Innovation timing games_ a general framework with applications

Innovation Timing Games:A General Framework with

Applications1

Heidrun C.Hoppe2 University of Bonn and CEPR Ulrich Lehmann-Grube3 University of Hamburg

April2003

1We would like to thank Benny Moldovanu for very helpful comments and suggestions.

2University of Bonn,Department of Economics,Chair of Economic Theory II,Lennéstr.37,D-53113Bonn,Germany,e-mail:hoppe@wiwi.uni-bonn.de,Tel:49-228-737942,Fax:49-228-737940.

3University of Hamburg,Department of Economics,Von-Melle-Park5,D-20146Hamburg, lehmann@hermes1.econ.uni-hamburg.de

Abstract

We o?er a new algorithm for analyzing innovation timing games.Its main advantage over the traditional approach is that it applies to problems that had previously been intractable. We use the algorithm to examine two classical innovation problems.In both settings,we ?nd that the competition takes the form of a waiting game with a second-mover advantage if R&D costs are high.Moreover,both models predict that the second-mover advantage is monotonically increasing in the costs of R&D.We also illustrate how our approach can be used to assess welfare properties of speci?c R&D policies.

Key words:Simple timing games,Preemption,Waiting,Innovation,R&D.

JEL Classi?cation:L13,O31,O33.

1Introduction

In capitalist reality as distinguished from its textbook picture,it is not[the

price or quantity]competition which counts,but the competition from the new

commodity,the new technology[...]-competition which commands a decisive

cost or quality advantage and which strikes not at the margins of the pro?ts

and the outputs of the existing?rms but at their foundations and their very

lives.

Joseph Schumpeter,Capitalism,Socialism and Democracy,1942

Economists have paid much attention to the analysis of technological competition.Yet, despite important advances in the understanding of?rms’incentives to innovate,there are at least two ways in which the existing literature may be extended.The?rst concerns the traditional approach to model technological competition as a race.The second is methodological.

Traditionally,technological competition is modeled as a race to be?rst in marketing a new product or technology.In these games,the?rm that innovates?rst is assumed to be the only?rm that receives any returns from research and development(R&D).All other ?rms receive nothing.1However,recent empirical research has cast serious doubts on the reasonableness of the traditional view.In their study of50di?erent markets,Tellis and Golder(1996),for instance,discover that the failure rate for pioneers is high.In contrast to the traditional view,their results suggest that technological competition is governed by the following trade-o?:being?rst may yield monopoly pro?ts until a rival?rm innovates, while moving second may yield higher duopoly pro?ts due to better technology.

The aim of our paper is to investigate optimal innovation timing in the presence of this trade-o?.Our study will focus on the following questions:When will?rms engage in a race to be?rst and when will they wait to be best?How are the equilibrium innovation dynamics related to more primitive data,such as the costs of R&D?When is policy intervention in technological competition desirable,and what form should it take?

These issues appear to be particularly relevant in markets where?rms make a tech-1For a more extensive discussion of these so-called patent race games,see,e.g.,Baye and Hoppe(2003).

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nology choice at the time of market entry,thereby determining their technological level of production for subsequent product market competition.Examples include technological platforms choice in the aircraft,high-speed train,and automobile industries.A famous example for a successful second mover in these industries is Boeing.In the commercial jet aircraft market,the company surpassed the pioneer De Havilland Comet1through innovation-by building a safer,larger,more powerful jet.2Other examples can be found in markets where new products are de?ned by certain standards,such as the computer software or hardware industries,or markets where product characteristics must be ap-proved by government agencies.For instance,SmithKline Beecham’s Tagamet,pioneering the anti-ulcer drug industry,was overtaken in terms of sales by Glaxo’s Zantac,which was e?ective in smaller dosages and had fewer side-e?ects.3For the computer industry, the Economist recently observed that“even in industries where the winner takes all(also rarer than you might think),having the best product matters more than being?rst to market”.4

The second extension of the existing literature is methodological.Our study of optimal innovation timing can be based on a fairly small literature in which technological competi-tion is formulated as a simple timing game-i.e.,a game in which each?rm chooses at any point in time whether to make a single,irreversible move(cf.Fudenberg and Tirole,1985, 1991[ch.4.5];Katz and Shapiro,1987;Dutta et al.,1995).5Two important results of this literature are the following.First,in their extension of Reinganum’s(1981)duopoly model of technology adoption,Fudenberg and Tirole(1985)show that a?rst-mover advantage is not supported by subgame-perfect strategies if?rms are unable to precommit to future actions.In their model,?rms decide at any point in time whether to adopt a cost-reducing new technology,knowing that adoption costs decline over time.By assumption,the in-2”What are the options for later entrants?”Financial Times,October19,1998.

