Military Technology Races
Vally Koubi
Introduction
Because of the nature of modern weapons, significant innovations in arms technology
have the potential to induce dramatic changes in the international distribution of
power. Consider, for example, the ``strategic defense initiative'' (SDI), a program
initiated by the United States in the early 1980s. Had the program been successfully
completed, it might have led to a substantial devaluation of Soviet nuclear capabilities
and put the United States in a very dominant position. It should not then come as
a surprise that interstate rivalry, especially among super powers, often takes the form
of a race for technological superiority. Mary Acland-Hood claims that although the
United States and the Soviet Union together accounted for roughly half of the world's
military expenditures in the early 1980s, their share of world military research and
development (R&D) expenditures was about 80 percent.1 As further proof of the
perceived importance of R&D, note that whereas the overall U.S. defense budget
increased by 38 percent (from $225.1 billion to $311.6 billion in real terms) from
1981 to 1987, military R&D spending increased by 100 percent (from $20.97 billion
to $41.96 billion).2 Moreover, before World War II military R&D absorbed on average
less than 1 percent of the military expenditure of major powers,3 but since then it
has grown to 11-13 percent.4 The emphasis on military technology is bound to become
more pronounced in the future as R&D becomes the main arena for interstate
competition.
In this article I examine the properties of international military R&D competition
when military technology affects the distribution of power. I develop a dynamic
I am grateful to Steve Brams, Bruce Bueno de Mesquita, Bob Grafstein, Harris Dellas, David Lalman,
two anonymous referees, and the editors of IO for many valuable comments. Part of this article was
written while I was an EU TMR fellow at CORE at the Catholic University of Louvain (Belgium).
1. Acland-Hood 1984.
2. Weinberger 1986, 313.
3. SIPRI 1974, 127.
4. Acland-Hood 1986, 23-30.
International Organization 53, 3, Summer 1999, pp. 537-565
r 1999 by The IO Foundation and the Massachusetts Institute of Technology
model in which two nations ``compete'' for the development of a new weapon in a
multistage race and where R&D is costly and its outcome uncertain. Although the
model is based on a well-known model of commercial R&D competition,5 when
factors related to distribution of power are considered, military and private technology
races differ significantly from commercial competition.
I address a set of questions pertaining to R&D spending as nations move successively,
whether simultaneously or not, through the various stages of a technological
race. In particular, I study how the amount of resources devoted to weapons R&D
changes as a nation pulls ahead, falls behind, or catches up with a rival or as a nation
moves closer to successfully completing the development process. Do nations spend
relatively more on weapons development when they are in the lead or when they are
lagging behind? How does the intensity of effort change as the technological gap
increases? Does technological parity encourage or discourage R&D efforts? How do
imitation possibilities affect spending in the various stages of development? Providing
answers to these questions will enhance our ability to explain and predict the
intensity as well as the results of existing and future military rivalries in terms of
observable technological (and other) factors.
The practical importance of this analysis can hardly be overestimated. For example,
how will weapons development programs in India and Pakistan be affected by
China's effort to close its military technological gap with the United States?Will the
intensity of the nuclear development programs in the Middle East accelerate or decelerate
if the Arab states close the technological gap with Israel? Will China's emergence
as a challenger to the United States intensify weapons development programs?
The importance of these questions for the distribution of military capabilities and
hence the probability of war (as determined, for instance, by such theories as the
balance of power) is obvious. As far as I know, no formal models yet exist in the
literature to address such questions.
The predicted race dynamics vary considerably with the type of weapon considered
and the characteristics of the nations involved in the race. The typical patterns
for races where preemption by the winner is ruled out and catching up technologically
is of critical importance for the distribution of power are as follows: The race
begins with the two competitors spending modest amounts on R&D.Abreakthrough
by one nation stimulates higher spending in both nations, with spending increasing
dramatically in the laggard nation. The laggard intensifies effort even more if it falls
further behind or if the leading nation completes development. If, on the other hand,
the laggard nation manages to catch up, it relaxes effort somewhat, whereas the
leading nation redoubles effort. Finally, a tied race becomes more intense the closer
the rivals get to completing development.
I evaluate the empirical relevance of this theory (and hence its practical usefulness)
by examining its ability to match the actual patterns observed during the U.S.-
Soviet missile and antiballistic missile (ABM) systems race. The key implication of
the model is how the level of R&D spending varies with changes in a nation's rela-
5. Grossman and Shapiro 1987.
538 International Organization
tive position in the race. In particular, the theory predicts that weapons development
programs in the United States accelerated as a result of perceptions of either falling
behind or losing its lead and decelerated as a result of perceptions of surging ahead of
the Soviet Union. The study of the actual race reveals a close match with the theory.
This finding suggests that concerns about the distribution of power are an important
determinant of military R&D spending. Existing theories of military R&D programs
(such as the theories of bureaucratic politics, military-industrial complex, and the
technological momentum) tend to discount the role of such ``national interest'' motives.
I first offer a brief literature review and then describe the model. Using numerical
simulations, I address the questions raised earlier regarding the intensity of R&D in
the different stages of the race. I conclude by discussing the empirical content of the
theory.
Literature Review
Interstate technological races have been studied mostly within the arms race literature.
Samuel Huntington draws a distinction between qualitative arms races, where
competition involves developing new forms of military force and generating technological
breakthroughs, and quantitative races, where competition simply involves
expanding existing forms of military capabilities.6 Huntington asserts that qualitative
races may be more desirable from the standpoint of international stability because of
their greater power of deterrence. Michael Intriligator and Dagobert L. Brito challenge
this view.7 They argue that quantitative races insure nations against the possibility
that some technological innovation will render them incapable of an effective
retaliatory second strike (that is, if they have a large number of weapons, enough will
probably survive a first strike to allow them to strike back). Technological races, on
the other hand, are dangerous because they may give rise to a revolutionary technological
breakthrough that renders a first strike attractive (to use the familiar Richardsonian
terms, technological improvement can lead to instability by both shrinking
the region of deterrence and expanding the region of war initiation). D. S. Sorenson
discusses how technological changes in weapon characteristics affect arms race stability.
8 He notes that since a major technological breakthrough is always possible,
competitors in an arms race will invest heavily in military R&D to minimize uncertainty,
a fact that may well lead to arms race instability and higher overall levels of
arms capacity. Finally, MichaelWolfson stresses how uncertainty can lead to escalation.
9
These studies are exclusively concerned with the implications of technology for
international stability and the probability of war and pay little attention to the charac-
6. Huntington 1958.
7. Intriligator and Brito 1976 and 1984.
8. Sorenson 1980.
9. Wolfson 1985.
Military Technology Races 539
teristics of technological rivalry and the process of weapons development. In particular,
there has been no discussion of such important issues as where new weapons are
coming from (namely, that they are the purposeful but uncertain outcome of R&D
efforts), what determines military R&D spending, or how a nation's relative and
absolute position in the technological ladder influences the intensity of competition
and hence the rate of introduction of new weapons. The only formal treatment of
technological choice in a dynamic setting that I am aware of is that byYukiko Hirao,10
who deals with one particular aspect of technological choice, namely, quantity versus
quality. Hirao assumes the existence of two steps in technological decisions: first, a
nation (or its defense establishment if the latter pursues its own interests) decides on
the quality of weapons to be acquired; second, it decides on the quantity of weapons.
The Theory
I study the behavior of two nations competing for the development of a particular
new weapon that can significantly affect military capabilities and the distribution of
military power. The weapon sought could be a single one or a system of related
weapons, offensive or defensive; it may represent an improvement of an existing
weapon or it may be a completely new system; it may be easily producible or not;
and so on. Typical examples are the atomic bomb, the MIRVs (multiple independent
reentry vehicles), theABM systems, and a space-stationed ``super-laser'' gun.
To understand the characteristics of the race one must specify (1) the ``supply''
side (that is, the cost and the technical aspects of the development process), (2) the
``demand'' side (that is, the benefits accruing from this particular weapon that make
its development desirable), and (3) the rules of the competition (that is, how each
side perceives and reacts to its rival).
The starting point for understanding the ``supply'' side lies in recognizing that
weapons development is a complex process (at least for major weapons) whose
completion typically requires going through various stages and overcoming important
intermediate hurdles, each presenting its own difficulties and uncertainties and
each serving as a prerequisite for the final product. In each of these stages, scarce
resources must be employed. Although the more one spends the greater the likelihood
of success, there is no guarantee that one's efforts will meet with success.
