Discussion Paper No. 110
Recurrent Preemption Games

Hitoshi Matsushima
February, 2020

Data Science and Service Research
Discussion Paper

Center for Data Science and Service Research

Graduate School of Economic and Management
Tohoku University27-1 Kawauchi, Aobaku

Sendai 980-8576, JAPAN1

Recurrent Preemption Games*
Hitoshi Matsushima
Faculty of Economics, University of Tokyo
Hongo, Bunkyo-Ku, Tokyo 113-0033, Japan
Email: hitoshi@e.u-tokyo.ac.jp
Tel and Fax: +81-3-58415506

February 13, 2020*

This research was supported by a grant-in-aid for scientific research (KAKENHI 21330043,
25285059) from the Japan Society for the Promotion of Science (JSPS) and the Ministry of
Education, Culture, Sports, Science and Technology (MEXT) of the Japanese government. Hitoshi
Matsushima is appointed as a visiting professor at Graduate School of Economics and Managementat Tohoku University.

2

Abstract
I consider a new model of an infinitely repeated preemption game with random
matching, termed the recurrent preemption game, wherein each player’s discount factor
depends on whether she wins the current game. This model describes sequential
strategic technology adoptions in which a company becomes outdated by failing tomaintain a position at the forefront of innovation. Assuming incomplete information
about the presence of a rival, I clarify how the prominence of the innovator’s dilemma
influences the degree of excessive competition in preemption. I also reveal interesting
properties demonstrated by the unique symmetric Nash equilibrium of the recurrent
preemption game.
Keywords: Recurrent Preemption Game, Strategic Technology Adoption, Innovator’s
Dilemma, Unique Equilibrium, Random Technology.
JEL Classification Numbers: C72, C73, L13, O30, O31

3

1. Introduction
This study investigates the recurrent preemption game, which is defined as a newversion of an infinitely repeated game with random matching. The component game is a
simple form of Bertrand duopoly that I term the preemption game. I assume incomplete
information in that each player does not know whether her rival is present or absent.
This game mainly departs from existing repeated games in that the discount factor (orthe survival rate) of each player is not constant across component games and instead
depends on the state and the consequence of the current component game, such aswhether a player’s rival is present and whether she wins the current game. This study
shows that the recurrent preemption game demonstrates the unique symmetric Nashequilibrium, and reveals various important properties of this equilibrium.
This study situates the recurrent preemption game within the following dynamics
of sequential strategic technology adoptions. A company attempts to develop and adopt
a new technology within a limited time interval. If the company successfully adopts thetechnology earlier than its rival, then it earns the corresponding time-dependent
adoption value as the winner’s payoff; otherwise, it earns nothing. By postponing thetiming of adoption, the company can further develop the technology and increase thisadoption value, but the risk of being preempted by its rival increases at the same time.
Just after the end of the time interval, the company can enter the next preemption
game with a positive probability; the rival in the next game is different from the rival in
the current game due to the randomness of matching. Importantly, only one of the twocompanies in the current game will progress to the next game. Hence, the twocompanies are rivals in two respects: 1) they are rivals in the scramble over the next
opportunity for technology adoption, and 2) they are rivals in the competition forpreempting the adoption of the current technology.
I specifically focus on the situation in which the likelihood of the winner entering
the next game is not the same as the likelihood of the loser entering the next game. The
likeliness of each outcome crucially depends on the relative importance between thefollowing winner’s advantage and disadvantage. The case in which the loser is more
likely to enter the next game than the winner corresponds to the prominence of the

