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Journal of Strategic Studies
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How does the offense-defense balance scale?
Ben Garfinkel & Allan Dafoe
To cite this article:  Ben Garfinkel & Allan Dafoe (2019) How does the offense-defense balance
scale?, Journal of Strategic Studies, 42:6, 736-763, DOI: 10.1080/01402390.2019.1631810
To link to this article:  https://doi.org/10.1080/01402390.2019.1631810
Published online: 22 Aug 2019.
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ARTICLE
How does the o ﬀense-defense balance scale?
Ben Gar ﬁnkel and Allan Dafoe
Future of Humanity Institute, University of Oxford, Oxford, United Kingdom of Great Britain
and Northern Ireland
ABSTRACT
We ask how the o ﬀense-defense balance scales, meaning how it changes
as investments into a con ﬂict increase. To do so we o ﬀer a general
formalization of the o ﬀense-defense balance in terms of contest success
functions. Simple models of ground invasions and cyberattacks that exploit
software vulnerabilities suggest that, in both cases, growth in investments
will favor o ﬀense when investment levels are su ﬃciently low and favor
defense when they are su ﬃciently high. We refer to this phenomenon as
oﬀensive-then-defensive scaling orOD-scaling .S u c hs c a l i n ge ﬀects may help
us understand the security implications of applications of arti ﬁcial intelli-
gence that in essence scale up existing capabilities.
KEYWORDS Oﬀense-defense theory; emerging technologies; strategic stability
Introduction
Consider two neighboring countries which, through steady investments,
grow their armies to twice their initial sizes. How should we expect this
development to a ﬀect each country ’s capacity to invade the other?
Similarly, if investments into cybersecurity and into cyberattacks both
double, should we expect successful attacks to become more or less
feasible? What about investments int o missile and anti-missile systems,
into terror and counter-terror groups, or into biological weapons and
biosecurity?
What unites these questions is a concern with how the balance
between o ﬀense and defense scales , meaning how it changes as invest-
ment levels grow. Such questions have not been well-studied in the ﬁeld
of international security. However, t hey have theoretical and practical
importance. Asking them, we will sh ow, reveals ambiguities in classic
formulations of the concept of an o ﬀense-defense balance. Answering
them may also clarify the future prospects for security in a number of
CONTACT Ben Gar ﬁnkel benjamin.gar ﬁnkel@philosophy.ox.ac.uk Future of Humanity
Institute, University of Oxford, Oxford, United Kingdom of Great Britain and Northern IrelandJOURNAL OF STRATEGIC STUDIES
2019, VOL. 42, NO. 6, 736 –763
https://doi.org/10.1080/01402390.2019.1631810
© 2019 Informa UK Limited, trading as Taylor & Francis Group
domains. The vertical proliferatio n of emerging weapons systems, such
as cheap drones for use in swarms, might favor either o ﬀense or defense
depending on the nature of the associated scaling e ﬀects.
This article is divided into four sections. In the ﬁrst section, we
consider the concept of an o ﬀense-defense balance and clarify what it
means for the balance to scale. Our discussion draws upon the concept
of a contest success function (CSF), which can be used to relate the
expected outcome of an attack to investments made by the attackerand the defender. It is then natural to interpret the o ﬀense-defense
balance associated with the attack as the relative e ﬃcacy of the two
actors ’investments. We consider a pair o f formal metrics that describe
this relative e ﬃcacy. We note that both metrics are themselves functions
of the current investment level, meaning that they may change asinvestments grow.
In the second section, we consider two special cases of con ﬂict: ground
invasions and cyberattacks that exploit undiscovered software vulnerabil-ities. We present models suggesting that, in both of these cases, scaling up
investments is likely to beneﬁ t the attacker when investment levels are
suﬃciently low and beneﬁ t the defender when investment levels are suﬃ -
ciently high. We refer to this pattern as OD-scaling, for Oﬀensive-then-
Defensive scaling.