To capture this process in a tractable way, I will require that the development of a
weapon must go through three successive stages. In the initial stage, stage 0, no
significant progress has been made. For instance, MIRVs were in stage 0 before
restartable rocket motors and vernier vehicles were developed.11 In the next stage,
stage 1 (whose completion is a prerequisite for moving to the final stage), a nation
10. Hirao 1994.
11. A MIRV is a missile that carries multiple warheads, each able to be separately aimed and targeted.
The warheads are placed on a ``bus'' (a post-boost vehicle that carries and releases the reentry vehicles
(RVs) at precise times and directions) that is equipped with its own inertial guidance system and several
motors that can be used to change its orientation and velocity. The restartable rocket motors allow each RV
540 International Organization
has achieved a breakthrough in the form of an intermediate result, but significant
uncertainty still remains concerning its ability to successfully complete development.
The intermediate result may or may not have any value on its own, depending
on the range of its applications (for instance, the restartable rocket motor that made
MIRVs possible was initially developed for NASA). In the final stage, stage 2, all
remaining hurdles have been overcome and the sought-after weapon has (or can
easily) become available.
A three-stage specification is adopted for two reasons. First, it is realistic. For
example, the development of a relatively small thermonuclear warhead in 1953 was
the necessary intermediate step that made long-range ballistic missiles possible, and
the construction of an operational ``bus'' mechanism was the key engineering development
necessary for MIRV systems. Second, it serves as a means of modeling a
nation's absolute position (how close it is to completing the project) and its relative
position (how far it is ahead of its rivals) in a race. Adding multiple intermediate
steps is feasible but very cumbersome and, as has been demonstrated elsewhere, does
not add any new insights relative to the single intermediate step specification.12
Finally, besides engaging in original R&D efforts, nations may employ another
means of arriving at the desired result, namely, imitation (``espionage''). I will assume
that nations may be able to, perhaps imperfectly, copy technologies developed
elsewhere.
Summarizing the ``supply'' side: a nation can develop a new weapon either with
original R&D or with imitation activities. R&D involves successive stages, is subject
to uncertainty, and its probability of success can be increased by additional spending.
Now we turn to the ``demand'' side. A new weapon is conceived either as a response
to a perceived threat or as a possible source of creating an advantage vis-a`-vis
one's competitors. In either case, there are payoffs to the development of the weapon
that motivate the race. These payoffs may capture anything that nation finds valuable
about the weapon, such as economic resources of other nations that may become
prey to the new weapon, personal prestige that may accrue to the nation as a result of
success (such as completing the first trip in space), or the ability to use the new
weapon as a ``bargaining chip'' in future arms control negotiations.
The ``value'' of the weapon depends on its type, on whether the other side has
already developed or is close to developing it, and, finally, on the characteristics of
the nation (location, type of political system, existing capabilities, other weapons
possessed, type of adversaries, and so on). There is no such thing as a weapon in
abstract; instead, a particular weapon (say, a hydrogen bomb) has certain features, is
possessed by a particular nation (say, an expansionist military regime), and may or
may not be possessed by that nation's adversaries. A weapon may be valuable to a
particular nation only as long as its rival does not have it (for example, a nuclear
to be placed on a different trajectory (the main engine must stop and start again). The vernier vehicles
allow the trajectory of the MIRV to be adjusted.
12. Harris and Vickers 1987. In other words, three stages suffice to define absolute and relative position
in the race as well as distance from the finish line.
Military Technology Races 541
bomb to Nazi Germany); or its value may not be significantly compromised by simultaneous
foreign possession (for example, a nuclear bomb developed purely for deterrence
purposes). Although it may seem that different models would be needed for
different types of weapons, it will become clear shortly that all of the cases described
earlier can be analyzed within a single model by choosing the appropriate payoff
structure. I find this property of the model very appealing.
Finally, concerning the rules of the race, I make two assumptions about perceptions
and interactions. First, I assume that each side knows how far the other has
advanced. Allowing for incomplete information is straightforward. It can be done by
replacing actual with expected values as long as strategic interaction in the form of
signaling, belief manipulation, and so on, is ruled out (such considerations are certainly
interesting, but their introduction would create formidable technical difficulties13).
The most interesting aspect of expectations is that the patterns of weapon
races described in this article can be generated as a result not of an actual innovation,
but rather of a mistaken belief (self-fulfilling race dynamics). For instance, observers
have argued that much of the proliferation of R&D in the United States in the early
1960s resulted from misperceiving Russian successes.
The second assumption is that each nation behaves as if its intensity of R&D effort
does not influence R&D spending in its rivals. Although cross-nation interactions
may be modeled in alternative ways, this assumption is a simple, useful benchmark
in finite horizon games (it corresponds to the well-known Cournot competition) and
may not lack empirical content.
Finally, let me say that my analysis abstracts from issues concerning the optimal
menu of R&D activities, the trade-off between devoting resources to R&D rather
than actual production of existing weapons (qualitative versus quantitative races),
and issues of complementarities and substitutabilities. Attempting to account for these
elements would dramatically complicate the present analysis. In any case, it may not
be a major limitation, since the types of weapons I consider here are often perceived
as not having any close substitutes.
The Dynamics of the Race
The race begins with the two nations in stage 0. It ends when both of them have
developed the weapon or when one has dropped out while the other still continues. At
each point in time, a nation has a relative and an absolute position in the race. It either
leads, lags, or is tied with its rival; and it has either completed, is close to (has the
intermediate result), or is far from (does not have the intermediate result) developing
the weapon.
A nation tries to advance its position by spending on R&D, but the outcome is
uncertain. If it succeeds, then it moves to the next stage. Otherwise, it repeats the
current stage (if it chooses to remain in the race). I will postulate that the probability
of success (that is, completion of the current stage) is a function of the amount spent
13. For example, see Brams and Kilgour 1988.
542 International Organization
currently. I will also allow the relationship between spending and success to differ
across nations (to capture differences in economic and technological status, quality
of researchers, and so on) and to depend on one's absolute and relative position in the
race. This is useful for modeling intertemporal spillover effects within the nation that
successfully completes a stage (learning) or across nations (which will allow the
modeling of imitation in a simple manner).Aformal exposition of the structure of the
model and the resulting dynamics of the race can be found in the appendix. In this
section I offer a heuristic description of the decisions faced by the two rivals as they
move through the various phases of the race.
There are six phases (see the appendix for a more detailed explanation):
1. Both nations have completed development (stage 22).14
2. One nation (the winner) has already completed the project, whereas the other
one (the loser) only has the intermediate result (stage 21 for the winner and 12
for the loser).
3. Both nations have the intermediate breakthrough, but neither has completed
development yet (stage 11).
4. One nation has completed the whole project, whereas the other has not even
come up with the intermediate result (stage 02 for the loser and 20 for the
winner).
5. One nation has the intermediate result, whereas the other has nothing (stage
10 for the leader and 01 for the laggard).
6. Neither nation has the intermediate result (stage 00).
To describe the behavior of a nation at any given point in the race, we must know
the options the nation faces as well as their associated costs and benefits. A simple
way of formally summarizing all this information is through the use of the value
function, which simply computes the net benefits that accrue from pursuing development
in a particular stage. For nation A, it takes the general form
AVij52cA(Apij , AHij ) 1 AGij 1 Apij(1 2 Bpij ) AVi11, j
1 (1 2 Apij ) Bpi, j11 AVi, j11 1 Apij Bpij AVi11, j11 1 (1 2 Apij)(1 2 Bij)AVij
where AVij is the net benefit that nation A enjoys from being in stage ij and Apij is the
probability that nation A will complete stage i when its rival is in stage j if it spends
an amount equal to cA(Apij, AHij). Note that the probability of success depends not
only on the resources spent but also on the possibility of imitation (AHij).
According to this equation there are four possibilities:
14. Stage ij means that the nation under consideration is in stage i, and its rival is in stage j, where i,j 5
0, 1, 2.
Military Technology Races 543
1. Nation A succeeds, but nation B fails. In this case the game moves to the stage
i 1 1,j (from nation A's point of view). The likelihood of this event is simply
Apij(1 2 Bpij), and nation A's expected value associated with this development
is AVi11,j.