4winner’s disadvantage and is termed the innovator’s dilemma (Christensen 1997; Igami
2017): the winner is constrained by the current technology, and therefore tends to bebehind of the loser in starting the development of the next technology. Meanwhile, thecase in which the winner is more likely to enter the next game than the loser
corresponds to the prominence of the winner’s (i.e., the innovator’s) advantage: thewinner outweighs the loser in technical and information-gathering skills. This study
clarifies that the relative importance between the innovator’s dilemma and advantage iscrucial for understanding strategic behavior.
To be more concrete, I envision the following scenario behind the recurrent
preemption game. A commercial company generally wants to ensure its long-term
survival in emerging industries. To do so, it must constantly ride the tide of innovations
that create new markets through competition across existing industries. However, thepresence of a rival company can threaten its ability to remain at the forefront of the next
innovation. Once a company loses its forefront position, its growth can quickly slow
and it can become outdated.
Importantly, the company that wins the current race of technology adoption canalso succeed in developing talent with the ability to discover and develop the next
technology. However, the innovator’s dilemma recalls that the winner of the current
adoption race may not necessarily win the next opportunity—the loser of the current
adoption race may headhunt the talent developed by the winner. Moreover, thecompany that successfully enters the next opportunity may still have a rival in the formof either a new venture or a company that previously worked in a different industry.
With this scenario in mind, the recurrent preemption game sharply distinguishes
between the following two conflicts between the rivals in the current technology
adoption. The first conflict is about meeting the next opportunity for technology
adoption, which is expressed by the dependence of each player’s discount factor onwhether she wins the current game. The second conflict is the first-come, first-served
style of the race for current technology adoption, which is expressed by the current,
one-shot preemption game. This study is the first theoretical attempt to incorporate thisfirst conflict into strategic technology adoption.
For our purposes, it is important to note that this study is related to existing
literature on timing games (and specifically preemption games) in which various aspects

5of incomplete information are incorporated. Notable works here include those by
Fudenberg and Tirole (1986), Abreu and Brunnermeier (2003), Brunnermeier andMorgan (2010), Hopenhayn and Squintani (2011), Bobtcheff and Mariotti (2012), andMatsushima (2013; forthcoming). In contrast to this study, these works commonly
investigate static games.
The theoretical advantage of this study’s model is that it involves the uniquesymmetric Nash equilibrium. In this respect, this study is closely related to
Matsushima’s (2013; forthcoming), which assumed incomplete information implyingthat each player expects that her rival is absent (or irrational) with a positive probability,
and showed the uniqueness of the Nash equilibrium. This study extends this uniquenessto the recurrent preemption game by adding the symmetry constraint.
The unique symmetric Nash equilibrium in the recurrent preemption game makes
clear that the above-mentioned first conflict has the following significant effects on
strategic behavior. The prominence of the innovator’s dilemma eases preemptivecompetition and consequently promotes more efficient technology adoption, while theprominence of the innovator’s advantage worsens preemptive competition andconsequently promotes less efficient technology adoption. The players’ degrees ofcompetitiveness from the viewpoint of the second conflict depend on the difference
between the level of the current technology’s adoption value and the average level.
Importantly, the manner of this dependence substantially differs based on theprominences of the innovator’s dilemma and advantage. If the innovator’s dilemma is
prominent, then player competitiveness in the equilibrium will increase as the level ofadoption value increases. On the other hand, if the innovator’s advantage is prominent,
then player competitiveness in the equilibrium will decrease as the level of adoption
value increases. This difference is due to the fact that the higher the quality of thecurrent technology, the less important the first conflict.
In this respect, the work of Rotemberg and Saloner (1986) is related to this study,
although it does differ substantially. More specifically, Rotemberg and Saloner studied
a repeated oligopoly game with business fluctuations and showed that excessive pricecompetition (e.g., price war) takes place in a boom because business fluctuations
influence the relative importance between the instantaneous gains and the punishments
in future games. In contrast, this study does not consider such future punishments, and6instead considers the dependence of a player’s discount factor on whether she wins thecurrent component game: the excessive preemption competition takes place with a highlevel of adoption value if and only if the innovator’s dilemma is prominent.
This study assumes that the rival in the current game is different from the rival in
the next game. In this respect, this study is related to studies of repeated games such as
those by Kandori (1992) and Deb, Sugaya, and Wolitzky (forthcoming), whereinplayers are randomly matched across component games. These works investigate
community enforcements by assuming that a player can obtain information concerning
the opponent’s past history of play. In contrast, this study does not make thisassumption: a player cannot even confirm that her rival is really present until the current
game ends.
This study assumes that the development and adoption of the next technology do
not affect the adoption value of the current technology. Accordingly, this study does notconsider the replacement effect (Arrow 1962) that makes the incumbent hesitant toinnovate—while the replacement effect may be one cause of the innovator’s dilemma(Igami 2017), it is not the only cause (Christensen 1997). We conceptually distinguishbetween the innovator’s dilemma and the replacement effect based on theabove-mentioned headhunting scenario.
The organization of this study is as follows. Section 2 introduces the basic
formulation of the recurrent preemption game, where I assume that the level of adoption
value is constant across component games. Section 3 incorporates uncertainty about
future technologies into the recurrent preemption game. Section 4 generalizes theformulation of the winner’s payoffs without substantial changes to the contents outlined
in Sections 2 and 3. Section 5 concludes the paper.