3”Glaxo and EMI:a tale of two companies”Financial Times,August1,1997.

4”First will be last”,The Economist,September28,2002,p.84.

5There is a relatively large,related strand of literature,which has,however,a completely di?erent focus. In contrast to our paper,which considers the technology choice problem at the time of market entry when ?rms are initially symmetric,this literature focuses on asymmetric situations,i.e.,where one or more?rms have entered the market,already to analyze the?rms’incentives to improve existing products(e.g.,Budd et al.,1993;Athey and Schmutzler,2002;Bergemann and V?lim?ki,2002).

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crease in pro?ts due to innovation is greater for the?rst adopter than for the second.As the authors show,this potential?rst-mover advantage stimulates preemption up to a point where the extra?ow pro?t for the?rst mover just equals the extra costs of speeding up adoption.Second,Katz and Shapiro(1987)demonstrate that a potential second-mover advantage may give rise to subgame-perfect equilibria in which preemption and payo?equalization do not occur.In their model,payo?s to di?erent?rms are asymmetric.Re-sults for a symmetric setting are provided by Dutta et al.(1995)who demonstrate that a potential second-mover advantage may indeed prevail as the subgame-perfect equilibrium outcome.

However,the approach to simple timing games analysis used in this literature requires rather restrictive assumptions.In particular,the?rst-mover’s equilibrium payo?must be single-peaked in the times that?rms may move?rst.For some timing problems this condition may be satis?ed,but in many cases it is very di?cult or impossible to verify single-peakedness of the?rst-mover’s equilibrium payo?analytically.In fact,we encoun-tered such problems in our analysis of speci?c innovation timing games.

The present paper aims to?ll that gap by o?ering a more general approach for sim-ple timing games when the?rst-mover’s problem is not necessarily well-behaved.The approach generalizes the existing results of Fudenberg and Tirole(1985,1991),Katz and Shapiro(1987),and Dutta et al.(1995).Our central result is a theorem asserting gen-eral conditions under which simple timing games possess a unique equilibrium outcome. One property of this outcome is that it never involves a?rst-mover advantage.That is, depending on the speci?c context,the competition is structured either as a preemption game with payo?equalization in equilibrium or as a waiting game with a second-mover advantage.We also provide an algorithm for determining whether a speci?c game is one of preemption or one of waiting.

The framework o?ered in this paper is particularly useful for the analysis of innovation timing games.This is illustrated by applying it to two dynamic innovation problems,one of product innovation and another of process innovation.For product innovation,we consider a dynamic version of a vertical product di?erentiation model adapted from Dutta et al.(1995)and Tirole(1988).Firms choose at any date whether to bring the currently

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available product to the market or whether to wait and market a product of higher quality. Once both?rms have entered,they compete in prices.Second,we consider a dynamic version of the classical process innovation story as in Dasgupta and Stiglitz(1980)and Reinganum(1981,1982).Firms choose a level of variable costs over time,which determines their cost position during the subsequent Cournot quantity competition.To the best of our knowledge,the game of process innovation has not been studied before.6 Our treatment of these models enables us to draw conclusions about the dynamic nature of the competition,i.e.whether it takes the form of a preemption game or a waiting game,and about e?ects of certain parameter changes that had previously been intractable. In contrast to the traditional view,we?nd for both settings that technological competition may be structured as a waiting game with a second-mover advantage in equilibrium. Moreover,in both games,the second-mover advantage increases monotonically as R&D becomes more costly.

