considered that Apple has adopted aconscious strategy that exploits second (or later) mover advantage1. Thismay have been due to its embarrassing losses from the Apple Newton, anearly Personal Digital Assistant. Since then, it has delayed entry into keymarkets to learn from the mistakes of earlier entrants. The Apple iPod was

*The authors would like to thank Dale Stahl, Tom Wiseman and Max Stinchcombe forhelpful comments. We thank Parviz Gheblalivand and Matthew Sibley for able researchassistance. Comments by the Editor and three referees were also most helpful.

†Authors’ affiliations: TNK Capital Partners Hanoi, VietNam.e-mail: [email protected]

‡Department of Economics, University of Texas at Austin, Austin, Texas, 78712, U.S.A.e-mail: [email protected]

§Gould School of Law, University of Southern California, 699 Exposition Boulevard,Los Angeles, California, 90089, U.S.A.e-mail: [email protected]

1 Boddie, J. [2005] ‘Behind Apple’s Strategy: Be Second to Market’, Harvard BusinessSchool, Working Knowledge. Hbswk.hbs.edu/archive/4970.html, August 29, 2005.

THE JOURNAL OF INDUSTRIAL ECONOMICS 0022-1821Volume LX September 2012 No. 3

© 2012 Blackwell Publishing Ltd and the Editorial Board of The Journal of Industrial Economics

517

not the first portable music player to market, but now dominates themarket. Similarly, the iPhone was a late entrant into the smartphonemarket, behind Palm, Blackberry and Nokia. However, it now earns overhalf of the profits of the handset industry. Its current entry into cloudcomputing is also described in these terms.2

Dynamic entry with second mover advantage has been addressed in theworks of Mariotti [1992], Kapur [1995], Caplin and Leahy [1998], Hoppe[2000], Smirnov and Wait [2004] and Kopel and Loffler [2007]. Relatedworks also explore second mover advantage in R&D (Reinganum [1985]and Hoppe and Lehmann-Grube [2001]), in investment decisions (Chamleyand Gale [1994]), in war of attrition games (Hendricks, Weiss and Wilson[1988] and Ghemawat and Nalebuff [1990]) and in herding behavior (Choi[1997] and Banerjee [1992]).

Stemming from different forms of second-mover advantage, these papershave two features in common. First, in continuous time they all involvemixed strategy equilibria for the dates at which actions are taken. Second,they all imply entry time distributions that lead stochastically to sociallylate entry. They leave open the question of how firms’ entry timing deci-sions are affected by the size of the second mover advantage and of othermarket characteristics, such as size and production cost. The closest paperto ours is Mariotti [1992]. Assuming a fixed number of firms, Mariotti[1992] establishes that as the number of firms contemplating adoption of anew technology increases, the individual equilibrium probability of adop-tion decreases. He also examines the impact on timing of the size of thesecond mover advantage.

The literature ignores the possibility that the number of firms may beendogenous. This possibility is the focus of the present model. However,there is a tension between assuming second mover advantage and assumingfree entry. The former requires the existence of positive rents for laterentrants, whereas the latter implies zero profits. To model free entry, wemake a distinction between the number of potential entrants, which isexogenous, and the number of actual entrants (participants), which can beendogenous. We introduce a sunk participation cost, which can either bezero or strictly positive. In order to enter this dynamic entry game, firmshave to pay this cost up front. If the sunk participation cost is zero, allpotential entrants will play the entry game (become participants). If thesunk participation cost is strictly positive, the number of participants isendogenously determined by a zero expected profit condition. For exposi-tional purposes, we will describe the model in terms of decisions involvingan initial R&D cost and entry. Certainly other examples will fit our

2 ‘iCloud: Apple’s Late Mover Advantage in Cloud Computing’ www.nqlogic.com/2011/001.

VINH DU TRAN, DAVID S. SIBLEY AND SIMON WILKIE518


framework. For example, each firm might have to do its own marketresearch study before entering the market.

Within this framework, there are four possible extensive forms to con-sider over the pairs (entry decision, R&D cost incurred):

(i). The entry time decision and the R&D cost is incurred simultaneouslyat time zero;

(ii). The entry time decision is made at time zero, but the R&D cost can bedeferred until the time of entry;

(iii). The entry decision can be made at any time and the R&D cost isincurred simultaneously; and

(iv). The entry date can be chosen at any time and R&D done at any timeup to the date of entry.

We emphasize models (i) and (ii) in this paper. Model (iii) is uninterest-ing. Under this timing assumption, firms are free to incur the R&D aftersomeone else has moved first. With free entry, all followers make zeroprofit. But, this requires that the first mover make a negative profit, imply-ing that entry never occurs. Model (iv) is similarly uninteresting, because itdiffers from model (iii) only in allowing firms to incur R&D costs prior toentry by some firm. If nobody enters, the R&D cost is wasted. If somebodydoes enter, free entry leads to zero profits for new entrants who have not yetincurred R&D costs and negative profits for those who have but have notyet entered. Hence, the strategies allowed by model (iv) that are not allowedin model (iii) are weakly dominated and (iv) reduces to (iii). This leavesmodels (i) and (ii) as the only interesting timing assumptions. We focus onmodel (i), but will show below that its implied equilibria are essentially thesame as those in model (ii).