2. Basic Model
2.1. Formulation
I investigated a two-player infinitely repeated game with random matching that I
termed the recurrent preemption game. The component game is given by ( N , A, u ) ,7where N  {1, 2} , A  A1  A2 , u  (u1 , u2 ) , and ui : A  R for each i  {1, 2} . Each
player i  {1, 2} simultaneously selects an action ai  Ai , and obtains the instantaneous
payoff ui (a ) , where a  (a1 , a2 ) . I specify the component game by

A1  A2  [0,1] ,ui (a)   ai and u j (a)  0

if ai  a j ,andu1 (a)  u2 (a) t

if t  a1  a2 ,2where j  i and   0 . I call this the preemption game, and it corresponds to thesimple Bertrand duopoly. Each player i wins and obtains the winner’s payoff  ai if
her action choice ai is smaller than the rival’s action choice a j .
The preemption game involves the unique Nash equilibrium. The unique Nashequilibrium incentivizes each player to select ai  0 , which exhausts her potential
benefit. To calm this inefficiency, I assume incomplete information: each player i  Ndoes not know if her rival j is present or absent; she expects that with a positive
probability   0 , her rival j  i is absent and therefore she automatically wins andobtains  ai . With this incomplete information, the preemption game demonstrates theunique mixed strategy Nash equilibrium. This unique equilibrium incentivizes eachplayer to select positive actions with certainty.
A mixed strategy—or, to it put less formally for my purpose, a strategy—for
player i is defined as a cumulative distribution function qi : Ai  [0,1] , where qi (t )is non-decreasing and right continuous in t  [0,1] . I simply wrote qi  t when player

i selects ai  t with certainty. A strategy profile ( q1 , q2 ) is said to be symmetric if
q1  q2 . I simply wrote q instead of ( q1 , q2 ) when q1  q2  q . According to myearlier works (2013; forthcoming), the following strategy is the unique Nashequilibrium in one-shot preemption games:q * (t )  q * (t |  )  1 

and (1  t )for all t  [ ,1] ,(1   )t8

q* (t )  q* (t |  )  0 for all t  [0,  ] .
The interpretation of the preemption game is strategic technology adoption. A
company has an opportunity to develop and adopt a technology within a time interval[0,1] . If the company successfully adopts the technology at a time t  [0,1] , then it

earns  t , where  expresses the level of adoption value. Meanwhile, if thecompany’s rival adopts the technology before it does, then the company earns nothing.
While postponing the time of adoption allows a company to further develop its
technology, it increases the risk that the company’s rival will win the adoption race.
I introduce the recurrent structure as follows. Just after terminal time 1, thecompany can enter the next preemption game with a positive probability. I assume thatthe rival in the next game is not the rival in the current game. To be sure, if there is norival in the current game, then the company enters the next game with certainty.
However, if there is a rival in the current game, then the company enters the next gamewith a probability of W  [0,1] if it wins the current game, and a probability of L  1  W  [0,1] if it loses the current game.
Hence, the recurrent preemption game is a substantial modification of a standard
infinitely repeated game with random matching in that the discount factor of a player
depends on whether she wins the current game and also on whether there is a rival in thecurrent game. To be more precise, the discount factor equals 1 if the rival is absent in
the current game, W if the rival is present and the player wins the current game, and1  W (   L ) if the rival is present and wins the current game.
In the context of strategic technology adoption, the innovator (i.e., the winner) is at
risk of her competitor (i.e., the loser) taking her opportunity to participate in the next
race for technology adoption. The case that the loser will enter the next game more
likely than the winner ( W  1 ) corresponds to the prominence of the innovator’s
2
dilemma that the winner is constrained by the current technology, and therefore tends tobe behind of the loser in starting the development of the next technology. Meanwhile,the case in which the winner is more likely to enter the next game than the loser