Our framework may also be used to evaluate the impact of speci?c R&D policies.In the case of product innovation,our numerical results imply that an R&D subsidy will cause?rms to innovate later,but that this always leads to higher welfare(as measured by the present value of the sum of the intertemporal stream of?rm pro?ts and consumer surplus).In the case of process innovation,our results suggest that the appropriate policy depends critically on the magnitude of the second-mover advantage in equilibrium.A tax is found to be optimal if the second-mover advantage is small,and a subsidy if it is large.Moreover,if R&D costs are so high that no?rm would innovate at all,we ?nd that a subsidy,inducing one of them to innovate-and consequently monopolize the industry-,can improve social welfare.Interestingly,our results stand in contrast to policy implications derived from traditional patent race models(cf.Dixit,1988).

In Section2,we describe the general framework and give two speci?c examples of innovation timing games.In Section3,we illustrate why the existing approach fails to provide solutions to these games.Also,in Section3the existence and uniqueness results for simple timing games are formally stated and proved.In Section4,we apply our approach to the two examples and analyze the e?ects of R&D cost changes on the equilibrium 6Dutta et al.(1995)and Hoppe and Lehmann-Grube(2001)present analytical results for polar cases of the product innovation model.

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innovation dynamics.In addition,we examine welfare e?ects of speci?c R&D policies.We provide concluding remarks in Section5.

2Simple timing games

2.1Game form

We consider a class of simple timing games,Γ,characterized by the following structure: There are two?rms,i=a,b.At any point in time t∈R+,each?rm can choose whether to make an irreversible stopping decision,conditional on the history of the game.We will interpret the stopping decision as the?rm’s choice to adopt a currently available new technology.Let t a,t b denote the?rms’respective adoption dates.A?rm’s payo?depends on its own and its rival’s adoption date:πa(t a,t b)andπb(t b,t a).If?rm i chooses t1while ?rm j(j=i)chooses t2>t1,then i is called the leader and j the follower.Throughout the paper,we will make the following basic assumptions:

(A1)Time is continuous in the sense of‘discrete but with a grid that is in?nitely?ne’(cf.Simon and Stinchcombe,1989).That is,any continuous-time strategy pro?le will be restricted to an arbitrary,increasingly?ne sequence of discrete-time grids,and the continuous-time outcome will be de?ned to be the limit of the discrete-time outcomes.7 (A2)There exist two piecewise continuous functions

π1,π2:{(t1∈R+)×(t2∈R+)|0≤t1≤t2}→R+

withπ1(t1,t2)=π2(t1,t2)if t1=t2,and

πi(t i,t j)=(π1(t i,t j)if t i

π2(t j,t i)if t i≥t j

7Simon and Stinchcombe(1989)identify conditions under which the discrete-time outcomes converge to a unique limit that is independent of the particular sequence of grids.Roughly,the conditions require (i)an upper bound on the number of moves,(ii)that strategies depend piecewise continuously on time, and(iii)that actions later in the game are not“too sensitive”in a certain sense to the precise times at which earlier moves have been made.As it will turn out,the simple timing games considered in the present paper satisfy these conditions.

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for(i=j)∈{a,b}.

(A3)If a?rm is indi?erent between the leader’s and follower’s role at any date t,then it attempts to become the leader.If,additionally,the leader is indi?erent between adopting at two di?erent points in time,then it chooses the earlier one.Furthermore,if t a=t b, then we assume that only one?rm-each with probability1/2-actually adopts at that time and becomes the leader,while the other?rm becomes the follower and may postpone its adoption.8