2. Nation A fails, but nation B succeeds. In this case the game moves to stage
i,j 1 1. The likelihood of this event is simply (1 2 Apij) Bpi,j11, and nation A's
expected value associated with this development is AVi,j11.
3. Both nations succeed (something that happens with probability Apij Bpij. In this
case the game moves to stage i 1 1, j 1 1, and nation A's valuation is
AVi11, j11.
4. Neither nation succeeds (something that happens with probability (1 2 Apij)
(1 2 Bpij)), in which case the status quo is maintained (AVij).
Finally, the term AGij represents the benefits (losses) that accrue within this period
simply from having to spend this period in stage ij.
I now turn to a more detailed description of the various stages of the game. The last
stage (stage 22) is of no interest from the point of view of racing, because both
countries have completed the project. Let me then describe stage 12. In this stage,
one nation has already developed the weapon (the winner), whereas the other (the
loser) only has achieved the intermediate result. The former no longer needs to devote
any R&D resources to this particular project. Moreover, as long as the laggard is
still without the weapon, the winner enjoys an improvement in the distribution of
power that translates into a per period benefit (payoff), K.15 Although this benefit is
eliminated when the loser catches up, a nation may still draw some benefits from
possession of this weapon--say, W . 0--even if others have it too (perhaps because
of deterrence reasons or because there are technological spillovers across different
weapon systems). For the sake of generality, I will also allow for the possibility that a
nation may be worse off when both nations have the weapon than when neither have
it (W , 0). The loser must decide whether to concede the race and accept the new
distribution of power or to try to catch up (and how much effort to expend in the
process).
Staying in the race is costly because a nation must devote scarce economic resources
to the development of the weapon. This direct cost of R&D depends, among
other factors, on the intensity of effort, the quality of the human and other resources
employed, and the availability of the appropriate intermediate goods. It may also
depend on whether possession of the intermediate breakthrough makes research easier
during the final stage of the project (intertemporal spillovers)--that is, whether success
breeds success, possibly because of learning--and on the effectiveness of espionage
activities that may provide solutions to some of the technical problems encountered
in this stage (imitation).
15. For an example of this type of per period cost (the scenario of Russia reverting to communism and
secretly developing a dominantABM system), seeWeinberger, Schweizer, and Thatcher 1998.
544 International Organization
At the same time, dropping out is also costly for two reasons. First, the laggard
nation suffers a loss in each and every period (denoted by R2) as long as the distribution
of power has worsened. Second, if the development of the weapon is of value to
a nation independently of how far behind one finishes (W . 0), then dropping out
means forgoing these benefits.
Consequently, the laggard selects a level of R&D spending that balances these
costs and benefits. If R&D effort proves successful, both nations have the weapon
and the race ends. If not, and the laggard decides to remain active, then the same
process is repeated again.
Let me describe some of the properties of absolute R&D intensity during this
phase. The assumption that the laggard nation suffers a per period loss as long as it
has not caught up (R2 . 0) has two important implications. First, a loser never
concedes a race. Second, as losses cumulate with each failed attempt to catch up, a
loser will try to restore the previous balance of power as soon as possible. This
translates into a high level of R&D spending. Obviously, the higher the R2, the higher
the laggard nation's R&D spending.
Is unilateral withdrawal from the race a possibility in the absence of such recurrent
losses? An incentive to remain in the race still exists even with R2 5 0, as long as
some benefit accrues to crossing the finish line independent of order (that is, when
there is a direct reward from developing the weapon even if one is not the first to do
so,W.0). But in this case, there is no great urgency on the part of the laggard nation
to complete development immediately, because losses are not cumulative. Unlike the
earlier case (with R2 . 0), which corresponds to a weapon that may be critical for the
distribution of power, this case (with R250 andW.0) may correspond to a weapon
that is developed for prestige or with the objective of being used as a possible future
bargaining chip. Obviously, R&D intensity is increasing in W.
The preceding discussion concerning the specification of the gains and losses associated
with the race highlights one key difference between my model and the standard
patent race model.16 Models of commercial R&D patent races often make a
winner-takes-all assumption (in our model, this corresponds to setting R2 5 W 5 0).
This assumption implies that the race ends when one of the two competitors crosses
the finish line, and it gives rise to dynamics that are different from those described in
this article. For instance, it makes competition more intense when the contestants are
even and has the leader outspending the follower.
The winner-takes-all assumption may be realistic for commercial races (because
of patents or the possibility of undercutting competition through predatory pricing),
but it does not seem to capture the incentives and actions observed in military technology
races. In such races there are significant gains from catching up (or losses
from failing to do so), so losers typically continue their development efforts. This is
true even in situations where the reward from belated success is smaller than that of
winning the race. For example, the development of the A-bomb and the H-bomb by
the United States did not deter the Soviet Union from developing its own nuclear
16. Grossman and Shapiro 1987.
Military Technology Races 545
weapons. Similarly, the Soviet success in launching Sputnik and testing an intercontinental
ballistic missile (ICBM) in 1957 stimulated rather than discouraged large
U.S. programs in ballistic missiles and satellite technology and led in the long run to
the initiation of such new weapon systems as MIRVs and strategic cruise missiles.
The other phases of the race can be analyzed similarly.At each point a nation must
decide whether it will stay in the race and how much to spend on R&D, knowing (or
having a perception of) where its opponent is in the race. It does so based on calculations
of the benefits and losses associated with its relative and absolute position. I
postulate that it is costly not only to lose the race but also to fall behind (because, for
example, the bargaining position of the leader is strengthened or other nations switch
alliances as a result of observing a technological advantage). I also allow for spying
activities concerning the intermediate breakthrough; for leapfrogging, that is, getting
the intermediate and the final results in one step; and for preemption,17 that is, the
lead nation using its advantage to prevent the laggard from persisting with its efforts.
Naturally, the dynamics of the race are significantly affected by each of these possibilities.
The main findings are reported in the following section. My main objective is to
characterize the intensity of the race (the amount spent on R&D) as a function of the
two nations' absolute and relative positions in the race as well as of their other
characteristics.
The Main Patterns
The model is too complex to be solved analytically, and so I have resorted to numerical
solutions (see the appendix). Naturally, the solutions depend critically on the
values of the parameters of the model, which in turn reflect the characteristics of both
the weapon sought and the nations involved in the race. To avoid having to deal with
a myriad of cases I focus on weapon systems and nations that have the following
characteristics: (1) winning the race does not lead to a preemptive strike to prevent
others from developing the weapon under consideration (because of political and/or
military limitations); (2) the unilateral development of the weapon changes the distribution
of power; and (3) nations are defense oriented--that is, they are more preoccupied
with not losing rather than with winning a race (nevertheless, winning a race
is always beneficial to the winner).
Table 1 is the benchmark case. The values selected for the benchmark case give
this weapon the characteristics described earlier. The key feature is defense orientation.
The per period loss from losing a race, R2, is greater than the corresponding gain
from winning a race, K. The benchmark case has two additional characteristics. First,
a nation is better off having a weapon even when its rival also has it (W . 0; recall
17. Leapfrogging and preemption in the context of commercial patent races are analyzed in a multistage
game by Fudenberg et al. 1983. For an extension of this model allowing for variable effort, see
Grossman and Shapiro 1987; and Harris and Vickers 1987.
546 International Organization
that W is the payoff when both have developed the weapon). Unilateral possession,
of course, is even better (K . W). Second, even the intermediate breakthrough can
induce a change in the distribution of power against the laggard, but it is not as bad as
that resulting from unilateral full development (R1 , R2; R1 denotes the loss suffered
by the laggard when its rival only has the intermediate result).
Tables 2-8 show how particular parameters affect the intensity of R&D efforts
during the various stages of the race. For instance, in Table 4 the parameter of interest
is the damage suffered by the laggard nation when it finds itself in stage 12--that is,
without the weapon--but its rival has already completed development (R2). Two
points are worth emphasizing. First, being behind is associated with more intensive
R&D relative to being in the lead or in a position of parity (p12, p01, and p02 all
identify stages in which the nation under consideration is lagging behind its rival;
p10 represents a lead; and p00 and p11 represent positions of parity in the race).