9( W  1 ) corresponds to the prominence of the innovator’s advantage, that is, that the2winner obtains better technical and information-gathering skills than the loser.
I use V (q)  V (q |  , W ,  ) to denote the expected payoff for a player in therecurrent preemption game when all players commonly follow a strategy q . The
winner and the loser receive t  W V (q) and (1  W )V (q ) ,as their respective payoffs, where t denotes the winning time. If a player selectst  [0,1] , then she wins the current game against her rival with a probability of1

q (t )  lim q ( ) t

2

and therefore receives the expected payoff given byq ( )  q (t )  lim t
V (t , q )  (1   )  t  W V (q ) 1 

2




(1  W )V (q)q(t )  lim q( )  t   t  V (q ) ,2
that is,(1)

q(t )  lim q( )  t
V (t , q)   t 1  (1   )

2



q (t )  lim q ( )   t (1   ) W  (1  2W )    V (q) .
2


 

From (1), I define the expected payoff in the recurrent preemption game as1
q (t )  lim q( )  tV (q)   V (t , q)dq(t )    t 1  (1   ) dq(t )
2
t 0
t 0



1


q (t )  lim q ( )   tV (q )  (1   ) W  (1  2W )    dq (t ) ,2
t 0 


 

1

that is,10

q(t )  lim q( )  t
 dq(t )
2
t 0


V (q ) .
1 q(t )  lim q( )   t
1   (1   ) W  (1  2W )    dq(t )
2

 
t 0 
1  t 1  (1   )(2)

Definition 1: A strategy q is said to be a symmetric Nash equilibrium in the recurrent

preemption game if
V ( q )  V (t , q ) for all t  [0,1] .

2.2. Specification and UniquenessI specify a strategy qˆ  qˆ ( | W ,  ) as follows:(3)qˆ (t )  1 

W (1  t )for all t  [tˆ,1] ,(1   ) W t   (1  2 W )andqˆ (t )  0 for all t  [0, tˆ) ,where the critical time tˆ  tˆ(W ,  )  (0,1) is defined as(4) (1  W )
tˆ 

if

tˆ  1

if 0  W 

W

1 

 W  1,

and


1 

.

Note that q̂ is independent of  . The following theorem shows that q̂ is the uniquesymmetric Nash equilibrium irrespective of  . Since a player has an incentive to selectterminal time 1 , I reason thatV ( qˆ )  V (1, qˆ )     V ( qˆ )  (1   )(1  W )V ( qˆ ) ,that is,(5)

V ( qˆ ) .
(1   ) W

11

Theorem 1: The strategy q̂ is the unique symmetric Nash equilibrium in the recurrent

preemption game.
Proof: Consider an arbitrary symmetric Nash equilibrium q , where I assume that q (t )is continuous in t  [0,1] , and there exists t  [0,1) such that q (t ) is increasing int  [t ,1] and q ( t )  0 . From (1) and q(t )  lim q( ) , the following first order tcondition is necessary and sufficient for the Nash equilibrium: for every t  [t ,1] ,V (t , q)   {1  (1   )q(t )}   t  (1  2W )V (q ) (1   )q(t )  0 ,tthat is,(1   ) q(t ).
1  (1   ) q (t )  t  (1  2 W )V ( q )From q (1)  1 , for every t  [t ,1) , I can deriveq (t )  1 

 (1  t ),(1   )  t  (1  2 W )V ( q )which, along with (3), (4), and (5), implies thatq (t )  1 

W (1  t ) qˆ (t ) for all t  [t ,1) ,(1   ) W t   (1  2 W )andt  tˆ .

Hence, q̂ is the unique Nash equilibrium that satisfies continuity and increasingness.
I can prove that if a strategy q is a symmetric Nash equilibrium, then q (t ) is
continuous in t  [0,1] , and there exists t  [0,1) such that q (t ) is increasing int  [t ,1] and q ( t )  0 . Meanwhile, if q (t ) is not continuous, then there exists    0such that lim q ( )  q ( ) . By selecting a time slightly earlier than   , a player can  

dramatically increase her probability of winning. This implies that she does not select  , which is a contradiction. Hence, q (t ) is continuous.
From continuity, I can define  max   [0,1) : q ( )  q (0) .