Assumption A1circumvents the problem that backwards induction cannot be applied in continuous time.That is,we regard discrete-time with a very?ne grid as a convenient mathematical construction to represent the notion of‘continuous time’.Assumption A2 imposes symmetry between?rms.Assumption A3is used to formalize the idea that?rms will be able to avoid coordination failure as an equilibrium outcome.That is,?rms will not choose to move at the same instant of time if they would regret this move afterwards.As observed by Fudenberg and Tirole(1985),an equilibrium involving a positive probability of coordination failure cannot be obtained in the polar case of a continuous-time game without a grid,where equilibria are de?ned to be the limits of discrete-time mixed-strategy equilibria.By contrast,in the limit of a discrete-time game where the period length converges to zero,coordination failure is a possible equilibrium outcome.Hence,if one uses a discrete-time game with very short time lags to represent the notion of‘continuous time’(as we do),one needs to make an assumption that explicitly rules out the possibility of coordination failure.Several alternative assumptions can be made:(i)a randomization device as it is used here,in Katz and Shapiro(1987),Simon and Stinchcombe(1989),and Dutta et al.(1995);(ii)alternating-moves as in Riordan(1992)and Harris and Vickers (1985);(iii)?rm-speci?c lags between observations and decisions as in Gilbert and Harris (1984).

8In the limit,adoption by one?rm may result in an instantaneous follow-on adoption by the other?rm, i.e.,the two?rms adopt‘consecutively but at the same instant of time’,and both?rms obtain the same payo?.

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2.2Innovation timing games-two examples

In this subsection,we present two speci?c examples:?rst,a game of product innovation timing and,second,a game of process innovation timing.In both games,there are two ?rms who have the opportunity to develop a new product.At each point in time,each ?rm chooses whether to bring the new product to the market,using the so far developed technological potential for the rest of the game,or whether to continue to invest in research and development(R&D)to obtain a better technology.Let k(t)be each?rm’s R&D costs per unit of time at time t.9

The monopoly revenue per unit of time that is associated with the leader’s entry at time t1is R M(t1).The leader’s and follower’s equilibrium duopoly revenues per unit of time as functions of the leader’s and follower’s entry times are R1(t1,t2),and R2(t1,t2), respectively.The respective payo?s are thus given by

π1(t1,t2)=Z t2t1e?rτR M(t1)dτ+Z∞t2e?rτR1(t1,t2)dτ?Z t10e?rτk(τ)dτ(1)π2(t1,t2)=Z∞t2e?rτR2(t1,t2)dτ?Z t20e?rτk(τ)dτ(2) with t1≤t2,and R1(t1,t2)=R2(t1,t2)if t1=t2.

Product innovation timing:In the game of product innovation,?rms can choose to improve the quality of the new product before they enter the market.The available product quality s(t)is increasing in time t by means of a deterministic and possibly costly research technology.We assume,as in Dutta et al.(1995),that s is proportional to t, and without further loss of generality that t=s.After a?rm has entered the market,the quality of its product is?xed.Each?rm’s R&D costs per unit of time areλs,withλ≥0. Variable costs of production are independent of quality and zero.

9Notice that we allow for positive R&D costs per period of time,but ignore any additional?xed costs of market entry.As a consequence,at any date of market entry,all R&D expenditures are sunk.Furthermore, we assume that a?rm that stops its R&D activity and is indi?erent between entering the market at that date and staying out forever will choose to enter.Thus,entry deterrence is no issue in our paper.

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For the demand side,we use a model adapted from Tirole (1988).Each period,each consumer buys at most one unit from either ?rm 1or ?rm 2.Consumers di ?er in a taste parameter θ,and they get in each period a net utility if they buy a quality s i at price p i of U =s i θ?p i ,and zero otherwise.A consumer of ”taste”θwill buy if U ≥0for at least one of the o ?ered price-quality combinations,and she will buy from the ?rm that o ?ers the best price-quality combination for her.Consumers are uniformly distributed over the range [0,1].Without loss of generality,we choose physical and money units such that inverse (?ow)demand p (q )for s given quality units is:p (q )=s (1?q ),where q denotes aggregate quantity.

The equilibrium revenue ?ows per unit of time for the monopolist and the duopolists from price competition with vertically di ?erentiated goods are,respectively:10

R M =14t 1,

R 1=t 1t 2t 2?t 1(4t 2?t 1)2,R 2=4t 22t 2?t 1(4t 2?t 1)2.(3)

Process innovation timing:In the game of process innovation,?rms can choose to reduce the cost of producing the new product before they enter the market.The total cost of producing q i units of output is c i q i for ?rm i =1,2,where c i is constant.These costs decline over time by means of a deterministic and possible costly research technology:c i (t )=e ?αt i ,where α>0is the rate of technological progress.Note that the cost-reducing technology is characterized by diminishing returns per period.A natural form for the R&D cost function in this context is k =λ,with λ≥0for all t .11

The demand side for the new product is characterized by a simple linear inverse (?ow)demand function.Money units are normalized such that inverse demand per unit of time is p =1?q where q represents the aggregate quantity.