Model (i) is appropriate for cases in which the R&D phase representswork that cannot easily be licensed to or from another firm. In model (ii),we allow for the possibility that a firm might wait until it enters and thenlicense needed technology from an IP holder. Without discounting, thisoffers no advantage over the timing in model (i), but with positive timediscounting, it might be advantageous to adopt the model (ii) timing.

With model (i), we have a two-stage approach as follows:

• Stage 1. Starting with an arbitrary number N of potential entrants, inorder to play a dynamic entry game in the subsequent stage, a firm hasto pay a sunk participation cost, k2 � 0.

• Stage 2. Participating firms play the dynamic entry game. Each of themcontemplates an entry time t. The first entrant incurs an extra cost, k.Once the first entry has occurred, all other participants instantly learnhow to avoid the cost, k, that the first entrant incurred. They enterinstantaneously and compete in an oligopoly stage game with each otherand with the first mover.

SECOND MOVER ADVANTAGE AND ENTRY TIMING 519


In Stage 1, we assume that firms cannot observe which other firms under-take the sunk cost. Therefore, we model Stage 1 as a simultaneous movegame. In Stage 2, we assume that a firm’s decision to actually market aproduct can be observed and learned from by other firms who have paid k2.

We show that if the number of participants is exogenous and equal to N(k2 = 0), then increased attractiveness of the market (large size, lower cost)leads to earlier entry. This is because of the opportunity cost of delay.When the number of participants is endogenous (k2 > 0), this effect isreversed if N is treated as continuous. In this case, increased market attrac-tiveness leads to more entry, competing away any increased rents. Theincreased number of entrants causes each firm to delay slightly, becausethere is an increased likelihood of another firm going first. If N is treated asan integer, the effects of market attractiveness on entry are not monotone,because the opportunity cost effect and the increased entry effect are bothpresent and work in opposite directions.

In Section 2, we describe the technical setup of the model under themodel (i) timing. Section 3 presents the equilibrium analysis of our modelunder that timing assumption. Section 4 analyzes equilibrium behaviorunder the timing assumption in model (ii). Section 5 presents conclusions.

II. THE BASIC SETUP

Consider an industry with N potential participants, N > 0. There is a fixedsunk cost of entry, denoted by k2 ≥ 0. After paying this sunk cost, firms playa game of entry timing, in which each firm contemplates a time t to enter anew market. Time is continuous and bounded from above by T, which isnormalized to 1. For convenience, we ignore discounting. Whoever entersfirst has to pay an additional cost, k, which persists until the game ends.

Assume that the first entrant cannot effectively conceal the lessons thatit has learned, so that others can benefit and do not incur the cost k. Thesecond mover advantage becomes available to other firms instantaneously.

Regarding the post-entry profit, we assume that when n firms enter themarket, n ≤ N, each firm gets a reduced form profit p(n, x), where n is thenumber of competitors and x is a set of demand and cost shifters. p(n, x) isdecreasing with respect to n and limn→�p(n, x) < k. p(n, x) is increasing withrespect to x if we interpret x as a favorable shifter such as market sizeexpansion or improvement of cost-saving production technologies anddecreasing with respect to x if we interpret it otherwise3. We assume that Nand x are such that p(N, x) ≥ k. The profit p(n, x) may come from asymmetric equilibrium outcome of a simple Cournot quantity competition

3 Such a reduced form profit function is consistent with underlying models such as Cournot(homogeneous products) or a differentiated products model with logit choice (see Andersonand DePalma [1992]).



game, a Bertrand game with product differentiation or some other standardoligopoly model. If a firm enters at time t, it receives a post-entry profit ofp(n, x) for each instant of remaining time. Thus, the value of its post-entryprofit is (1 - t)p(n, x), ignoring discounting. Mariotti [1992] assumes thatpayoffs do not depend on the number of firms. Otherwise, his model is verysimilar to our model for the case k2 = 0.

Regarding the second-mover externality, we model a setting in which thesecond-mover externality recurs at every instant, like a fixed cost: If entryoccurs at time t then the leader gets (1 - t)[p(n, x) - k] while a follower gets(1 - t)p(n, x). Thus, the cost of being a leader is (1 - t)k, decreasing overtime. We have examined an alternative setting in which the k is a sunk costand does not persist. The qualitative results of such a model are the same asin the present one.

Now consider the game of entry timing. Each firm has two actions,‘enter’ and ‘wait.’ Once a firm chooses ‘enter,’ it cannot choose again. Theaction ‘wait’ means ‘do not enter unless someone else has entered first.’ Adecision node in this game is a point in time paired with a completedescription of the past activity in the game. At a point in time t ∈ [0, 1],denote a history with no entry being made by h0(t) and a history with atleast one entry by h1(t). A strategy is a function mapping each of thesenodes {t, h(t)}, to a point in the action space A ≡ {‘wait,’ ‘enter’}.

Our analysis relies on the concept of subgame perfect equilibrium (SPE).In a SPE, each agent’s strategy must be a best response to other agents’strategies, starting from every node in the game. Based on the SPE concept,at node t whereh(t) = h1(t), the best response for every player who has notentered yet is to enter immediately. Therefore, a characterization of a purestrategy on equilibrium path(s) can be reduced to a decision about when toenter given h0(t), "t ∈ [0, 1]. Entry time t = 1 indicates that the player waitsuntil the end of the game unless someone enters first and t ∈ [0, 1) indicatesentering at time t, given no entry has been made before. In other words, onthe equilibrium path, a pure strategy can be sufficiently characterized by anentry time t ∈ [0, 1].