12

Meanwhile, if q (t ) is not increasing in [ ,1] , then, there exist    [ ,1] and   [ ,1] such that      , q ( )  q ( ) , and the selection of   is a best response.
Since a player does not select any  in the open interval ( , ) , it follows fromcontinuity that by selecting   instead of   , she can increase the winner’s payoff
without decreasing her winning probability, which is also a contradiction. Hence, q (t )is increasing in t  [ ,1] , where I can set   t .
Q.E.D.

2.3. Discussion
The unique equilibrium strategy q̂ depends on the frequency with which thewinner enters the next game, W . The following proposition states that the more
frequently the winner enters the next game (i.e., the greater W ), the morecompetitively the players behave in equilibrium; that is, the less efficiently technologies
are adopted. Hence, the prominence of the innovator’s dilemma eases preemptivecompetition and consequently promotes more efficient technology adoption.



,1) ,
Proposition 2: For every W  (1 qˆ (t | W ,  ) 0 for all t  (t ,1) , W
V ( qˆ ( |  W ,  ))tˆ( W ,  )0.
 0 , and
 W
 W

Proof: From (3),qˆ (t | W ,  ) 2 (1  t )

 0,
W
{(1   )W t   (1  2W )}2
implying that qˆ (t | W ,  ) is increasing in  . From (4) and (5), both tˆ(W ,  ) andV (qˆ |  , W ,  ) are decreasing in W .
Q.E.D.

13

From (3) and (4), in the medium case in which the winner and the loser have thesame opportunity to enter the next game ( W  1 ), the corresponding unique2equilibrium strategy is given by (1  t )for all t  [ ,1] .
qˆ (t )  qˆ (t | 1 ,  )  1 
2(1   )tThis is the same as the unique equilibrium strategy in the one-shot preemption game,that is, I have q̂  q* in the medium case. From the specification of qˆ  qˆ ( | W ,  ) , Ihaveqˆ (t | W ,  )  q* (t |  )for all t  (tˆ,1)if W  1 ,
2qˆ (t | W ,  )  q* (t |,  )for all t  (0,1)

if W  1 .
2

while

Hence, if W  1 , then the introduction of recurrent structure makes players more
2
competitive, while if W  1 , then the introduction of recurrent structure makes2
players less competitive.
The following proposition shows that the greater possibility of the rival being
absent (i.e., the greater  ) makes technology adoption more efficient.



,1) ,
Proposition 3: For every   (0,1) and W  (1 qˆ (t | W ,  )0

for all t  (tˆ,1) ,V (qˆ ( | W ,  ))tˆ(W ,  ) 0.
 0 , and
Proof: From (3), (4), and (5), it is clear that qˆ (t | W ,  ) is decreasing, tˆ(W ,  ) is

increasing, and V (qˆ |  , W ,  ) is decreasing in  .
Q.E.D.

14

From the specification of q̂ , I find thatlim qˆ (t | W ,  )  1 for all W  (0,1] and t  [0,1] .
 0This implies full inefficiency in that the technology is immediately adopted without
development if a player expects that a rival is (almost) certainly present. From thespecification of q̂ ,lim qˆ (t | W ,  )  0 for all W  (0,1] and t  [0,1) .
 1This implies full efficiency in that the technology is adopted after it is fully ripe, if aplayer expects that a rival is (almost) certainly absent.

3. Random Technology
3.1. Formulation
I have assumed that the level of adoption value  is constant across component
games. This section considers the situation in which  is determined randomly andindependently across component games according to a cumulative distribution functionp : (0, )  [0,1] , where p ( ) is differentiable and increasing in  , lim p( )  0 , 0and lim p ( )  1 . Players are informed of the adoption value of the current technology 

but not informed about the future.
In the recurrent preemption game with random technology, I define a strategy asr  (r ( |  )) (0, ) , where r ( |  )  r ( |  , W ,  ) :[0,1]  [0,1] denotes a cumulative
distribution function according to which a player selects a time in any preemption gameassociated with  . I denote by V (r )  V (r | W ,  ) the ex-ante expected payoff in therecurrent preemption game with random technology. If the current game is associatedwith  and a player selects t  [0,1] , then she obtains the expected payoff given by(6)

r (t |  )  lim r ( |  )  t
V (t , r ,  )   t 1  (1   )

2



15


r (t |  )  lim r ( |  )   t (1   ) W  (1  2W )    V (r ) .
2


 

Let V ( r ,  ) denote the expected payoff associated with 

when all players

commonly follow q , which is defined as1

V (r, )   V (t , r, )dr (t |  )

(7)

t 0

1
r (t |  )  lim r ( |  )  t   t 1  (1   ) dr (t |  )
2
t 0




r (t |  )  lim r ( |  )   tV (r )  (1   ) W  (1  2W )    dr (t |  ) .
2
t 0 
 
1I define the ex-ante expected payoff in the recurrent preemption game with random
technology as(8)

V (r )  E [V (r , )] 1 
r (t |  )  lim r ( |  )  tE    t 1  (1   ) dr (t ) 2
 t 0 


.