In the third section, we explain this commonality. The general mechanism
is this: First, initial investments increase the attacker ’s ability to exploit
points on an attack surface where the defender has provided relativelyless coverage ( gap exploitation ). Second, beyond a certain investment
level, the defender begins to saturate the attack surface, reducing any
opportunities the attacker derives from di ﬀerences in the two actors ’pat-
terns of coverage ( defensive saturation ). We describe a set of abstract con-
ditions that are suﬃ cient for OD-scaling to emerge and use them to analyze
additional cases. We argue that the conditions are unlikely to be satis ﬁed in
the context of missile defense but may be satis ﬁed in the context of defense
against drone swarms.
In the ﬁnal section, we argue that scaling e ﬀects are one of the core
pathways by which technological progress can shift the relative e ﬃcacy
of oﬀense and defense. They may also play an increasingly large role in
producing future shifts. We note that progress in arti ﬁcial intelligence
and in robotics will, in e ﬀect, scale up the number of weapons plat-
forms that actors deploy, as well as the number of software vulner-
abilities that they can discover. The role of this progress in reducing orTHE JOURNAL OF STRATEGIC STUDIES 737
exacerbating threats to security will depend to a signi ﬁcant extent on
the associated scaling e ﬀects.1
Understanding the o ﬀense-defense balance
The o ﬀense-defense balance
Informally, the oﬀense-defense balance refers to the relative ease of carrying out
and defending against attacks.2Nearly all accounts, including both classic and
contemporary ones, agree on this much. Accounts then diverge on a number of
points, particularly on the concept ’s domain of applicability.3The earliest
authors to discuss the concept used it to speci ﬁcally describe the ease of
capturing an opponent ’s territory through the defeat of their army. However,
it has become increasingly common to apply the concept to other objectives
and forms of con ﬂict. Most notably, the cybersecurity and cyberwarfare litera-
ture includes frequent discussions of the ‘cyber o ﬀense-defense balance. ’4
We will adopt this more ﬂexible conception, under which it is sometimes
more appropriate to speak of ‘an’oﬀense-defense balance than ‘the ’
oﬀense-defense balance. An o ﬀense-defense balance is de ﬁned relative to
a particular objective, form of con ﬂict, and set of actors. Most discussions of
how ‘the ’oﬀense-defense balance has evolved over time, for instance, can
be interpreted as discussions of the o ﬀense-defense balance speci ﬁcally for
European states attempting to capture territory.5
1There are of course many other pathways, beyond shifting the o ﬀense-defense balance, by which
emerging technologies might alter threats. Other pathways also discussed in this special issue
include shifting o ﬀense-defense distinguishability (discussed by Volpe), creating ﬁrst-strike instability
(discussed by Horowitz), and shifting the likelihood of con ﬂict escalation (discussed by Talmadge).
Todd Sechser, Neil Narang, and Caitlin Talmadge, ‘Emerging Technologies and Strategic Stability in
Peacetime, Crisis, and War ’,Journal of Strategic Studies 42/6 (2019); Tristan Volpe, ‘Dual-Use
Distinguishability: How 3D-Printing Shapes the Security Dilemma for Nuclear Programs ’,Journal of
Strategic Studies 42/6 (2019); Michael C. Horowitz, ‘When Speed Kills: Autonomous Weapons Systems,
Deterrence, and Stability ’,Journal of Strategic Studies 42/6 (2019); Caitlin Talmadge, ‘Emerging
Technologies and Intra-War Escalation Risks: Evidence from the Cold War, Implications for Today ’,
Journal of Strategic Studies 42/6 (2019). For a discussion of the role of arms control in managing such
threats, also in this issue, see Heather Williams, ‘Asymmetric Arms Control and Strategic Stability:
Scenarios for Limiting Hypersonic Glide Vehicles ’,Journal of Strategic Studies 42/6 (2019).
2Robert Jervis, ‘Cooperation under the Security Dilemma ’,World Politics 30/2 (1978).
3Charles L. Glaser and Chaim Kaufmann, ‘What is the O ﬀense-Defense Balance and How Can We
Measure It? ’,International Security 22/4 (1998).