III. EQUILIBRIUM ANALYSIS

The fixed sunk cost of entry, k2 ≥ 0, is a critical factor in determining thenumber of actual players in the timing-of-entry game. If k2 = 0, every firmwill join the game (n = N). If k2 > 0, it may not be profitable for every firmto join the game. In the following analysis, we will treat the two casesseparately. We will use the notation n to denote the number of players.Thus, n = N when k2 = 0 and n ≤ N when k2 > 0. In the former case, thenumber of entrants is n = N, which is exogenously determined. In the lattercase, n ≤ N, which is determined endogenously by a zero expected profitcondition described below.



III(i). Market Entry with an Exogenous Number of Entrants

We first consider the benchmark case where k2 = 0. With k2 = 0, becausep(N, x) ≥ k, n = N potential participants will join the game of entry timing.Because the net payoff for the first entrant is always lower than the payofffor a follower, a second mover advantage exists. There is a class of asym-metric pure-strategy SPE which involves one and only one firm’s enteringimmediately at the beginning of the game (the leader), and other firmsfollow instantaneously (the followers). The strategy profile which supportssuch an equilibrium is described as following:

• The leader’s strategy: Enters immediately at t = 0. If not, then at everynode {t, h(t)}, t > 0, chooses ‘enter’ regardless of h(t).

• The follower’s strategy: At every node {t, h(t)}, chooses ‘wait’ ifh(t) = h0(t) and ‘enter’ if h(t) = h1(t).

Given that every other firm plays the follower’s strategy, the leader’s bestresponse is to play the leader’s strategy, since he receives p(n, x) - k if hefollows the leader’s strategy, and only (1 - t)[p(n, x) - k] if he deviates andenters at a later time, and zero otherwise. Given that there exists a firmwhich plays the leader’s strategy, the optimal strategy for other firms is toplay the follower’s strategy. This class of equilibria—which exhibits asym-metries in firms’ optimal strategies and payoffs—is argued by many to bearbitrary and unexplained (Dixit and Shapiro [1986]), ignoring the coordi-nation issues (Farrel and Saloner [1988]), not convincing (Crawford andHaller [1990]) and uninteresting (Smirnov and Wait [2004]).

Since firms are symmetric ex-ante, the more interesting equilibrium isthe symmetric mixed strategy equilibrium. As mentioned earlier, a purestrategy can be sufficiently characterized on the equilibrium path by anentry time t ∈ [0, 1]. Thus, a mixed strategy can be sufficiently character-ized on equilibrium path by a distribution of entry times on [0, 1].Suppose that each firm plays a mixed strategy F(t), where t is the sup-posed time of entry given h0(t), and consider the problem faced by firm i.Let the cumulative distribution of the smallest value of t among theremaining (n - 1) firms be G(t). The relationship between G(t) and F(t)can be characterized as:

(3.1) 1 1 1− ( ) = − ( ) −G t F t n[ ]

which implies

(3.2) F t G t n( ) = − − ( ) −1 11

1[ ]

Before characterizing the mixed equilibrium strategies in detail, we firstclaim that their upper and lower supports are 0 and 1.



Lemma 1. The support of F (t) and of G (t) is the closed interval [0, 1].

Proof. See the Appendix.

From Lemma 1, the payoff associated with playing t for player i can becharacterized as:

(3.3) V t n x s g s ds G t t n x kit

( ) = − + − − −∫ π π( , )( ) ( ) [ ( )]( )[ ( , ) ]1 1 10

where g(t) = G’(t). Maximizing Vi (t) with respect to t gives the followingfirst order condition:

(3.4) g tn x k

k tG t

n x kk t

( ) + ( ) −−( )

( ) = ( ) −−

π π, ,( )1 1

Equation (3.4) is an ordinary first order differential equation. SolvingEquation (3.4) gives:

(3.5) G t C tn x k

k( ) = + −−

1 1( )( , )π

Note that since G(0) = 0, the constant C = -1. Thus, the symmetricequilibrium mixed strategy is:

(3.6) F t tn x k

k n( ) = − −⎡

⎣⎢⎢

⎤

⎦⎥⎥

−−1 1 1( ) .

( , )( )

π

It is easy to verify that 0 ≤ F(t) ≤ 1, "0 ≤ t ≤ 1, F(0) = 0 and F(1) = 1.When each firm plays this mixed strategy F(t), the distribution of theequilibrium entry time for the industry is

(3.7) H t F t tnn x k nk n( ) = − − ( ) = − −

⎡

⎣⎢⎢

⎤

⎦⎥⎥

( )−[ ]−1 1 1 1 1[( ] ( ) .