The equilibrium revenue ?ows per unit of time for the monopolist and the duopolists from Cournot quantity competition with homogenous goods are,respectively:

?R M =1(1?c 1)2,?R 1=max n 19(1?2c 1+c 2)2,0o ,?R 2=min n 19(1?2c 1+c 2)2,14(1?c 1)2o .(4)10See Hoppe and Lehmann-Grube (2001)for a more detailed description of this example and for the analysis of the limiting case in which the R&D costs parameter λtends to in ?nity.

11This form for the R&D cost function is,for instance,used by Reinganum (1982).

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Note that there are two cases to distinguish:c2≥2c1?1,i.e.the innovation by the follower is nondrastic,and c2<2c1?1,i.e.it is drastic.If the innovation becomes drastic,the follower will set its monopoly price and the leader will shut down.Clearly,

?R

M(c1)and ?R

i(c1,c2)can be written as functions of time.Hence the structure of the

payo?s is of the form(1)and(2).

3Solutions to simple timing games

The natural solution concept for simple timing games is subgame-perfect equilibrium.As in Simon and Stinchcombe(1989),we restrict attention to pure strategies and invoke the additional concept of iterated elimination of weakly dominated strategies.Here,the re?nement is however only used to exclude some uninteresting,rather pathological cases.

Following Fudenberg and Tirole(1985),we take the relevant choice variable to be the time that?rms may choose to move?rst.Let R(t):R+→R+be the best response

function of the follower,de?ned on the set of times t that?rms may choose to move ?rst.If such a best response function R(t)exists,we can specify the leader’s and the follower’s payo?s as functions of t alone:de?ne L(t)=π1(t,R(t))and F(t)=π2(t,R(t)), respectively.12To ensure that R(t)is a well-de?ned function of the leader’s choice t,we assume that if the follower is indi?erent between moving on di?erent dates,it will choose the earlier one.

The aim of this section is to isolate a class of simple timing games for which some equilibrium outcome can be uniquely identi?ed and is easily described.We begin by brie?y discussing the existing approach to simple timing games analysis.

3.1Approach with single-peaked L curve

The existing literature on simple timing games(cf.Fudenberg and Tirole,1985,1991 [ch. 4.5];Katz and Shapiro,1987;Dutta et al.,1995),requires that the leader payo?L(t)satis?es one major assumption:L must have a unique maximum.The assumption is made to ensure an analytical determination of the exact location of the global maximum 12We use t instead of t1to denote the leader’s adoption time whenever there is no ambiguity.

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of L(t).13The approach is suggested in Figure1.

t* 1t* 1

t*

1

*

Figure1

Consider?rst the situation depicted in Figure1a.Suppose that the L curve is single-peaked at t??1.The solution can then be obtained by applying the following argument due to Fudenberg and Tirole(1985).14It is clear that each?rm would like to move?rst at t??1. Knowing this,however,each?rm also has an incentive to preempt its rival by adopting slightly before t??1.Hence,?rst adoption at t??1cannot be an equilibrium.Similar reasoning can be applied to any t1∈]t?1,t??1[,where t?1denotes the intersection point between the L curve and the F curve.This yields?rst adoption at time t?1and equal payo?s for both ?rms as the unique subgame-perfect equilibrium outcome.Next,consider the situation depicted in Figure1b.Suppose that the L curve is single-peaked at t?1.Since the F curve lies above the L curve at any t1≤t?1,it is clear that no?rm has an incentive to preempt its rival before date t?1.In fact,the unique subgame-perfect equilibrium(up to relabelling of?rms)involves?rst adoption at t?1and a higher payo?for the second mover.15

13Alternatively,as in Fudenberg and Tirole(1985)where the?rst-mover’s equilibrium payo?may have two peaks,the location of the global maximum is assumed to be known.The model is then essentially treated as a game in which the?rst-mover’s equilibrium payo?is single-peaked.