4Rebecca Slayton, ‘What is the Cyber O ﬀense-Defense Balance? Conceptions, Causes, and Assessment ’,
International Security 41/3 (2017); Lucas Kello, ‘The Meaning of the Cyber Revolution: Perils to Theory
and State-Craft ’,International Security 38/2 (2013); Erik Gartzke and Jon R. Lindsay, ‘Weaving Tangled
Webs: O ﬀense, Defense, and Deception in Cyberspace ’,Security Studies 24/2 (2015); Jacquelyn
Schneider, ‘The Capability/Vulnerability Paradox and Military Revolutions: Implications for
Computing, Cyber, and the Onset of War ’, Journal of Strategic Studies 42/6 (2019).
5This more ﬂexible conception can also accommodate both dyadic and systemic o ﬀense-defense
balances. A dyadic balance is deﬁned relative to a concrete pair of actors (such as England and
France). A systemic balance is deﬁned relative to a system of actors (such as 19th century Europe) and
describes the relative ease of o ﬀense and defense for something like a typical or idealized pair of
actors in the system.738 B. GARFINKEL AND A. DAFOE
A number of scholars have argued that o ﬀense-defense balances play a
signiﬁcant role in determining the risk of con ﬂict breaking out ( ‘crisis stabi-
lity ’) and the incentives to arms race ( ‘peacetime stability ’). Jervis argues that
when the o ﬀense-defense balance in a system tilts toward o ﬀense, this
exacerbates the security dilemma.6The easier o ﬀense is relative to defense,
the more actors should feel threatened by one another and the more
inclined they should be toward the use of preventative attacks as a means
of reducing threats. Van Evera lists several other reasons why an o ﬀensively-
tilted balance may produce instability.7
On the other hand, other authors have also suggested that o ﬀense-
dominance within particular domains might actually support stability.8We
will not weigh in on this debate, other than to note that on a number of
diﬀerent views o ﬀense-defense balances can play signi ﬁcant roles in shap-
ing the frequency and character of con ﬂict.
Formalism
Contest success functions
In his foundational article on the subject, Jervis suggests that there are two
ways to describe the relative ease of o ﬀense and defense.9First, there is the
relative e ﬃcacy of investments used to conduct attacks and investments
used to defend against attacks. Second, there is the advantage that one
gains by being the actor who moves ﬁrst in a con ﬂict.
Most subsequent analyses of the concept have focused on only the ﬁrst
of these two components. Glaser and Kaufman argue that the oft-ignored
second component, ‘ﬁrst-strike advantage, ’ought to be separated out as an
analytically distinct concept.10We accept this suggestion and place our
focus on the e ﬃcacy of investments.
The o ﬀense-defense balance is then a partial characterization of the
relationship between o ﬀensive investments, defensive investments, and
attack outcomes. Before attempting to give the concept a more precise
deﬁnition, we ﬁnd it useful to ﬁrst describe what a more complete char-
acterization would entail.
LetAbe a measure of a potential attacker ’s investments. Let Dbe a
measure of a potential defender ’s investments. Let Sbe a measure of attack
6Jervis, ‘Cooperation under the Security Dilemma ’.
7Stephen Van Evera, ‘Oﬀense, Defense, and the Causes of War ’,International Security 22/4 (1998).
8For an argument that o ﬀense-dominance on land might increase the variance of military outcomes,
and thus cause risk-averse leaders to refrain from initiating war, see James D. Fearon, ‘The O ﬀense-
Defense Balance and War Since 1648 ’, draft paper (1997). For an argument that o ﬀense-dominance in
cyberspace can support defense in physical domains, see Erik Gartzke, ‘The Myth of Cyberwar:
Bringing War in Cyberspace Back Down to Earth ’,International Security 38/2 (2013): 67.
9Jervis, ‘Cooperation under the Security Dilemma ’.
10Glaser and Kaufmann, ’What is the O ﬀense-Defense Balance and How Can We Measure It? ’.THE JOURNAL OF STRATEGIC STUDIES 739
success. Then we can consider a function F(A,D) that describes, for any pair
of investments, the degree of success that the attacker can expect to
achieve.