,( )

π

As noted by Hendricks, Weiss and Wilson [1988], the game describedin this section is called a noisy game of timing, because the payoff to aplayer depends only on when the first entry is made. This reflects theassumption that a player who plans to wait until time t to move does nothave to commit himself to not moving until t. Instead, if he observesentry before t, his optimal reaction is to enter at once, which is independ-ent of what he had planned to do had the other player not moved beforetime t. Thus, a decision to move at time t is actually a decision to movefirst at time t, given that none of the other players have already moved bythat time. Because of this feature of our mixed strategy equilibrium, it is



desirable to verify that the symmetric equilibrium derived above issubgame perfect.

Equation (3.6) implies that, for any t < 1, there is a positive probabilitythat it will be reached in the symmetric mixed equilibrium. It followsimmediately that the symmetric mixed equilibrium characterized above issubgame perfect.

The result below establishes that as the number of players (n) increases,the expected time of first entry also increases. Intuitively, as the number offirms increases, each firm contemplates the increased likelihood of anotherfirm’s being the first entrant and paying the cost k. Hence, it pays to delayentry slightly, so as to increase the odds that this will happen. Thus, F(t,n, k) stochastically dominates F(t, n + 1, k). Also, as n increases each firmdelays slightly, contemplating that it is more likely that some other firmwill enter first and incur the entry cost. When all n firms do this, the dateof first entry becomes stochastically later. This result is in Mariotti [1992].An additional effect present in our model is that because p(n, x) isdecreasing in n, the greater is n, the lower is the opportunity cost ofdelaying entry. Both effects work in the same direction and reinforce eachother.

Proposition 1. Suppose the number of entrants is exogenous (n = N) andthe second mover advantage exists (k > 0). A higher number of firmsimplies a higher unilateral incentive to wait; and thus a more delayedexpected entry industry-wide, all else equal.

Proof.

(3.8)

∂∂

=

−( ) −−

( ) −

( )−−

H t n k xn

t t

k nn x k

n x k nk n

( , , , )

ln ( )

( ),

[ , ]( )1 1

1

1

2

π

π −− −( ) ∂ ( )∂

⎡⎣⎢

⎤⎦⎥

<n nn xn

1 0π ,

.

This result also implies that the effect of an increase in the number ofplayers on expected consumer welfare is ambiguous. Once entry occurs,higher n means a more competitive market. However, it also means delayedentry.

The main contribution of this section involves the effects of marketcharacteristics on the timing of entry. We denote the vector of demand andcost shifters by x. The effect of x on the post-entry equilibrium are wellknown for a variety of oligopoly models, such as Cournot, Bertrand andrelated models, with features such as endogenous sunk costs, imperfectinformation, etc. The effects of x on entry time have been less studied,particularly when entry timing is affected by second mover advantage. Our



next result sheds light on the effect of demand and cost shifters on theindustry’s expected entry time:

Proposition 2. Suppose the number of entrants is exogenous (n = N) andthe second mover advantage exists (k > 0). A bigger market size or a lowerproduction cost post-entry increases the unilateral incentive to enter themarket earlier, which leads to earlier expected entry time industry-wide.

Proof. From Equation (3.6), the partial derivative of H(t, n, k, x) is givenby:

(3.9) ∂∂

= − −( ) −−

∂∂

( )−−H t n k x

xt tk n

n xx

n x kk n( , , , ) ln ( )

( )( , )

,( )1 1

1

1π

π

Because ln(1 - t) < 0, the effect of x on the equilibrium strategy F has theopposite sign of its effect on post-entry profit. Thus, an increase in marketattractiveness or a reduction in cost increases unilateral incentive to enterthe market earlier. Also because n is fixed, from Equation (3.7), ∂

∂Hx

has thesame sign as ∂

∂Fx

. If we interpret x as market attractiveness, then an increasein market attractiveness shifts H up, implying earlier equilibrium entrytime. If x is a cost shifter, the result is reversed. The intuition is that theopportunity cost of delay in entering a more desirable market exceeds thecost of delay in entering a less desirable market. Hence, the more desirablemarket will be entered sooner (on average).

As we will see below, the result in Proposition 2 changes as the numberof entrants changes from being exogenous to being endogenous. Becauseendogenous numbers of entrants arise naturally in many economic situa-tions, we believe that it is striking to observe that market characteristicssuch as size and cost have completely different effects in the two scenarios.

III(ii). Free Entry: the Continuous Case

It is not straightforward to model free entry with second mover advantage.In order to get the comparative statics effects described in Propositions 1 and2, it is necessary that both first and later movers get positive rents. On theother hand, free entry implies that rents are competed away. To illustrate,suppose that a first mover incurs an entry cost k > 0 and second movers incura lower entry cost. Treating N as continuous, free entry leads to zero profitsfor second movers, implying negative profits for the first mover.

This problem arises because the zero profit condition applies to thedecision of whether or not to enter the market. We will deal with this byadding a prior stage in which each of the N potential participants mustincur a sunk cost k2 > 0 in order to be eligible to enter the market. Uponpayment of k2, participating firms play the dynamic entry game describedabove. In this setting, the zero profit constraint applies to the entire two



stages of the game. Therefore, firms that have paid k2 can earn positiverents from the market entry stage of the game.

Clearly, this is a special assumption about timing. However, the frame-work fits many interesting market situations, such as those described in thispaper. In any case, our results extend to the model (ii) timing assumption,as described below.