Second, increasing the cost of losing the race (from R2 5 1.5 to R2 5 2.5) intensifies
effort and spending in all but the initial stage. For instance, the probability of success
in stage 12 increases from.57 to .68, which requires that spending must increase by
about 70 percent.18
In addition to these findings, there are several other interesting patterns. First,
consider a weapon that offers an advantage to the winner as long as its rival does not
possess it (K . 0) but has no value (W 5 0) or even carries a deadweight loss (W ,
0) if both have it. If the rewards and losses are fully understood from the beginning,
this weapon may never be developed (p00 5 0 in Table 5). If for some reason a
nation initiates the development of such a weapon (for example, if it miscalculates its
18. The cost of R&D is calculated by plugging the probability of success into the cost function. In this
case, c(0.57) 5 4 3 (0.57)3 5 0.74 and c(0.68) 5 4 3 (0.68)351.25, a 69 percent increase. The cost
function is described in the appendix. The parameter values are from the benchmark case.
TABLE 1. The benchmark case
p12 p02 p01 p10 p11 p00
0.63 0.63 0.62 0.48 0.57 0.34
Note: K 5 1, W 5 0.5, R2 5 2, R1 5 0.5, a 5 b 5 g 5 1, h 5 4, z 5 3, where
K 5 payoff to the winner of the race
W 5 payoff when both have developed the weapon
R2 5 per period loss to a laggard nation whose rival has completed development
R1 5 per period loss to a laggard nation whose rival possesses the intermediate result
b 5 spillovers from the intermediate to the final result within a nation
g 5 imitation of the intermediate result
a 5 imitation of the final result
h and z 5 parameters of the R&D cost function
c( pij) 5 h( pij)z
pij 5 effort level in stage ij
Military Technology Races 547
rival's technical abilities), and if in addition its unilateral possession is important for
the distribution of power, then the rival will have no choice but to follow suit.
Second, the intensity of R&D increases as the cost of resources used for military
R&D decreases (Table 2). Moreover, if the cost of developing a particular weapon is
excessive, then the two rivals may abstain from pursuing this weapon (in Table 2,
p00 is 0.0000001 for z 5 2).19 The relative cost of R&D--in terms of consumption
forgone--tends to decrease in a fast-growing economy. For the same reason, it is
lower in more affluent societies. This tendency implies that nations such as China or
India (who have grown fast relative not only to the United States but also to nations
19. Note that a higher ``z'' means a lower cost, because p , 1.
TABLE 2. The cost of R&D
p12 p02 p01 p10 p11 p00
z 5 4 0.64 0.64 0.63 0.52 0.60 0.51
z 5 2 0.00a
Note: z 5 parameter of the R&D cost function; c( pij) 5 h( pij)z; pij 5 effort level in stage ij.
aAn entry of 0.00 indicates a number approximately equal to zero.
TABLE 3. The laggard nation's loss when the leader only has the intermediate
result
p12 p02 p01 p10 p11 p00
R1 5 0 0.63 0.63 0.60 0.48 0.57 0.29
R1 5 0.8 0.63 0.63 0.63 0.48 0.57 0.37
Note: R1 5 per period loss to a laggard nation whose rival possesses the intermediate result; pij 5
effort level in stage ij.
TABLE 4. The laggard nation's loss when the leader has completed development
p12 p02 p01 p10 p11 p00
R2 5 1.5 0.57 0.57 0.57 0.47 0.52 0.36
R2 5 2.5 0.68 0.68 0.66 0.50 0.62 0.32
Note: R2 5 per period loss to a laggard nation whose rival possesses the final result; pij 5 effort level
in stage ij.
548 International Organization
like Pakistan during the last thirty years) may have found military R&D spending
more affordable and hence may have done more of it. If these high growth rates
persist--as they are widely expected to--then the military R&D spending in these
two nations is likely to increase significantly in the future.
Third, as the gains from winning the race (K) increase, so does the effort of those
close to the finish line (whether they are in the lead or tied; see Table 6). Similarly,
increasing the penalty for losing a race (R2) intensifies effort throughout the race
(Table 4). Finally, improving the effectiveness of espionage (lowering ``a'' and/or
TABLE 5. The reward when both have developed the weapon
p12 p02 p01 p10 p11 p00
W 5 1 0.63 0.63 0.62 0.48 0.57 0.41
W 5 0 0.00
W521 0.00
Note: W 5 payoff when both nations have developed the weapon; pij 5 effort level in stage ij.
TABLE 6. The reward to the winner of the race
p12 p02 p01 p10 p11 p00
K 5 2 0.63 0.63 0.63 0.55 0.59 0.45
K 5 0.8 0.63 0.63 0.62 0.47 0.57 0.30
Note: K 5 payoff to the winner of the race; pij 5 effort level in stage ij.
TABLE 7. Imitation of intermediate and final result
p12 p02 p01 p10 p11 p00
Imitation of the intermediate result
g 5 0.8 0.63 0.68 0.67 0.48 0.57 0.28
g 5 0.6 0.63 0.71 0.70 0.48 0.57 0.21
Imitation of the final result
a 5 0.8 0.68 0.63 0.61 0.47 0.54 0.32
a 5 0.5 0.79 0.63 0.60 0.44 0.49 0.29
Note: g 5 imitation of the intermediate result; a 5 imitation of the final result; pij 5 effort level in
stage ij.
Military Technology Races 549
``g'') improves the prospects of success for the laggard nation and decreases its R&D
spending. For instance, Table 7 suggests that if research on the intermediate breakthrough
can be combined with espionage activities (say, ``g''5 0.6), then the probability
of developing the intermediate result for the laggard nation (p01) increases
from 0.62 to 0.70, but the amount spent declines by about 25 percent (4 3 0.6 3
0.73 2 4 3 0.623, see footnote 18).
One can use the findings reported in the tables to address several questions of
interest concerning how relative and absolute positions in the race affect the intensity
of R&D effort; for example,
1. Who devotes more resources to developing new technology: the nation leading
the technological race or the one lagging behind? The technological laggard
tends to devote significantly more resources to the development process
than the leader; that is, p10 , p01 (recall that pij gives the probability of success
--and hence the intensity of effort--when the county under consideration
is in stage i while its rival is in stage j, where ij 5 00, 01, 10, 02, 12, 11).
2. Starting from a position of parity at the beginning of the race, if a nation falls
behind its competitor, does it increase its efforts in order to catch up, or does
it get discouraged and lower R&D spending? Moreover, what happens to the
intensity of the laggard's effort as the distance from the leader increases? Falling
behind increases effort in order to catch up, that is, p01 . p00. Moreover,
the intensity of the laggard's effort tends to increase somewhat as the distance
from the leader increases, that is, p01 , p02.
3. If a nation that was ahead is caught up with from behind, does it intensify its
effort in order to pull ahead again? An interesting way of restating this question
is, is the rate of new weapons development higher when the two nations
are competing neck to neck or when a nation develops the weapon from a
position of technological advantage? A nation that is caught up with from
behind accelerates spending, that is, p11 . p10. Moreover, the former laggard
relaxes effort once it has restored parity, that is, p11 , p01. The total effect is
that neck-to-neck competition is more intense.
4. If, from a position of parity, a nation moves ahead of its rival, does it increase
effort or become complacent? If, starting from a position of parity, a nation
TABLE 8. Intertemporal, intranation spillovers (learning)
p12 p02 p01 p10 p11 p00
b 5 0.8 0.68 0.63 0.64 0.54 0.70 0.38
b 5 0.6 0.75 0.63 0.66 0.61 0.95 0.49
Note: b 5 spillovers from the intermediate to the final result within a nation; pij 5 effort level in stage
ij.
550 International Organization
moves ahead of its rival, then it increases spending in order to take advantage
of its lead, that is, p10 . p00.
5. Are R&D efforts higher when nations are close to completing their projects or
when they are in the beginning of the race before any significant results have
been achieved? In positions of parity, effort is lower in the beginning stages,
that is p11 . p00.