ES½/C138¼FA ;DðÞ
This function is a generalization of two-player contest success functions
(CSFs ) that economists sometimes use to study competition and con ﬂict. We
will therefore adopt the name.11
There will typically be more than one plausible measure that can be used
to describe the outcome of an attack. For instance, for an attempt to seize
territory from another state, one might use the area of territory captured,
the economic value of the territory captured, or simply a binary variable that
represents whether or not the attack was a ‘success. ’
Broadly, there are also three varieties of measures one can use to
describe investments into a con ﬂict. First, one can use a measure of the
resources invested into the con ﬂict. Second, one can use a measure of the
capabilities invested into the con ﬂict. Third, one can use a measure of the
utility lost through investments into the con ﬂict.
Consider the case of a ground invasion. The two actors ’investments
could be measured in dollars spent, in Armored Division Equivalents
(ADEs), or in the utility cost. As there are trade-o ﬀs associated with each,
no measure is universally more useful than the others. For instance, utility
lost is often the quantity most relevant to strategic analysis, but is hard to
measure. By contrast, dollars spent is easier to measure, but may require
additional assumptions to be theoretically relevant.
Metrics of the o ﬀense-defense balance
The most common metric for the relative e ﬃcacy of o ﬀensive and defensive
investments is the minimum investment ratio that would allow the attacker
to achieve a particular degree of success. One example of this approach is
provided by Lynn-Jones:
[The o ﬀense-defense balance] is the o ﬀense/defense investment ratio required for
the o ﬀensive state to achieve victory. If, for example, a state must invest $3
million in military capabilities in order to mount a successful o ﬀensive against a
state that invests $1 million in its military capabilities and adopts a defensive
strategy, the o ﬀense-defense balance is 3:1 .12
11Although they have not played a signi ﬁcant role in the o ﬀense-defense balance literature, CSFs have
been applied to the study of military con ﬂict. The standard formalism for CSFs di ﬀers from the one
we use primarily by assuming a binary success metric. Michelle R. Gar ﬁnkel and Stergios Skaperdas,
‘Economics of Con ﬂict: An Overview ’,Handbook of Defense Economics 2 (2007); James D. Fearon,
‘Cooperation, Con ﬂict, and the Costs of Anarchy ’,International Organization 72/3 (2018).
12Sean M. Lynn-Jones, ‘Oﬀense-Defense Theory and its Critics ’,Security Studies 4/4 (1995).740 B. GARFINKEL AND A. DAFOE
Another example is provided by Glaser and Kaufman:
We prefer to de ﬁne the o ﬀense-defense balance as the ratio of the cost of the
forces the attacker requires to take territory to the cost of the forces the defender
has deployed .13
To extend these de ﬁnitions to cover additional forms of con ﬂict, such as
cyberattacks, one can simply replace their conquest-speci ﬁc language.
Roughly, this is the approach taken by many authors who discuss the
‘cyber o ﬀense-defense balance. ’14One can also extend them to cover non-
monetary measures of investment.
However, even with such modi ﬁcations, these de ﬁnitions are not su ﬃ-
ciently general. They contain strong implicit assumptions about the forms of
the contest success functions that underlie con ﬂict.
First, Lynn-Jones ’deﬁnition implies that there is some absolute invest-
ment ratio that is necessary and su ﬃcient for victory. If $3 million in
oﬀensive investments is just enough to o ﬀset $1 million in defensive invest-
ments, then, given that the ratio is the same, it follows that $6 million must
be just enough to o ﬀset $2 million. Glaser and Kaufman ’sd eﬁnition man-
ages to avoid this assumption, by ﬁxing the ratio to a particular level of
investment made by the defender. Although they do not highlight this
point, their de ﬁnition transforms the o ﬀense-defense balance from a con-
stant to a function of what the defender invests.
Both de ﬁnitions also contain a second implicit assumption, that success is
a binary variable and that investment levels produce success deterministi-
cally. While this may often be a useful simplifying assumption, there are also
many cases where it is inappropriate. An attempt to disrupt another coun-
try’s infrastructure with cyberattacks, for instance, might be most naturally
described in the language of uncertainty and degrees of success.