Suppose now that there is a sunk participation cost k2 > 0. With k2 > 0, itmay not be profitable for every firm to participate. Denote the number ofparticipating firms by n, n ≤ N. Participating firms have to pay the sunkcost k2 up front before they are eligible to enter. Upon paying k2, the nparticipating firms face the same game played by n players described above.Once n is determined, the mixed strategy, from Equation (3.6), is given by:

(3.10) F t n k x tn x k

k n, , ,,

( ) = − −( )⎡⎣⎢

⎤⎦⎥

( )−−( )1 1 1

π

Because t = 0 belongs to the support of this mixed strategy, the expectedprofit for a participating firm is

(3.11) V n k k x n x k ki , , , , .2 2( ) = ( ) − −π

The equilibrium number of entrants, n, is such that

(3.12) V n k k x n x k ki , , , , .2 2 0( ) = ( ) − − =π

Thus, the equilibrium number of firms, n, is a function of k + k2 and x,

(3.13) n n k k x= +( , ).2

It is straightforward that n is decreasing with respect to k + k2 andincreasing with respect to x, if x is interpreted as market size. Given theequilibrium n, the distribution functions F and H can be rewritten as:

(3.14) F t k k x tk

k n k k x, , , ( ) ,,2

11 12

2( ) = − − +( )−[ ]

(3.15) H t k k x tk n k k x

k n k k x, , , ( ) ,,

,2

11 12 2

2( ) = − −+( )

+( )−[ ]

As noted, n is an increasing function of x, if x is market attractiveness.Thus, from Equation (3.6) above, an increase in x leads to a decrease in F.The reverse is true if x refers to production cost. Although given n, x alsoaffects post-entry profits, p(n, x), free entry competes them away, so thatp(n, x) = k + k2, a constant. Hence, x affects equilibrium play only through



its effect on n. Proposition 3 states that the effect of market characteristicsx on the industry’s expected entry time is opposite to the result stated inProposition 2.

Proposition 3. Suppose the number of entrants is endogenous (n ≤ N) andthe second mover advantage exists (k > 0). A bigger market size or a lowerproduction cost post-entry leads to delayed equilibrium entry times.

Proof

(3.16) dH t k k xdx

t t

k n

nx

k nk n, , , ln ( )

( ).

( )2

1

2

1 1

1

2

( ) = −( ) −−

∂∂

⎡⎣⎢

⎤⎦⎥

−

Since ln(1 - t) < 0 and ∂∂ >nx 0, Equation (3.16) implies that dH t k k x

dx( , , , ) .2 0<

The difference between results in Propositions 2 and 3 arises from twocontrasting effects of an improvement in market characteristics, x. Holdingn fixed, a more attractive market raises the opportunity cost of delay,making entry times earlier. This lies behind Proposition 2. On the otherhand, if n is endogenous, entry dissipates the higher profits made possibleby a better market. Hence, there is no opportunity cost effect. Instead, thesole remaining effect of a better x is that rents are dissipated and n is larger.From Proposition 1, the larger the number of firms, the later is entry, whichis the content of Proposition 3.

The effect of sunk participation cost, k2, on expected entry time is unam-biguous. One can easily verify that the higher the sunk participation cost,the smaller the number of entrants, which ultimately leads to an earlierexpected entry time. However, an increase in the sunk cost of participation,by reducing n, has ambiguous effects on economic efficiency. Post-entrycompetition is reduced because the number of firms is reduced, loweringwelfare. However, entry does occur earlier, so that the net effect is unclear.

Our results have clear testable implications. To illustrate, suppose thatone is concerned with the introduction of a new product by n firms acrossa number of cities. Assume that these cities are equally costly to serve, butdifferent in size. To introduce the new product requires a sunk cost. Ourmodel might be used to test hypotheses about which cities are entered firstand which are entered later on.

Propositions 2 and 3 imply that the comparative statics of timing withrespect to demand and cost shifters produce qualitatively different resultsdepending on whether n is exogenous or endogenous. In the former case,the n firms would enter the cities in stochastic order of their market size, thelargest market being entered first. If n is endogenous, however, we getthe reverse prediction. It may be possible, of course, to determine whetherthe number of participants is exogenous or not from the data. If so, then



our theory yields clean predictions. This may be hard to do convincingly.However, even if we cannot test n for endogeneity, there is still an impor-tant sense in which the model is testable. The relationship between the sizeof a market, the number of entrants, and the timing of entry has fourpossibilities. As the size of a market increases, (1) n increases and largermarkets are entered earlier; (2) n increases and smaller markets are enteredearlier; (3) n is unchanged and larger markets are entered earlier; and (4) nis unchanged and smaller markets are entered earlier. Our model withcontinuous N implies, whether n is endogenous or not, possibilities (1) and(4) are inconsistent with a theory of entry timing based on second moveradvantage. Hence, to this extent, our theory is testable empirically, what-ever the endogeneity or exogeneity of the number of participants.