14In fact,a similar argument has already been made by Karlin(1959,chapter6),however,without using the concept of subgame perfection.

15This equilibrium is asymmetric.That is,the competitors’expectations about the rival’s strategies determine the equilibrium outcome.If,for example,?rm i believes that j never enters?rst,i may choose to be the?rst entrant.Likewise,if j has the reputation of being likely to enter?rst,it may be optimal for

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Note that one consequence of assuming that the L curve is single-peaked is that the problem of determining a“terminal subgame of the game”where one can begin apply-ing Fudenberg and Tirole’s preemption argument is simple:by examining the?rst-order condition for maximizing L,it is typically easy to determine the accurate location of the maximum of L.Then,what is left to check is whether the L curve is above or below the F curve at that point.If L is above F(as in Figure1a),the equilibrium involves preemption and payo?equalization.If L is below F(as in Figure1b),?rms wait until the maximum point of the L curve,and there is a second-mover advantage in equilibrium.

However,as we illustrate in the following,it may be very di?cult or impossible to verify single-peakedness of the L curve for speci?c simple timing games.

Product innovation timing:For the game of product innovation described above,the derivative of the leader payo?curve L(t1)with respect to t1is given by:

dL dt1=

1

4

e?rt1?e?r R(t1)

r

+

d R

dt1

e?r R(t1)μ14t1+1r?R1?t2?R1??14t1e?rt1 +

?R1

?t1

1

r

e?r R(t1)?λe?rt1t1

where the best response of the follower,R(t1),is the solution to the?rst-order condition of the follower’s maximization problem,

64λrt32?48λrt1t22+12λrt2t21?λrt31

?16t22+12t1t2?8t21+16rt32?20rt22t1+4rt2t21=0(5) and the reaction coe?cient d R/dt1is given by

d R dt1=?

12t2?16t1?20rt22+8rt2t1?48λrt22+24λrt2t1?3λrt21?32t2+12t1+48rt22?40rt2t1+4rt21+192λrt22?96λrt2t1+12λrt21(6)

Obviously,verifying single-peakedness of L is no longer analytically tractable.

i to wait until j has entered.In the case where the game is structured as a waiting game,there is also a continuum of mixed-strategy equilibria which are not considered here.

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Process innovation timing:Similarly as for product innovation,it turns out to be impossible to analytically verify single-peakedness of the L curve for the process innovation model described above.Moreover,additional complications arise here due to the fact that the best response of the follower,R(t1),may involve drastic innovation,nondrastic innovation or joint adoption.That is,the L curve need not be di?erentiable and continuous everywhere.

3.2Approach without single-peaked L curve

We now present an approach to simple timing games analysis that allows for the possibility that the L curve is not single-peaked.We shall also provide an algorithm for determining the dynamic nature of the game,i.e.,whether it is a one of preemption or one of waiting.

De?ne date T1by:

T1:=min?τ:L??t1(τ)￠≥F(τ)a(7) where

?t

1(τ):=max??t:?t=arg max[0,τ]L(x)?(8) That is,T1denotes the earliest point in time where the F curve just falls below the maximum value that the L curve achieves over the range[0,t].In the next lemma,we state conditions under which a simple timing gameγ∈Γhas a unique point T1.The lemma is proved in the appendix.

Lemma1Assume that a simple timing gameγ∈Γful?lls the following conditions:

1.There exists a best response function R(t).

2.There exist some points t0,t00∈[0,∞)with t0L(t0)≥0and

F(t00)≤0.

Then there exists a unique T1.

Our strategy in the present paper is to replace the problem of determining the global maximum of the L curve by the problem of determining point T1,provided such a point exists.Our central result is a theorem asserting two rather mild conditions under which a

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simple timing gameγ∈Γhas a unique equilibrium outcome that is easily obtainable by analyzing the game only for the range[0,T1],irrespective of the shape of the payo?curves after T1.We have incorporated these conditions in Figure2:(1)the L curve is continuous, and(2)the F curve is continuous and non-increasing.In this?gure,L has multiple peaks, which rules out an application of the existing approach.The thick curve,which we call the“envelope L curve”,gives the maximum value of L over the range[0,t],for any given t.As we shall show below,these two conditions on the payo?functions imply that point T1is the?rst intersection point between the envelope L curve and the F curve.This,in turn,implies that T1is a boundary point of the set of times that?rms will move?rst.16

t* 1T

1

Figure2

Before we state our theorem,we will introduce the two conditions on the payo?func-tions informally and illustrate what happens if they are violated.