To remove this limitation, we o ﬀer the following metric as a general-
ization of these authors ’conceptions of the o ﬀense-defense balance. For a
given contest success function Fand threshold of expected success S*,w e
deﬁne the ratio-for-success R S∗as a function of the defender ’s investments.
RS/C3ðDÞ;D
minfA:FðA;DÞ¼S/C3g
This represents the ratio of the defender ’s investment to the minimum o ﬀen-
sive investment that would allow the attacker to secure some expected level of
success. A larger value corresponds to an easier attack. Beyond a reversal of the
13Charles L. Glaser and Chaim Kaufmann, ‘What is the O ﬀense-Defense Balance and How Can We
Measure It? ’.
14Patrick. J. Malone, ‘Oﬀense-Defense Balance in Cyberspace: A Proposed Model ’, Master ’s thesis, Naval
Postgraduate School (2012); Keir Lieber, ‘The O ﬀense-Defense Balance and Cyber Warfare ’,Cyber
Analogies (2014); Ilai Saltzman, ‘Cyber Posturing and the O ﬀense-Defense Balance ’,Contemporary
Security Policy 34/1 (2013).THE JOURNAL OF STRATEGIC STUDIES 741
numerator and denominator, the Glaser and Kaufman de ﬁnition represents a
special case of the ratio-for-success metric, for a binary CSF characterizing the
success of a ground invasion and a success threshold set at certain success
(S∗=1 ) .15
Ratio-for-success is not the only metric one can use to assess the relative
eﬃcacy of o ﬀensive and defensive investments. For instance, Van Evera
notes that, in the context of conquest, another useful metric might be ‘the
probability that a determined aggressor could conquer and subjugate a
target state with comparable resources. ’16
We introduce success-from-a-ratio as a generalization of this additional
metric, to cover other varieties of attack, non-binary success functions, and
various degrees of comparability. For a given contest success function Fand
investment ratio R∗, success-from-a-ratio FR∗describes how well the attacker
can be expected to do when the defender ’s investment is R∗times its own.
FR/C3ðDÞ;F1
R/C3/C3D;D/C18/C19
We regard ratio-for-success and success-from-a-ratio as complementary
metrics, describing closely related aspects of the o ﬀense-defense balance.
Ratio-for-success answers the question: ‘What ratio of investments would
allow the attacker to expect a certain level of success? ’Success-from-a-ratio
answers the question: ‘How successful could the attacker expect to be if
there was a certain ratio of investments? ’
Scaling
When we ask how a metric of the o ﬀense-defense balance scales , we are
asking how it changes as investment levels increase. Speci ﬁcally, given that
both of our proposed metrics take the defender ’s investment Das their
independent variable, we are asking how the metric changes as Dincreases.
If either metric increases with D, favoring o ﬀense, then we say that metric
undergoes oﬀensive scaling . If the metric decreases, favoring defense, then
we say that it undergoes defensive scaling . If it undergoes o ﬀensive scaling
for su ﬃciently low investment levels and defensive scaling for su ﬃciently
high investment levels, then we say that, overall, it exhibits OD-scaling .
Both metrics will typically scale in similar ways, since they capture inter-
twined properties. It should thus often be su ﬃcient to analyze scaling
phenomena using only one of the two metrics.
15We have chosen to de ﬁne the metric in terms ofD
Arather thanA
Din order to accommodate the
intuition that an increase in the o ﬀense-defense balance corresponds to a tilt toward o ﬀense.
16Van Evera, ‘Oﬀense, Defense, and the Causes of War ’.742 B. GARFINKEL AND A. DAFOE
Despite often being neglected in theoretical analyses, scaling can in
principle have a dramatic impact on the o ﬀense-defense balance. Take the
idealized case of a defender whose preparations for a war include building a
long wall along its border. Given that the attacker has the ability to focus its
attacks on any gaps in the wall, a defensive investment that is su ﬃcient to
cover its entire border will be much more than twice as e ﬀective and
diﬃcult to o ﬀset as an investment that only su ﬃces to cover half its border.
The defender, therefore, can decrease the local o ﬀense-defense balance
simply by spending more.