III(iii). Free Entry: the Integer Case

In the case where N is an integer there is a similar sort of modeling problemregarding free entry as that discussed above. Suppose that all secondmovers incur a sunk cost of entry f, but that the first mover incurs a highersunk cost, K > f. Free entry limits a second mover’s rent to f, therebydetermining the equilibrium number of participants, n. Also assumep(n, x) - f < 0 and p(1, x) - f > 0 Assume that firms decide simultaneously(1) whether to incur the sunk cost k2, and (2) when to enter, conditional on(1).This is model (iii), described above. Because n is an integer, this allowsall participants to earn the non-negative rents required by the model as longas K - f is not too large. This seems appealing at first sight, but to avoid theproblems with model (iii) outlined above, we need to assume that allpotential second movers coordinate among themselves to see which n - 1enter and which N - n stay out. This type of coordination is unappealing onintuitive grounds and would also violate antitrust laws. Hence, we willadopt the integer version of the 2 stage game described above.

Thus far, we have shown that the comparative statics of entry timing areof opposite sign, depending on whether the number of participants is fixedexogenously or is a continuous variable determined endogenously by a freeentry condition. In this section we take seriously the fact that n is an integer.If n is an integer, elements of both Propositions 1 and 2 exist. As the marketattractiveness variable, x, rises, n will increase in a stepwise fashion, so thatentry times are reduced by Proposition 1as long as n is locally unaffected.But once n increases by 1, then from Proposition 2, entry times are delayed.Clearly, the comparative statics of entry times with respect to x are notmonotone.

This might appear to imply that the integer version of the model is verydifficult to test empirically. Surprisingly, this is not the case and the mainpurpose of this section is to present a simple empirical test of the integerversion of the model.



To treat n as an endogenous integer means that the free entry conditionEquation (3.12) must be modified. Instead, for n to be the equilibriumnumber of firms with free entry in our two-stage model x must be such that

(3.21) π π πN x n x k k n x, , ( , )( ) ≤ +( ) ≤ + ≤1 2

From this set of inequalities it should be clear that for x such that allthese inequalities are strict, then as x increases, entry times must stochas-tically decrease (Proposition 2). However, once x increases to the point thatp(n + 2, x) ≤ k + k2 ≤ p(n + 1, x), then entry times are stochastically delayed(Proposition 1). To characterize equilibrium play, we first define the regionsof x where n is constant. Because profit increases in x, we can define anincreasing sequence {xn}, where xn is defined by p(xn, n) = k + k2. We definethe semi-open interval In by In ≡ [xn, xn+1).

For x ∈ In, equilibrium behavior is defined by Equation (3.7) as

(3.22) H t x n k k tn

n, , , , ( )2 11 1( ) = − − −θ

where

(3.23) θ π= ( ) −x n kk

,

For simplicity, we will omit explicit mention of k and k2 in H(t,x,n,k,k2),and write H(t,n,x) hereafter. To describe equilibrium behavior in the two-stage model we define the function H*(t,x), where

(3.24)

H t x H t x

H t x H t x

H t x H t

x I

x I

* for

* for

*

, , ,

, , ,

, ,

( ) = ( )( ) = ( )

( ) =

∈∈

2

32

3

�nn x x In,( ) ∈for

Thus, H*(t, x) represents equilibrium behavior when n is an integer butadjusts to x step-wise.

From the definitions of H*(t, x), xn and q, it can be shown that:

(3.25) inf H t x inf H t xx I x In n∈ ∈

( )> ( )+

* *, ,1

That is, as x increases in xn where n is constant, H*(t, x) increases fromProposition 2. But as x moves from xn+1 - e to xn, H*(t, x) shifts downwarddiscontinuously from H*(t, xn+1 - e) to H*(t, xn+1); this follows from Propo-sition 1. Figure 1 shows H*(t), with downward discontinuities at x3, x4, ... ,etc.



To relate the behavior of H*(t, x) to entry timing, define the equilibriumexpected entry time, conditional on x, as given by:

(3.26) t x tdH t x dt H t x dt* * *( ) = ( ) = − ( )∫ ∫, ,0

1

0

11

Hence, as x increases within the interval In, t*(x) decreases. However, whenx passes from In to In+1, H*(t, x) has a downward discontinuity, meaningfrom Eq. 3.26 that t*(x) has an upward discontinuity. Figure 2 shows t*(x),with upward discontinuities at x3, x4, x5. . ., xn-1.

Equations 3.25 and 3.26 give us a sense in which t*(x) is strictly mono-tone with respect to x:

(3.27) sup t x sup t xx I x In n∈ ∈

( ) > ( )−

* *1

.

Nonetheless t*(x) is generally not monotone in x. This very non-monotonicity yields a simple testable implication of our mode. FromProposition 1 and Eq. 3.27, we can make the following statement:

Consider any two observations of x, xi and xj, where xi > xj. Thus, eithert*(xi) < t*(xj) or the number of firms associated with xi is larger than thenumber of firms associated with xj. Any other fact pattern is inconsistentwith the model.

To see how this might work empirically, consider the problem of multi-market entry analyzed in Section 3.2. Order cities by size x. The actual

Figure 1



entry time for each city is, of course, the realization of the random variablet. Therefore, approximate t*(x) by calculating the mean entry time for allcities of size x, t x( ). The test of our model then proceeds as just described,replacing t*(x) by t x( ). For xi > xj, either t x t xi j( ) < ( ) or there are morefirms in cities of size xi than in cities of size xj.