Continuous L curve:The following?gure illustrates the possible problem arising from a discontinuous L curve.In the?gure,the L curve jumps upwards at some date later than T1.One can verify that in this case there are two di?erent subgame-perfect equilibrium outcomes:(i)both?rms trying to be?rst at T1,and(ii)both?rms waiting until the point of discontinuity.Thus,discontinuities of the L curve may give rise to multiple equilibrium outcomes.

16The approach of?nding solutions via intersection point arguments is thus somewhat similar as the methodology used by Vives(1990)in his analysis of supermodular games.

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The condition is clearly restrictive,and especially not suitable if additional ?xed-costs of market entry and hence entry deterrence are an issue.17However,apart from such cases,continuity of the L curve is typically satis ?ed in games with ongoing technological progress and where the best response function of the follower,R (t )does not change discontinuously,such as in the two innovation games considered in the present paper.Moreover,as we will show below,our algorithm may still be applicable when the condition is relaxed.

= T 1

t*1Figure 3

Continuous and non-increasing F curve :The second class of games that we ex-clude are those in which the F curve is discontinuous and/or increasing.Restricting F to be a continuous function,turns out to be a rather mild assumption,since this function gives the best-response payo ?to any leader’s choice t,and is obtained by integrating with respect to time some instantaneous ?ow pro ?ts,which typically depend only on time and the fact that the other ?rm has adopted already.Requiring F to be non-increasing in the leader’s choice,is more restrictive.In the context of innovation timing,the assumption has,however,a rather natural interpretation:in the presence of ongoing technological 17Entry deterrence plays no role in the examples studied in this paper,since R&D expenditures are sunk at any date where the entry decision has to be made.Thus,entry is at any moment in time e ?ectively costless.Also,note that no ?rm can force its rival into an eventually pro ?tless situation,since any ?rm could decide to be the leader at any point in time.We leave the issue of entry deterrence for future research.

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progress,earlier innovation means usage of a less advanced technology.Thus,if the leader innovates earlier,it will be in a weaker technological position during product market com-petition.An earlier leader’s choice thus implies higher duopoly pro?ts for the follower. In fact,the assumption of a non-increasing F curve turns out to be typically satis?ed for games of innovation timing where?rms enter a new market,such as in the two examples considered in this paper.

Without the condition,one can still establish the existence of subgame-perfect equi-libria.However,when the L curve has multiple peaks,the equilibrium outcome is not necessarily unique.The analysis of subgame-perfect equilibria then requires to examine a considerably larger set of subgames,with a“terminal node”speci?ed in a way which depends on the particular nature of the speci?c game under consideration.

Theorem1(Unique equilibrium outcome)Consider a simple timing gameγ∈Γthat satis?es the conditions stated in Lemma1and in addition:

1.L(t)is continuous.

2.F(t)is continuous and non-increasing.

Then the game has at least one subgame-perfect equilibrium in undominated pure strategies and the equilibrium outcome is unique(up to relabelling of?rms):one ?rm adopts at t?1,the other?rm follows at R(t?1),where t?1:=?t1(T1),as above;

equilibrium payo?s are L(t?1)=F(t?1)if t?1=T1,and L(t?1)

?T

1(t)≡min?τ:L??t1(t,τ)￠≥F(τ)a(9)

?t

1(t,τ):=max??t:?t=arg max[t,τ]L(x)?(10) That is,?T1(t)denotes the point in time where the F-curve just falls below the maximum level that the L-curve obtains from some t up to?T1.Correspondingly?t1(t,?T1)is the point in time where the L-curve attains that maximum.