Before concluding this section, let me briefly comment on the implications of
including preemption in the model. Suppose that it is feasible--politically and militarily
--for the winner of the race to use force to prevent its rivals from continuing
with their efforts to develop the weapon under consideration (a preemptive strike
such as those carried out by Israel). This means that the last phase of the race described
earlier (stage 12 for the loser and 21 for the winner) is eliminated. This case
corresponds to the winner-takes-all specification (where the race ends when one
nation crosses the finish line) and gives rise to the standard dynamics found in commercial
patent races described earlier. The laggard is now discouraged by the success
of its rival. This is because the expected return to the laggard from persisting with
intermediate-stage R&D is now lower because the leader is closer to the finish line
and a preemptive strike--if the weapon is developed--will turn the laggard's investments
to waste. This result means that a nation like Iraq may not try to develop
nuclear weapons as vigorously as it would if it were not concerned about Israeli
preemptive strikes (that is, the Israeli strategy of preemption may have discouraged
nuclear development programs in the Middle East).
Finally, nothing in the model limits the number of nations participating in the race.
Adding a third nation increases the number of possible configurations considerably
but does not affect the qualitative results.
Empirical Aspects
The model has generated interesting predictions concerning the characteristics of
military R&D races. Its main implication is that a nation will tend to spend considerably
more when it lags behind or is tied with its rivals than when it leads. I now
examine the empirical support for this proposition by studying two cases: The U.S.-
Soviet technological rivalry and the recent Indian-Pakistani nuclear development
programs. I conclude this section with a discussion of some additional predictions.
The U.S.-USSR Rivalry
The key pattern I examine regards how military R&D spending varies as a function
of a nation's relative position in the race (the lack of relevant data poses a great
hurdle to examining the other predictions). In particular, I ask whether weapons
development programs in the United States accelerated as a result of U.S. perceptions
of either falling behind or losing its technological lead and decelerated as a
Military Technology Races 551
result of perceptions of surging ahead of the Soviet Union. In particular I focus on the
U.S.-Soviet race to develop missile andABM defense systems during the 1950s and
1960s.
I define the weapon under development as a combination of an offensive missile
and anABM system that gives a decisive strategic advantage to one of the two rivals,
for instance, a system that makes a first strike a winning proposition. Obviously, this
is a multistage development process, where generations of successive, individual
missile andABM systems represent intermediate steps that are valuable on their own
(the R1 term in equation (13) in the appendix). Moreover, unlike the model, where the
end point is fixed (there is a known finish line), this race may have an uncertain
ending point as the technological possibility frontier is pushed further out stochastically.
The race ends when missile development can no longer contribute to military
capabilities.
The close of World War II was marked by two significant technological developments:
the nuclear bomb and ballistic missiles. These two innovations created the
possibility of producing a major new weapon system, namely, the nuclear armed,
intercontinental guided ballistic missile.
The United States initiated several programs aimed at missile development (the
Snark, Navajo, and Redstone are some of the early missiles developed). These programs
intensified significantly after 1952 (six new crash programs were initiated)
mostly as a result of intelligence information that the Soviet Union had not only
made progress in the development of large long-range rockets but also enjoyed a
head start of several years in this area (a conclusion reached by the Von Neumann
committee).20 These efforts led to the development of missiles such as the Atlas,
Titan, Minuteman, and Polaris.
In August 1957 the Russians launched a test ICBM that traveled the length of
Siberia; and two months later, in October 1957, the first artificial satellite (Sputnik)
went into orbit. These events sent shock waves throughout the rest of the world as
they created the impression of a significant Soviet lead in missile development, the
so-called missile gap. This gap consisted of a hundredfold weight gap between the
first U.S. and Soviet satellites and a time gap in the launching of ICMBs (sixteen
months) and first manned orbital flights (Gagarin's flight took place ten months before
Glenn's).21 How did funding for related R&D programs in the United States
behave around the time of these two events? In the summer before Sputnik was
launched, U.S. spending on ballistic missile research and space programs had been
significantly curtailed. After Sputnik, Congress immediately passed a supplementary
defense budget that restored reductions in the missiles programs and increased the
budget of space programs beyond what it had been before the cuts.22 A number of
``exotic'' missile-space projects were also funded (such as Dyna-Soar and the Aerospace
Plane).
20. York 1970, 86.
21. Ibid., 109.
22. Ibid., 126.
552 International Organization
As a result of new intelligence information confirming that the Russians were not
enjoying a lead in missile deployment, many programs were phased out or cancelled
late in the Eisenhower administration or very early in the Kennedy administration.23
For instance, funding for the Aerospace Plane decreased from $200 million to $25
million and was discontinued in 1961 (Semi Automatic Ground Environment [SAGE]
suffered a similar fate). But this lull did not last long. In 1961 the Soviet Union
initiated deployment of an ABM system around Leningrad (in Griffon) and began a
series of high-altitude nuclear tests (tests of existing ABM war heads and also tests
aimed at developing a new X-ray-intensive ABM warhead). These activities led the
United States to conclude that large ABM deployment was imminent.24 U.S. fears
were exacerbated in 1962 when another possibleABM site near Moscow was sighted
(Khrushchev's statement in June 1962 that Soviet missiles could hit a fly in space
may also have contributed to U.S. fears). The initiation of the MIRV programs was
the direct consequence of these developments.
The picture changed again in early 1963 when it was discovered that the LeningradABMs
were too slow and poorly maneuverable to be of any effectiveness against
U.S. missiles (such as the Polaris A-3). At the same time, construction at the Moscow
site seemed to have run into problems. These events led to a slowdown in U.S. MIRV
development.25 For example, the Polaris B-3 program was postponed for at least a
year, and the Mark 12 program was delayed and nearly cancelled.
The pendulum swung in the other direction in late 1963 and early 1964 as a result
of new intelligence findings indicating Soviet progress. New SovietABM sites were
observed (in Tallin and elsewhere), old ones were upgraded with more advanced
systems (Moscow), and tests of an improved high-altitude interceptor took place at
Sary Shagan.26 As a result, U.S. MIRV development accelerated dramatically during
1964 (for example, the Mark 12 programs were accelerated and reconfigured, the
B-3 warhead was made bigger, and it was decided to develop MIRVed front ends for
Poseidon and Minuteman missiles). The MIRV development ``for the Poseidon programs
started mainly because of the uncertainty of the Tallin threat.''27
The rest of the decade continued in a similar fashion with one apparent exception.
Intelligence information in 1967 suggesting that the sophistication of Soviet ABM
systems fell short of prevailing perceptions did not lead to a slowdown in MIRV
development in the United States. Although one may interpret this behavior as reflecting
the influence of political and bureaucratic forces tied to an ongoing project, Ted
Greenwood argues that it resulted partially from U.S. fears that the Soviets might
introduce new, more advanced systems and partially from the fact that the Soviets
were developing a new interceptor and upgrading theirABM radar.28
23. Ibid., 147.
24. Greenwood 1975, 97.
25. Greenwood 1975, 99.
26. See ibid.; andWeber 1991, 189.
27. Senate Committee on Armed Services 1968 (quoted byWeber 1991, 190).
28. Greenwood 1975, 102.
Military Technology Races 553
In conclusion, the MIRV seems to be ``a prime example of an interactive, action-
reaction process driving the nuclear arms race.''29 This pattern is precisely what the
theory predicts.
It is worth noting that this pattern may not be easily accounted for by strong
versions of some other theories of R&D spending that draw a sharp distinction between
national and special group interests and emphasize solely the role of the latter
(for example, theories of bureaucratic politics, the military industrial complex, and
technological momentum). This inability is because these theories tend to predict
that R&D spending follows mostly its own course and is relatively unresponsive to
developments in rival nations (at least when reductions are necessitated as a result of
establishing a clear lead over the competitors). The empirical evidence, though, seems
to refute the thesis that the dynamics of weapons development programs are unrelated
to perceptions of national interest. By ``national interest'' here I mean the value
of the game for the nation in each phase as described in the appendix.