IV. MODEL (II)

So far, we have assumed that each firm chooses its date of entry and incursthe sunk R&D cost both at time zero. In this section, we allow for each firmto delay its R&D spending until its entry date. We also allow for timediscounting. Note that Part A of Lemma 1 works for this timing assump-tion, as well as for (i), and is unchanged by discounting. This means that atthe product introduction stage (whether or not R&D is done at the sametime), each firm mixes over [0,1] when choosing its entry time. In thesequential approach implied by model (i), since immediate entry is in thesupport of each firm’s strategy, F(t), expected profit is given by:

(4.1) π n x k k,( ) = + 2

In the simultaneous move approach implied by model (ii), each firm canstill move at t = 0, so that the zero profit condition is:

(4.2) π n x kk

Iwhere I t e dsrs

t,

( ),( ) = + ( ) = −∫2 1

0

Figure 2



where ‘r’ is the discount rate. Note that with no discounting (r = 0), equa-tions Equation (4.1) and Equation (4.2) are the same. Even with r > 0, it isobvious that the comparative statics of ‘x’ on ‘n’ will be of the same signunder either timing assumption.

The R&D cost, k2, is incurred at the time of entry. With some abuse ofnotation, we will refer to the distribution of entry times for the other n-1firms as G(t) in both cases. Expected profits are given by:

(4.3) V t I s n x g s ds G s I t n x k k eit rt( ) = ( ) ( ) ( ) + − ( )[ ] ( ) ( ) −[ ]−∫ −π π, ,0

21

This gives rise to the following differential equation characterizing G(t)in the simultaneous move case:

(4.4) g tG t n x k

ke I t

rk

ke I t

n x k

ke I trt rt rt( ) + ( ) ( ) − + = ( ) −−

[ , ]

( ) ( )

,

( )

π π2

Note first that with r = 0 (no discounting), equations Equation (4.4) andEquation (3.4) are identical, as are the two zero profit conditions.

Substituting equation Equation (4.2) into Equation (4.4), we get:

(4.5) g tG t k

I ke I trk

k r

I ke I trt rt( ) + ( ) + = −22

2

0

1

0( ) ( )

( )

( ) ( )

Note that any explicit dependence of the solutions to equations Equation(4.4) and Equation (4.5) on x is absent at this point. Because

1 1 1− ( ) = − −G t F tn

n( ) , where F(t) is the strategy of the individual firm, thenthe equilibrium distribution of first entry times is given by:

(4.6) H t G tn

n( ) = − − ( ) −1 1 1[ ]

In equation Equation (4.6), G(t) is the solution to Equation (4.5). Equa-tion Equation (4.6) is obviously true for either model (i) or model (ii). Aswith model (i), the market descriptor, x, enters F(t) only through n/(n-1).From equations Equation (4.1) and Equation (4.2), the effects of x on n arequalitatively the same under either approach. Clearly, the comparativestatics of x on H(t) are the same qualitatively under either timing assump-tion (and are exactly the same if r = 0).

V. CONCLUSION

Based on second mover advantage and endogenous participation, thispaper has analyzed the effects of market attractiveness on entry timing. Themain contributions concern the modeling and effects of free entry. From a



conceptual standpoint, the existence of second mover advantage is intension with the zero profit free entry condition. To motivate initial entry,the first mover must be able to earn non-negative profits, implying thatsubsequent entrants earn rents, due to second mover advantage. But this isnot compatible with a zero profit condition, if that condition applies tofirms that are all market participants. Timing assumptions (i) and (ii) aredifferent ways of resolving this difficulty.

Under both assumptions, we have focused on the comparative statics ofentry timing with respect to market attractiveness. One’s initial intuitionwould probably be that a more attractive market would be entered soonerthan a less attractive market. This intuition is correct if the number ofparticipants is fixed. However, with free entry, the potentially larger rentsin the more attractive market are dissipated through increased entry. Witha larger number of market participants, entry dates are delayed, notspeeded up.

If the number of firms is large, and each firm is small relative to themarket, it may be appropriate to treat the number of potential entrants ascontinuous. In that case, the rent dissipation effect of entry dominates: themore attractive the market, the later will it be entered, reversing the initialintuition.

However, if the number of potential entrants is best treated as an integer,the effects of market timing are not monotone. Over ranges of marketattractiveness that do not change the number of firms, increased attractive-ness leads to earlier entry. However, if an increase in market attractivenessincreases the number of participating firms, then entry occurs later.

Despite the non-monotonicity of this relationship, there is a clear testablerelationship between market attractiveness and entry timing: either therelationship is negative or more attractive markets contain more firms thanless attractive markets. Any other pattern in the data falsifies the model.