1 

r (t |  )  lim r ( |  )   t
1  E   (1   ) W  (1  2W )   dr (t ) 2
 t 0 


 

Definition 2: A strategy r is said to be a symmetric Nash equilibrium in the recurrent

preemption game with random technology if for every   0 ,V ( r ,  )  V (t , r ,  ) for all t  [0,1] .

3.2. Specification and UniquenessI specify a strategy rˆ  (rˆ( |  )) (0,1] as follows. Let us specify tˆ( )  [0,1] ,V ( rˆ)  0 , and ˆ  0 according to the following five equations:(9)(1   )(1  2W )V (rˆ)
tˆ( )   
,



16



(10)

V (rˆ) (11)

V (rˆ) 

(12)

ˆ 

2  ˆ  dp ( )

(1   ) 1  (1  2W ){1  p (ˆ)}
ˆ



0



(1   )   dp ( )  2  ˆ  dp ( )(1   ) 1  (1  2W ){1  p (ˆ)}

(1   )(2W  1)V (rˆ)



if W  1 ,2

if W  1 ,2
if W  1 ,
2

and
(13)

ˆ  (1  2W )V (rˆ)

if W  1 .
2

Based on these specifications, for every   ˆ , let(14)rˆ (t |  )  1 

 (1  t )for all t  (tˆ( ),1] ,(1   )  t  (1  2 W )V ( rˆ )

andrˆ(t |  )  0 for all t  [0, tˆ( )] .

For every   (0, ˆ] , let(15)rˆ(t |  )  1 for all t  [0,1]if W  1 ,2

rˆ(t |  )  0 for all t  [0,1)if W  1 .
2

and
(16)

The following theorem shows that it is the unique symmetric Nash equilibrium.
Theorem 4: The specified strategy r̂ uniquely exists, and it is the unique symmetric

Nash equilibrium in the recurrent preemption game with random technology.
Proof: I show that r̂ uniquely exists. For each   ˆ , tˆ( ) must satisfy thefollowing equation:rˆ(tˆ( ) |  )  1 

{1  tˆ( )} 0,(1   )  tˆ( )  (1  2W )V (rˆ)

which implies (9). For r̂ to satisfy the Nash equilibrium property, the selection ofterminal time 1 must be the best response to rˆ( |  ) for each   ˆ . Hence,

17

V ( rˆ,  )  V (1, rˆ,  )    (1   )(1  W )   V ( rˆ) .

(17)

Consider W  1 . Equation (15) makes clear that for each   (0, ˆ] , a player
2
must have an incentive to select 0, which implies

V (rˆ, )  V (0, rˆ, ) 1 V (rˆ) .
2

Hence, from (17),V (rˆ)  V (rˆ | W ,  )  E [V (rˆ, )]1  p (ˆ)  (1   )(1   W )    1  p (ˆ)  ,   ˆ  dp ( )  V ( rˆ) 

2







which implies (10). Since tˆ(ˆ)  0 and (9), I have (12). From (10) and (12),ˆ 1  (1  2W ){1  p (ˆ)}  2(1  2W )  ˆ  dp ( )  0 ,

(18)



which determines ˆ . Such ˆ uniquely exists because the left-hand side of (18) is
continuous in ˆ , increasing in ˆ , negative for ˆ  0 , and positive for sufficient ˆ .
Hence, as per Equations (9), (10), and (12), tˆ( ) , V (rˆ) , and r̂ are all uniquely
defined.
Consider W  1 . Equation (16) makes clear that for each   (0,ˆ) , no player
2
must have an incentive for a time before 1, which implies

V (rˆ, )  V (1, rˆ, ) 1   V (rˆ) .
2

Hence, from (A-1),
V (rˆ)  V (rˆ | W ,  )  E V (rˆ,  ) 