The Indian-Pakistani Nuclear Development Programs
I now turn to an important recent development, namely, the nuclear tests conducted
by India (May 1998). It may appear that the most plausible rationalization for these
tests is their possible political benefits for the ruling party in India. Can my model
account for this situation? The model predicts that a nation accelerates significantly
the process of weapons development when either (1) it loses its technological lead, or
(2) it finds itself lagging behind its rivals. Moreover, the latter condition is associated
with the greatest intensification. If one takes these tests to be part of the Indian-
Pakistani race, the model fails to justify them, since neither the first condition nor the
second seem to have occurred recently. This is not the case, though, if one views the
tests as part of the Indian-Chinese race (where India has a significant lag) rather than
the Indian-Pakistani race (where the rivals seem to be more or less in a position of
parity or perhaps India enjoys a slight lead). According to the Wall Street Journal,
``China has always been the focus of India's nuclear program. . . . The Indian Prime
Minister Atal Bihari Vajoayee made clear that long-unspoken truth in his letter to
U.S. President Bill Clinton and other world leaders explaining why India conducted
the initial, triple test. China got top billing, though he did not mention it by name.''30
If this is true (and my model actually validates this interpretation), then Southeastern
Asia is likely to experience a major military technology race in the future, with China
being the driving force behind it. As China becomes wealthier, it is likely to try to
challenge the United States' military hegemony. Being the technological laggard
vis-a`-vis the United States, China will have to increase R&D spending significantly
(and hence the rate of its weapons development). But this will tend to increase China's
lead over India, making India redouble its efforts to prevent this from happening.
Undoubtedly, Pakistan will be forced to follow suit to prevent India from surging
ahead.
29. Greenwood 1975, 104.
30. ``Fear of China Drives India Tests,'' The Wall Street Journal, 15 May 1998, 1.
554 International Organization
This interpretation is also consistent with India's (and hence Pakistan's) refusal to
sign the nonproliferation treaty, suggesting that the heart of the problem may be that
India lags behind China technologically rather than Indian-Pakistani distrust and
insurmountable monitoring difficulties.
Additional Empirical Observations
Some of the other questions posed in the introduction can be addressed in a similar
fashion. For instance, the model predicts that reducing Israel's technological lead
will increase Israeli R&D spending significantly and decrease somewhat the intensity
of effort by Israel's rivals. If one identifies the rate of introduction of new weapons
with p10 and p11 (the probability that a nation will develop the weapon), then the
net effect will be that the pace of introduction of new weapons in the Middle East will
accelerate. To see this, consider the benchmark case (Table 1). Israel being in the lead
translates into a probability of introducing a new weapon of 0.48 (p10). A tied race,
though, is associated with a probability of 0.57 (p11), a 20 percent increase.
The question of whether military R&D programs will accelerate when China challenges
the United States has an affirmative answer, since the theory predicts that a
competing laggard spends a lot and a challenged leader accelerates significantly.
I conclude this section by describing another possible use of my model. In the
introduction I argued that there has been a large increase in the share of post-World
War II defense budgets devoted to R&D. Although the model is not designed to
account for this pattern directly (the model contains no other types of military spending,
so spending shares cannot be calculated), it can offer some indirect insights
concerning the factors behind the large increase in the level of spending.
The model predicts that R&D intensity will increase if the rate of return to R&D
becomes high, that is, if the perceived payoffs from leading or winning (or the losses
from falling behind in) a major race go up (Table 6). The payoffs associated with the
weapons pursued/developed in the post-World War II period may indeed have this
characteristic since they have the potential to bring about significant changes in the
distribution of power. The nuclear bomb, MIRVS,ABM, and SDI systems all had the
property that a unilateral, successful development could allow a nation to dominate
world military affairs.
It must be noted that commercial technologies--which often represent spin-off
military technologies--have had very high rates of ``return'' (measured in financial
terms, market shares, and so on) from the 1950s through the early 1970s. The main
reason for these high rates of return (and hence for the desirability of R&D) may be
found in the plethora of important advances that took place duringWorldWar II and
that made additional innovations easier to generate (that is, they increased the productivity
of R&D activities). This effect is captured in my model by the parameter ``b''
(see equation (1) in the appendix). A decrease in ``b'' leads to higher R&D intensity
and spending. In addition, the frequency and extent of technological breakthroughs
in the early post-World War II period also implies that the two super powers found
themselves in the intensive R&D stages 11, 10, 01 rather than in the less intensive
stage 00.
Military Technology Races 555
Note that the share of military R&D in defense spending has stabilized over the
last twenty-five years (with the exception of the early 1980s). This may partly reflect
the realization that additional major innovations may not be feasible. According to
Herbert F. York, it has become ``harder to invent anything that can make a real
difference.''31Asimilar pattern has been observed in commercial R&D and has been
blamed for the post-1974 slowdown in productivity in the United States and other
industrialized nations.
Conclusions
Political science is becoming increasingly interested in constructing models that can
be used to forecast important political phenomena. In this article I have built a rigorous,
quantitative framework that may prove useful for explaining the dynamics observed
in interstate military technological rivalries. Admittedly, the model is stylized.
Nonetheless, it produces clear predictions regarding the intensity of effort (the use of
resources) in the nations participating in a technological race. For weapons that are
critical for the distribution of power (but cannot be used in a preemptive strike), the
typical pattern involves a great effort to close a technological gap, relative complacency
when one has the lead, and an intense race in conditions of parity when the
nations are close to developing the weapon.
The predicted patterns seem empirically plausible. They are consistent with the
U.S.-Soviet missile race in the 1950s and 1960s and the recent conduct of nuclear
tests by India. Nevertheless, many important tasks remain. It would be interesting to
attempt to introduce multiple, interrelated research projects and to rank and correlate
them in terms of ``insurance'' and national security. It may also be interesting to
consider other forms of strategic interaction (for instance, to allow for signaling and
manipulation) as well as diverse national objectives that may differ across nations.
The methodology developed in this article seems quite promising and versatile for
addressing military technology issues that play a central role in the design of modern
defense policy and are likely to prove critical for the distribution of international
power. It can also be used to address issues surrounding arms control agreements.32
Appendix
Formal Description of the Race
ASSUMPTIONS
1. The race has two participants (nations or coalitions of nations), A and B, that
compete for the development of a new weapon system. Decisions concerning
the development of new weapon systems are reached by a single leader.33
31. York 1970, 165.
32. Koubi 1998.
33. For a justification of the unitary actor assumption, see Bueno de Mesquita 1981; and Bueno de
Mesquita and Lalman 1992.
556 International Organization
2. Technological competition is centered on a single project (weapon).
3. The race has three stages. Success during the first stage of R&D produces an
intermediate result that may have some value on its own (perhaps 0) and is
also a prerequisite for success in the second stage of R&D. Allowing for leapfrogging
is technically feasible, and I describe how it can be incorporated into
the analysis. Nonetheless, I believe that one usually has to take several steps
in succession by solving a number of intermediate problems before the final
stage can be completed. For example, various technological problems in aerodynamics,
propulsion, electronic control, and explosive yield had to be solved
before effective, unmanned long-range ballistic missiles could be developed.
4. The two participants behave according to the standard Cournot model.
5. Imitation (partial or full) of the intermediate and/or the final result may be
possible.
6. There is an advantage to winning the race, but losing is also rewarded if the
loser persists and manages to develop the weapon.
7. A nation can achieve a probability of success in period t, pt, if it spends an
amount equal to c(pt), where c(0) 5 0, c8(p) . 0, and c(p) is strictly convex.
I focus attention on a subgame perfect Nash equilibrium. In each period t, each
side observes the opponent's position and then chooses its investment in R&D based
on that observation. I compute the optimal choice of effort as well as the expected
value from R&D activities in each and every stage under all possible configurations
for the positions of the two rivals. There are five possibilities: (1) neither nation has
achieved the intermediate stage; (2) one nation has completed the intermediate stage
but not the final stage, and the other nation has not achieved the intermediate stage;
(3) both nations have achieved the intermediate stage but not the final stage; (4) one
nation has achieved the final stage and the other the intermediate; and (5) one nation
has achieved the final stage, and the other has not yet completed the intermediate
stage. I will use the subscripts i and j ( i 5 0, 1, 2; j 5 0, 1, 2) to denote the phase of
the race (0 for the initial stage, 1 for the intermediate, and 2 for the final result). For
instance, ij 5 10 means that the nation under consideration has achieved the intermediate
stage, and its rival has not; ij 5 02 means that the nation under consideration is
still in the initial stage, but its rival has completed the project; and so on. Subsequently,
Ap11 denotes the probability that nation A will complete the project when
bothAand B have already achieved the intermediate result; Bp01 denotes the probability
that nation B will achieve the intermediate result when this stage has already been
achieved by its rival (nation A), and so on. Similarly, I use sVij (s 5A, B) to denote
the value of the game to nation s in stage ij. Each nation is assumed to select a level of
effort that maximizes its value function, taking as given the behavior of its rival.