Note that T1=?T1(0)and t?1=?t1(0,T1),where T1is as de?ned above.By Lemma1, t?1and T1exist and are unique.

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We show that a gameγ∈Γthat satis?es the conditions of Lemma1has a subgame-perfect equilibrium consisting of the following pair of pure strategies:

Given no previous adoption,both?rms choose”No adoption”at any t

?”Adoption”if L(t)≥F(t)or[L(t)

and?rm j chooses

?”Adoption”if L(t)≥F(t),or

?”No adoption”if L(t)

If either?rm has already adopted at some timeτ,the other?rm adopts at R(τ).

First,note that the conditions stated in Lemma1ensure that each?rm?nds it optimal to wait at any t

(i)If L(t)≥F(t),the given strategies yield a payo?of1/2[L(t)+F(t)]for each?rm, while any deviation yields at most F(t),with F(t)≤1/2[L(t)+F(t)].

(ii)If L(t)?T1(t)with L(t0)> L(?t1(t,?T1(t))).It follows from Conditions1and2that in that case there must be an intersection between L and F at some t00for?T1(t)

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(iii)If L(t)

Thus,the described strategies are best responses to each other and constitute a subgame-perfect equilibrium in every simple timing gameγ∈Γthat satis?es Condi-tions1to2.If the strategies are played on an arbitrary discrete-time grid,the resulting equilibrium outcome is that the?rst adoption occurs weakly beyond t?1.In the limit,?rms adopt exactly at(t?1,R(t?1)),with L(t?1)=F(t?1)if t?1=T1and L(t?1)

To prove the second claim,we show that there exists no subgame-perfect equilibrium in pure strategies which does not implement?rst adoption at t?1and equilibrium payo?s of L(t?1)=F(t?1)if t?1=T1,and L(t?1)t?1does not change the equilibrium outcome since t?1remains the leader’s choice in the described equilibrium.

Consider now potential equilibria with?rst adoption at t1where t1>t?1.By Condition 2,we must have for any t1>t?1that either F(t1)≤L(t1)or L(t1)t?1cannot be the leader’s choice in an equilibrium in pure strategies. Now consider the case where F(t1)t?1is ruled out by assumption(A3).

Finally,consider possible equilibria with?rst adoption at t1where t1L(t1)must hold for all t1

The equilibrium strategies identi?ed in the proof of Theorem1have the following properties.The?rst adoption occurs at point t?1≤T1,and the second adoption at point

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R(t?1)≥t?1.If t?1=T1,?rms engage in a preemption game to be the?rst,and they obtain equal payo?s,i.e.,L(t?1)=F(t?1).If t?1

Note that our results reduce to those of the existing literature if the theorem is applied to problems with a single-peaked L curve.For example,in the situations depicted in Figure1,point T1,as de?ned above,exists and is the?rst intersection point between the envelope L curve and the F curve.In Figure1a,we have T1=t?1,so we know from Theorem1that the unique equilibrium outcome involves preemption and?rst innovation at t?1,with payo?equalization across?rms.In Figure1b,we have T1>t?1,so the unique equilibrium outcome involves waiting until one?rm innovates at t?1,and there is a higher payo?for the second mover.Theorem1thus generalizes the existing results by omitting the usual assumption on the?rst-mover’s equilibrium payo?.Furthermore,the theorem isolates a class of simple timing games for which the equilibrium outcome is unique and easily describable,thus providing a basis for numerical approaches to compute explicit solutions in particular games.

3.3Approach with possibly discontinuous L curve

This subsection shows that the application of our algorithm is not restricted to games with a continuous L curve.The following two corollaries to Theorem1are proved in the appendix.

Corollary1Consider a simple timing gameγ∈Γthat satis?es the conditions stated in Lemma1and in addition:

1.Let Q={t:L(t)>F(t)}.Then L(sup X)≤sup L(X)and L(inf X)≥inf L(X)

for every nonempty subset X of Q.

2.F(t)is continuous and non-increasing.

Then the unique equilibrium outcome is as described in Theorem1.

Corollary1reveals that the algorithm suggested in Theorem1continues to be applica-ble to simple timing games in which the L curve is discontinuous,but involves no upwards

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