ˆ



0



(1   )   dp ( )  2  ˆ  dp ( )(1   ) 1  (1  2W ){1  p(ˆ)}

which implies (11). Since tˆ(ˆ)  1 and (9), I have (13). From (11) and (13),
(19)





ˆ(1   ) 1  (1  2W ) 1  p (ˆ) ˆ(1  2W ) (1   )   dp ( )  2  ˆ  dp ( )   0 ,

0



,

18

which determines ˆ . Such ˆ uniquely exists because the left-hand side of (19) is
continuous in ˆ , increasing in ˆ , negative for ˆ  0 , and positive for sufficient ˆ .
Hence, as per Equations (9), (11), and (13), tˆ( ) , V ( qˆ ) , and r̂ are all uniquely
defined. Note that rˆ(t |  ) is continuous in t  [0,1] , irrespective of  .
The above observations prove that r̂ uniquely exists.
Notably, r̂ is the unique symmetric Nash equilibrium. Suppose W  1

2 andconsider an arbitrary symmetric Nash equilibrium r that I assume is continuous int  [0,1) and there exists   0 such thatr (0 |  )  1 for all   [0,  ) ,and, moreover, for each    , there exists t ( )  [0,1) such that r (t |  ) is
increasing in t  [t ,1] and r (t ( ) |  )  0 . Consider    . From Equation (6) andr (t |  )  lim r ( |  ) , the following first order condition is necessary and sufficient for tthe Nash equilibrium: for every t  [t ( ),1] ,V (t , r , )   1  (1   )r (t |  )t  t  (1  2W )V (r ) (1   )r (t |  )  0 .

As in the proof for Theorem 1, it follows from the unique specification of r̂ thatV ( r )  V ( rˆ) ,for every t  [t ( ),1) ,r (t |  )  1 

 (1  t ) rˆ(t |  ) ,(1   )  t  (1  2 W )V ( r )and t ( )  tˆ( ) , where, since a player has an incentive to select 1 if and only if   ˆ ,we have   ˆ . These observations imply that r̂ is the unique symmetric Nashequilibrium.
Suppose W  1

2 and consider an arbitrary symmetric Nash equilibrium

I assume is continuous in t  [0,1) and there exists   0 such thatr (t |  )  0 for all   [0,  ) and t  [0,1) ,r that19

and, for each    , there exists t ( )  [0,1) such that r (t |  ) is increasing int  [t,1] and r (t ( ) |  )  0 . In the same manner as in the case of W  1 ,2
r ( |  )  rˆ( |  ) for all    ,where, since a player has an incentive to select 1 irrespective of her rival’s strategy if
and only if   ˆ , we have   ˆ , and r (t |  )  0 for all t  1 and   ˆ . Theseobservations imply that r̂ is the unique symmetric Nash equilibrium.
As in the proof for Theorem 1, any symmetric Nash equilibrium satisfies
continuity and increasingness. Hence, the above work proves Theorem 4.
Q.E.D.

3.3. Discussion
The unique equilibrium strategy rˆ( |  ) crucially depends on  . The reason forthis dependence is that the relative importance of the future expected payoff after
entering the next game, compared to the winner’s payoff in the current game, decreases
as the current level of adoption value increases. Importantly, the manner of dependenceof rˆ( |  ) on 

substantially differs between W  1

  1 2 . This
2 and Wsubsection will show that if W  1 , then the greater the level of adoption value  ,2the less competitive a player will behave; meanwhile, if W  1 , then the greater the2level of adoption value  , the more competitive a player behaves.
Suppose that the level of adoption value is insufficient, that is,   ˆ . Accordingly,
Equation (15) suggests that if W  1 , then players are so competitive, act2
immediately, and therefore adopt the technology without development. On the other
hand, Equation (16) suggests that if W  1 , then players are so uncompetitive,2
postpone the timing to terminal time 1, and therefore adopt the technology with fulldevelopment. The following proposition shows that this tendency holds even if the levelof adoption value is sufficient.