APPENDIX

Proof of Lemma 1.

a. The support of any equilibrium set of mixed strategies is [0,1]. To see why, Supposethat the lower support is some t0 > 0 and the upper support is some ti < 1. Becauseit is feasible for any firm to enter at t0, the equilibrium ex-ante profit is given by(1 - t0)[p(n, x) - k]. However, by deviating to place unit mass on t = 0, a firm canraise its ex-ante profit to this amount plus the monopoly profit net of k for theinterval [0, t0). Similarly, if all firms are playing a mixed strategy with an uppersupport ti < 1, then any firm can avoid paying k by deviating to a unit mass onti + e, where e is as small as desired. Hence, the support of any equilibrium set ofmixed strategies is in the unit interval.

b. The support of F(t) does not have any mass point. To see why not, suppose that F(t)has a mass point at tm and that tm is in the interior of the support. Since F(t) has



a mass point at tm, G(t) also has a mass point at tm. Consider player i: there existsa probability Pr(t = tm) ≡ p > 0 that someone else will enter at time tm. For anyvalue of e > o, i is strictly worse off entering at tm than entering at tm - e, becauseif we let Vi(tm) and Vi (tm - e) be the payoffs for Firm i if it exits at tm and at tm - ethen limi→0Vi(tm) - Vi(tm - e) > 0, which contradicts the assumption that tm doesnot belong to the interior of the support, we consider a mass point at zero4. SinceF(t) has a mass point at tm ≡ 0, G(t) also has a mass point at tm ≡ 0. Player i isstrictly better off if he deviates from F(t) by removing the mass point at tm ≡ 0 andredistributing the mass equally to the whole support. By doing so, he increases hispayoff by eliminating the chance that he is the first entrant at tm ≡ 0 and scaling upthe expected payoff at tm > 0. Finally, we will prove that the support is the interval[0, 1] with no gap.

Suppose, otherwise, that there is a gap in [0,1] and that the gap is a closed intervaldenoted by [t0, t1] with t0 > 0 and t1 < 1. Consider a player i: since G(t) is constant on[t0, t1] and since t1 - t0 > 0, i strictly prefers entering at t0 to entering at t1. Givent1 - t0 > 0, we can always find a sufficiently small e such that i strictly prefers t0 - e tot0 + e. This contradicts the assumption that t0 - e and t0 + e belong to the support ofthe mixed strategy F(t) where i is supposedly indifferent between two entry timest ∈ [t0, ti] and ti ∈ [t0, ti]. Note that because the support of F(t) has no mass point, thisproof can be extended to the case where the gap is not a closed interval.

REFERENCES

Anderson, S. and de Palma, A., 1992, ‘The Logit as a Model of Production Differen-tiation,’ Oxford Economic Papers, 44, pp. 51–67.

Banerjee, A.V., 1992, ‘A Simple Model of Herd Behavior,’ The Quarterly Journal ofEconomics, 107(3), pp. 797–817.

Caplin, A. and Leahy, J., 1998, ‘Miracle in Sixth Avenue: Information Externalities andSearch,’ The Economic Journal, 108, pp. 60–74.

Chamley, C. and Gale, D., 1994, ‘Information Revelation and Strategic Delay in aModel of Investment,’ Econometrica, 62(5), pp. 1065–1085.

Choi, J.P., 1997, ‘Herd Behavior, the Penguin Effect, and the Suppression of Informa-tional Diffusion: An Analysis of Informational Externalities and Payoff Interdepend-ency,’ The RAND Journal of Economics, 28(3), pp. 407–425.

Crawford, V. and Haller, H., 1990, ‘Learning How to Cooperate,’ Econometrica, 58(3),pp. 571–595.

Dixit, A. and Shapiro, C., 1986, ‘Entry Dynamics with Mixed Strategies,’ in The Eco-nomics of Strategic Planning: Essays in Honor of Joel Dean, (ed.) L. Thomas III,Lexington Books, Lexington, Massachusetts, U.S.A.

Farrell, J. and Saloner G., 1988, ‘Coordination through Committees and Markets,’ TheRand Journal of Economics, 19(2), pp. 235–252.

Ghemawat, P. and Nalebuff, B., 1990, ‘The Devolution of Declining Industries,’ TheQuarterly Journal of Economics, 105(1), pp. 167–186.

Hendricks, K.; Weiss, A. and Wilson, C., 1988, ‘The War of Attrition in ContinuousTime with Complete Information,’ International Economic Review, 29(4), pp. 663–680.

4 In other cases, for example when there is a gap [t0, t1) in the support of F(t), we can provethat there is no mass point at t1 by using the same reasoning used in the proof for no masspoint at tm = 0.



Hoppe, H.C., 2000, ‘Second-Mover Advantages in the Strategic Adoption of NewTechnology under Uncertainty,’ International Journal of Industrial Organization,18(2), pp. 315–338.

Hoppe, H.C. and Lehmann-Grube, U., 2001, ‘Second-Mover Advantages in DynamicQuality Competition,’ Journal of Economics and Management Strategy, 10(3), pp.419–433.

Kapur S., 1995, ‘Technological Diffusion with Social Learning,’ The Journal of Indus-trial Economics, 43(2), pp. 173–195.

Kopel, M. and Loffler, C., 2007, ‘Commitment, First-Mover and Second-MoverAdvantage,’ Journal of Economics, 94(2), pp. 143–166.

Mariotti M., 1992, ‘Unused Innovations,’ Economics Letters, 38, pp. 367–371.Reinganum, J.F., 1985, ‘A Two Stage Model of Research and Development with

Endogenous Second-Mover Advantage,’ International Journal of Industrial Organi-zation, 3, pp. 275–292.

Smirnov, V. and Wait, A., 2004, ‘Second-Mover Advantage in a Market-Entry Game,’working paper, University of Sydney, Sydney, New South Wales, Australia, http://economics.ca/2005/papers/0413.pdf



Documents

Second Mover Advantage and Market Entry