I will carry out the analysis in a recursive manner, starting from the most advanced
stage, ij512, when one nation has already developed the weapon (say, nation B) and
the other only possesses the intermediate result (say, nation A). If in period t, nationA
chooses to finance an R&D intensity of Ap12, then the value of the game for nation A,
Military Technology Races 557
AV12, as a function of current and future optimal actions, is defined as follows:
AV1252aAbA cA(Ap12) 2 AR2 1 Ap12WA 1 (1 2 Ap12) AV12 (1)
where 1AV12 is the value the nation places on participating in the current phase of
the race, and Ap12 is the probability that this period's efforts will be met with success
(nationAwill end up with the desired weapon).34 The first term on the right-hand side
of equation (1) is nation A's current direct cost of R&D. The second term, AR2,
represents the loss suffered by nationAas long as the balance of technological power
remains tilted in favor of nation B. The third term, WA, captures the benefit from
catching up with a successful rival and thus restoring the distribution of power. The
fourth term corresponds to current failure to catch up--an event that occurs with
probability 1 2 Ap12--and is simply the status quo.
The parameters ``b'' and ``a'' will be used to capture spillover effects either across
stages for the same nation (learning) or across nations (imitation). The value of ``b''
(0 , b , 1) will measure the positive effects that successful intermediate-stage
research has on the effectiveness of final-stage research (a value of ``b'' less than
unity implies that success breeds success, possibly because of learning). I will use
``a'' (0 , a , 1), on the other hand, to capture the opportunity for imitating the final
result (similar opportunities will also be available for copying the intermediate result).
If ``a'' 5 0, then the weapon can be imitated at zero cost, whereas ``a'' 5 1
leaves no room for imitation. I will also allow for imitation opportunities of the final
result to differ from those of the intermediate one.
Note that if AR2 . 0, then nation A will never give up its pursuit of this particular
weapon (because if it does, it will keep on suffering a penalty indefinitely). Moreover,
the laggard nation wants to have this weapon as soon as possible in order to
stop suffering this loss (one could set R2 5 0 to model weapons of lesser importance
that would allow a nation to drop out unilaterally). Similarly, an incentive to remain
in the race exists even when AR2 5 0 as long as WA . 0 (that is, when there is a direct
reward from developing the weapon even if both have it). But since WA is a lump
sum, it does not induce any great urgency in the laggard nation's effort to complete
this project (so AR2 5 0 and WA . 0 may refer to a weapon that is developed for
prestige or with the objective of being used as a possible future bargaining chip).
Solving equation (1) for AV12 gives
AV12 5
2aAbA cA(A p12) 2 AR2 1 Ap12 WA
Ap12
(2)
The optimal choice of effort dictates setting dAV12/dAp1250 in equation (2), which,
after some manipulation that makes use of the definition of AV12 from equation (2),
34. In general, sVij, s 5 A, B, is the value that the nation attributes to staying in the race when it finds
itself in stage i and its opponent is in stage j (i, j 5 0, 1, 2).
558 International Organization
gives
AV12 5 WA 2 aAbA c8A(Ap12 ) (3)
where c8 is the marginal cost. Combining equations (2) and (3) gives an equation
in Ap12, namely
AR2 2 aAbA c8A(Ap12)Ap12 1 aAbA c(Ap12) 5 0 (4)
Equation (4) determines the optimal choice of effort, Ap12. Substituting this value
into equation (3) gives the corresponding maximized value of the value function,
AV12.
I now move back one step and calculate the optimal effort and the expected payoffs
when both nations have achieved the intermediate (but not the final) result. If, in
period t, nationAfinances an R&D intensity of Ap11, and nation B an intensity of Bp11,
then the value of the game for nation A, AV11, as a function of current and future
optimal actions is defined as follows:
AV1152bAcA(Ap11) 1 Ap11(1 1 Bp11)AV21
1 (1 2 Ap11)Bp11 AV12 1 Ap11 Bp11WA 1 (1 2 Ap11)(1 2Bp11)AV11 (5)
The first term on the right-hand side of equation (5) is nationA's current direct cost
of R&D. There are four possible outcomes following the investment of Ap11 and Bp11
by nationsAand B, respectively:
1. Nation A succeeds in developing the new weapon (an event that occurs with
probability Ap11) and at the same time nation B fails (an event that occurs with
probability 1 - Bp11). Nation A then receives a payoff of AV21, and the expected
value for nation A that is associated with this outcome is given by the
second term of equation (5). AV21 is given by the following expression:
AV21 5 (1 2 Bp12)KA 1 Bp12WA 1 (1 2 Bp12)AV21 (6)
where KA is nation A's per period gain from having developed the weapon while
nation B has not, and WA is the benefit to nationAwhen both nations have completed
the R&D process.
2. Nation A fails while nation B succeeds. The game then moves to the stage
described earlier, where the payoff to nation A is AV12 (the third term).
3. Both nations succeed (the fourth term). The payoff to nation A is then WA.
4. Both nations fail (the last term in equation (5)). The value of the game in the
next period will be identical to that of the current period because the two rivals
will find themselves in exactly the position in which they started out dur-
Military Technology Races 559
ing this period; that is, having achieved the intermediate result and still seeking
the final one.
Solving equation (5) for AV11 results in
AV115
2bAcA(Ap11)1Ap11(12Bp11)AV211(12Ap11)Bp11AV121Ap11Bp11WA
12(12 Ap11)(12Bp11)
(7)
NationAchooses an effort level Ap11 in order to maximize equation (6), taking Bp11
as given. Taking the derivative of AV11 in equation (7) with regard to Ap11, setting it
equal to zero, and using the definition of AV11 from equation (6) in the resulting
expression gives
AV11 5
2bA c8A
(Ap11) 1 (1 2 Bp11)AV21 2 Bp11 AV12 1 Bp11 WA
(1 2 Bp11)
(8)
Combining equations (7) and (8) gives an equation in Ap11 and Bp11. Using the
nation B counterpart to equations (7) and (8) produces another equation in Ap11 and
Bp11. Solving these two equations simultaneously gives the optimal values of Ap11 and
Bp11 (and substituting these values in equation (7)--or (8)--and in the corresponding
equations for nation B gives the maximized value of AV11 and BV11).
I now move back one step and calculate the optimal effort and the expected payoffs
when one nation has achieved the intermediate result but the other has not (the
former nation is said to lead the technological race). The expected value of the game
for the leader (say, nation A) is given by
AV1052bA cA(Ap10) 1 Ap10(1 2 Bp01)AV20 1 Ap10 Bp01 AV21
1 (1 2 Ap10)Bp01V11 1 (1 2 Ap10)(1 2 Bp01)V10 (9)
The first term on the right-hand side of equation (9) is the current cost flow. The
second term is the leading nation's expected benefit if it moves further ahead of its
rival, that is, if it completes the project (which happens with probability p10) and its
rival fails to come up with the intermediate result. Let us define its payoff in such a
phase, V20, as
AV20 5 K 1 Bp01 AV21 1 (1 2 Bp01)AV20 (10)
The third term in equation (9) is the leading nation's benefit if it completes the
project but at the same time its rival produces the intermediate result.
560 International Organization
If, on the other hand, nation A fails (which happens with probability 1 2 p10) and
the laggard succeeds, then the race becomes a tie and the game moves to the next
stage ij 5 11, which was described earlier (the third term in equation (9)). Finally, if
the leading nation fails to cross the finish line and the laggard does not achieve the
intermediate result (which happens with probability 1 2 p01), then the game in the
next period starts from the same position (the last term in equation (9)). Solving
equation (9) for AV10 gives
AV105
2bAcA(Ap10)1Ap10(12Bp01)AV201Ap10 Bp01 AV211(12Ap10)Bp01 AV11
12(12Ap10)(12Bp01)
(11)
The optimal choice of Ap10 satisfies
AV10 5
2bAc8A(Ap10) 1 (1 2 Bp01)AV20 1 Bp01 AV21 2 Bp01 AV11
(1 2Bp01)
(12)
Equating equation (12) to equation (11) gives an equation in Ap10 and Bp01.