20

Proposition 5: Suppose   ˆ . Then,1 W  2     rˆ(t |  )  0 and ˆ
t ( )  0  ,



1

 
W  2     rˆ(t |  )  0 and ˆ
t ( )  0  .



and

Proof: From (14), (1  t )2 (1  2W )V (rˆ)
rˆ(t |  ) 
2 ,
(1   )  t  (1  2W )V (rˆ)
which is negative (positive) if W  1

  1 2 , respectively). From (9),2 ( W(1   )(1  2W )V (rˆ)
 ˆt ( )  ,2which is positive (negative) if W  1

  1 2 , respectively).

2 ( W

Q.E.D.

Consider the medium case, that is, W  1 . Note that rˆ(t |  ) is independent of2

 , andrˆ( |  )  q* for all   0 .
In other words, the unique equilibrium strategy in the medium case is the same as theunique equilibrium strategy in the one-shot game. Hence, the incorporation of random
technology does not influence the players’ equilibrium behavior in the medium case.
Moreover, it is helpful to note that

lim rˆ(t |  , W ,  )  q* for all W  (0,1) and   (0,1) .
 The equilibrium strategy associated with a sufficiently high level  is approximatedby the equilibrium strategy in the one-shot preemption game. As in Section 2, ifW  1 2 , then the incorporation of recurrent structure makes players more competitive;

21

meanwhile if W  1 , then the incorporation of recurrent structure makes players less2
competitive.
From the specification of r̂ and V ( rˆ)  0 , the following proposition holds.
Notably, this proposition implies that the prominence of the innovator’s dilemma
promotes more efficient technology adoption. I write rˆ( |  )  rˆ( |  , W ,  ) .

Proposition 6: If W  1 , then2rˆ(t |  , W ,  )  q* (t ) for all t  (0,1) and   (0, ) ,while if W  1 , then2qˆ (t |  , W ,  )  q* (t ) for all t  (0,1) and   (0, ) .
In the same manner as in Subsection 2.3, the positivity of  plays a central rolein making players delay technology adoption. For every   0 and W  (0,1] ,lim rˆ(t |  , W ,  )  1 for all t  [0,1] , 0which implies full inefficiency in that the technology is immediately adopted without
development if a player expects that a rival is (almost) certainly present. For every  0 and W  (0,1] ,lim rˆ(t |  , W ,  )  0 for all t  [0,1) , 1which implies full efficiency in that the technology is adopted with full development if
a player expects that a rival is (almost) certainly absent.

4. Generalization
I can generalize the winner’s payoffs by replacing  t with v(t |  ) without any
substantial change, where I assume that v(t |  ) is continuous and increasing int  [0,1] and   (0, ) , v (0 |  )  0 , and v(1|  )   . In the recurrent preemption

game without random technology, I specify q   q  ( |  ,  , v( |  )) by22q  (t )  qˆ (As in Theorem 1, q v(t |  )

) for all t  [0,1] .

is the unique symmetric Nash equilibrium, and thecorresponding expected payoff is the same as V ( qˆ ) . In the recurrent preemption gamewith random technology, I specify r  byr  (t |  )  rˆ(v(t |  )|  ) for all t  [0,1] and   (0, ) .

As in Theorem 4, r  is the unique symmetric Nash equilibrium, and the corresponding
expected payoff is the same as V (rˆ) .

5. Conclusion
This study introduced and investigated the recurrent preemption game as theinfinitely repeated game with random matching, wherein each player’s discount factor
depends on whether her rival is present in the current game and whether she wins thecurrent game. Notably, the recurrent preemption game involves the unique symmetric
Nash equilibrium. This paper clarified the impact of the innovator’s dilemma on
equilibrium behavior from various viewpoints such as the degree of competitivenessand its dependence on the quality of technology.
It will be important for future research to generalize this study’s analysis in various
respects. For instance, the recurrent preemption game can be replaced with a richer
model that details a company’s internal organization and explains how the company
addresses the innovator’s dilemma. Moreover, in conversation with previous literatureon strategic technology adoption, my component game can be replaced with moregeneral games that incorporate various aspects of asymmetry, such as the replacement
effect (Arrow 1962), the informational spillover (Hoppe 2000; Mariotti 1992; Awaya
and Krishna 2019), the preemption effect (Gilbert and Newbery; 1982), the imitation(Katz and Shapiro 1987; Hendricks 1992), and the uncertainty of profitability (Jensen1982). To be sure, sound policymaking requires such future studies, and I hope that mystudy of the recurrent preemption game provides a solid basis for such work.

23

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