I now turn to the maximization problem faced by the laggard in the race (nation
B). Its value function is described by equation (13):
BV0152gBcB(Bp01) 2 BR1 1 Bp01(1 2 Ap10)BV11 1 Bp01 Ap10 BV12
1 (1 2 Bp10)Ap10 BV02 1 (1 2 Ap10)(1 2 Bp01)BV01 (13)
The first term in equation (13) is the direct cost of R&D. The parameter ``g''
captures the opportunity for imitating (copying) the intermediate result (which is
already available to the other nation). g 5 0 implies costless imitation (that is, the
result is achievable without the imitating nation doing any of its own R&D), and
g 5 1 implies no imitation at all. The second term in equation (13), R1, measures the
loss suffered by the laggard in the race. It is positive if the intermediate result can be
used to influence the balance of power (for instance, if other nations switch allegiance
toward the likely winner of the race). The third term describes the reward to
the follower from catching up with the leader (by developing the intermediate result).
The fourth term corresponds to success in both nations: The laggard gets the intermediate
result, but at the same time the leader completes the project. The last term
corresponds to the status quo (both nations fail in their respective projects); and the
fifth term represents the worst possible scenario for the laggard, namely, its falling
further behind the leader (the leader crosses the finish line, while the follower has yet
to come up with the intermediate result). The value function in the last case is BV02
and will be derived shortly.
Military Technology Races 561
Solving equation (13) for V01 gives
Taking the derivative of BV01 with regard to Bp01 and setting it equal to zero produces
BV01 5
2gBc8B(Bp01) 1 Ap10 BV12 1 (1 2 Ap10)BV11 2 Ap10 BV02
(1 2 Ap10)
(15)
Combining equations (14) and (15) results in another equation in Ap10 and Bp01.
This equation together with the one derived by combining equations (11) and (12)
can be solved for the optimal values of Ap10 and Bp01 (which can then be used to
derive the corresponding values of V10 and V01).
I now describe the optimization problem faced by a nation (say, B) that has fallen
two steps behind its rival. Its value function is
BV0252g*B cB(Bp02) 2 BR*2 1 (1 2 Bp20)BV02 2 Bp02 BV12 (16)
The imitation coefficient, g*B
, may now be different from that in the ij 5 01 case,
because the imitation set is different (both the intermediate and the final result are
now available in nation A). Similarly, the loss associated with the change in the
balance of technological power, R2*, may be different from that arising when the
follower has the intermediate result (R2). Solving equation (16) gives
BV02 5
2g*B cB(Bp02) 2 BR*2 1 Bp02 BV12
Bp02 (17)
And setting the derivative of BV02 with regard to Bp02 equal to zero
BV0252g*B c8B(Bp02) 1 BV12 (18)
The optimal level of effort of nation B is found by combining equations (17) and
(18).
Turning now to the initial phase of the race (ij 5 00) and using the same methods
applied earlier, we find that the value function of nationAis
AV005
2cB(Ap00)1Ap00(12Bp00)AV101Ap00B p00AV111(12Ap00)Bp00V01
12(12Ap00)(12Bp00)
(19)
BV01 5
2gB cB(Bp01) 2 BR1 1 Bp01 Ap10 BV12 1 Bp01(1 2 Ap10)BV11 1 (1 2 Bp01)Ap10 BV02
1 2 (1 2 Ap10)(1 2 Bp01)
(14)
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and that the optimal choice of Ap00 satisfies
AV005
2c8A
(Ap00)1(12Bp00)AV101Bp00 AV112Bp00 AV01
(12Bp00)
(20)
Equations (1)-(20), together with their nation B counterparts, determine Ap00, AV00,
Bp00, BV00.
The complete solution of the model is described by equations (1)-(20). These
equations are nonlinear, which makes the derivation of analytical solutions not feasible.
Although some comparative statics can be carried out even without analytical
solutions, many important questions require knowledge of the levels of effort in the
various stages of the race rather than just the direction of change. Consequently, I
have resorted to numerical methods. I have also relied on a symmetric equilibrium in
order to focus more clearly on the role played by relative and absolute position rather
than by asymmetries (results obtained in an asymmetric equilibrium are available on
request). In such an equilibrium no nation subscript is needed, and moreover Bp21 5
Ap125p12.
Preemption
Suppose that a preemptive strike is feasible both politically and militarily; that is, it is
feasible for the winner of the race to use force to prevent its rivals from continuing on
with their efforts to develop the weapon under consideration. This case can be studied
adequately within the present framework by choosing the appropriate parameter
values. An effective preemptive strike could be modeled by eliminating stage ij 5
12; that is, by assuming that once a single nation has developed the weapon the race
ends.
Leapfrogging
Such a possibility can be easily incorporated into the analysis by allowing R&D in
the beginning of the race (ij 5 00) to be associated with a positive probability, q, of
generating the final result. NationAthen can achieve the final result with probability
qA; the intermediate result with probability (1 - qA) Ap00; and neither with probability
(1 - qA)(1 - Ap00). The corresponding probabilities for nation B are qB, (1 - qB)Bp00
and (1 - qB)(1 - p00), respectively. Subsequently, equation (19) takes the form
AV0052cA(Ap00)1qA[qBK*1(12qB)Bp00K1(12qB)(12Bp00)K] (21)
1(1 2 qA)Ap00[qB AV12 1 (1 2 qB)Bp00 AV11 1 (1 2 qB)(1 2 Bp00)AV10]
1(12qA)(12Ap00)[qBAV021(12qB) Bp00AV011(12qB)(12Bp00)AV00]
Military Technology Races 563
Equation (20) can be used to study how the possibility of leapfrogging affects the
dynamics of the race (for arbitrary values of q).
Numerical Solution
To solve the model numerically, one must first parameterize it. The model contains
several parameters: the cost function, c; the rewards attained from the completion of
the R&D process (K, W); the losses that result from unfavorable developments in the
balance of technological power (R1, R2 and R2*); imitation opportunities (a, g, and
g*); and learning, b.
Unfortunately, nothing in the literature can help us to select realistic parameter
values (that is, to calibrate the model). Although some degree of arbitrariness is
inevitable, some choices are restricted not only by technological considerations (for
instance ``a'' must be between zero and unity) but also by the fundamental characteristics
of the technological race under consideration. For instance, consider the type of
weapon sought. If the weapon is such that it does not matter in the long run who
develops it first, as long as both nations develop it, then K and W ought to be comparable
in size. A good example is the nuclear bomb (in the absence of a preemptive
strike), which gave the United States only a temporary advantage. On the other hand,
if the first introduction of a new weapon can permanently change the balance of
power so that the winner's advantage is not eroded by the laggard's success, then the
benefits to the winner of the race must be set to exceed considerably the benefits from
belated success (K is large relative to W). Similarly, if nations mostly care about not
losing a race rather than winning one (defense orientation), then R2 ought to be larger
than K. If nations are worse off when both have the weapon than when both have it,
then W , 0. In a similar vein, one can argue that the loss suffered by the laggard
nation as a result of its rival's development of the intermediate result is likely to fall
short of that suffered when its rival achieves full development of the weapon (R2 .
R1).
I assume that the cost function takes the form cs(spij) 5 hs (spij)z, s 5A, B, i 5 0, 1,
j 5 0, 1, 2, h . 0, z . 1 (convexity). Given a set of values for the parameters of the
model, say G1, the model was solved numerically in a recursive manner as follows:35
I started with stage ij 5 12. Equation (4) was used to determine the optimal value of
Ap12 (and nation B's counterpart for Bp12). That value was then used in equation (3) to
compute AV12 (and BV12 was calculated similarly). I then moved one step back to
stage ij 5 11 and used equations (7) and (8) as well as their nation B counterparts--
together with the value already computed from the previous step value of AV12 and
BV12--in order to calculate Ap11, Bp11, AV11, and BV11. I then used the computed values
in the calculation of the optimal values of p and V in stage ij 5 02. I continued in a
similar fashion with ij 5 01, ij 5 10, and ij 5 00 until the complete time series of
35. The numerical analysis was carried out using MATHEMATICA. The program is available from the
author on request.
564 International Organization
spij(G1) was computed. I then repeated the process using a different configuration of
parameters, Gn, n 5 2, 3.
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