As a concrete analogue, we can consider Germany ’s invasion of France in
the Second World War. The successful German war plan took critical advan-
tage of the fact that the Maginot Line, the string of defensive forti ﬁcations
that France built along a portion of its Eastern border, was not extended to
cover either the France-Belgium or Belgium-Germany border. Historical
accounts suggest cost was a signi ﬁcant factor in the decision.17It is there-
fore plausible that a con ﬂict with double the resources invested by both
sides would have been notably more favorable to the defense.
Scaling e ﬀects for invasions and cyberattacks
In this section, we analyze the scaling e ﬀects associated with two varieties of
attacks and two forms of investment. We consider, in particular, the follow-
ing questions:
Invasions : How does an attacker ’s ability to break through a defender ’s lines
scale, as both actors devote more ground forces to holding or breaking
through the lines?
Cyberattacks : How does an attacker ’s ability to acquire “zero day ”exploits for
a defender ’s system scale, as both actors ’increase their investments into
vulnerability discovery?
We approach these questions by exploring idealized models, with our appendix
also considering a number of possible relaxations to these models. In both
cases, we ﬁnd that there are strong reasons to expect OD-scaling. Scaling up
investments ﬁrst bene ﬁts the attacker and then bene ﬁts the defender.
Invasions
Background
In this case, an attacker attempts to move ground forces into a defender ’s
territory. Once inside, the attacker may attempt to pursue a number of
possible objectives, such as conquest or liberation. The defender, in turn,
17Robert Jackson, The Fall of France: May-June 1940 (Oxford: Oxford University Press 2003), 25 –27.THE JOURNAL OF STRATEGIC STUDIES 743
attempts to prevent the attacker from successfully penetrating too deeply or
from achieving its objective.
Both actors have a range of possible strategies to choose from. Analysis is
simplest, though, if we make the assumption that the defender adopts a
forward defense. This means that the defender places a large portion of its
forces close to its border, with the aim of preventing the attacker from
penetrating its territory to any signi ﬁcant depth. When the defender adopts
a forward defense, the attacker must break through some section of the
defensive line in order to advance.
It is widely held that defenders hold a natural advantage in combat.
This advantage has a number of sources, including the fact that one is
naturally more vulnerable when advancing than when remaining
behind cover. Its practical signi ﬁcance is that the attacking force must
typically achieve a much greater concentration at a single point to
break through. A longstanding rule-of-thumb sets the necessary con-
centration ratio at 3-to-1.18
Although there are a number of ways an actor can make investments to
improve its prospects for success, we will focus speci ﬁcally on investments that
increase the size of ground forces. We are interested in how scaling up the sizes
of both ground forces will a ﬀect the attacker ’s ability to break through.
Theﬁrst analyst to consider this question in signi ﬁcant depth was Liddell
Hart, who concluded that greater force levels could signi ﬁcantly bene ﬁt the
defender.19His argument was that there was some force-to-space ratio
necessary to provide su ﬃciently unbroken coverage of a front. Below this
force-to-space ratio, the attacker could easily break through by concentrat-
ing its forces along weak points or gaps in the defensive line. Other analysts,
particularly Mearsheimer, later extended and popularized Hart ’s work on the
topic, often with a focus on estimating the size of the defensive force
necessary to repel a Soviet invasion of NATO countries.20
To our knowledge, Biddle et al. provide the most in-depth analysis of the
role of force-to-space ratios in determining the outcomes of ground
invasions.21They present a complex model, involving several dozen para-
meters, which predicts that increasing force levels above a certain point will
diminish the attacker ’s ability to capture territory. On the other hand,
increasing force levels will bolster the attacker ’s ability to capture territory
when force levels are su ﬃciently low. In our terminology, the model predicts
OD-scaling.
18John J. Mearsheimer, ‘Assessing the Conventional Balance: The 3:1 Rule and its Critics ’,International
Security 13/4 (1989).
19Basil L. Hart, ‘The Ratio of Troops to Space ’,Royal United Services Institution Journal 105/618 (1960).
20John J. Mearsheimer, ‘Numbers, Strategy, and the European Balance ’,International Security 12/4
(1988).
21Stephen D. Biddle et al., Defense at Low Force Levels: The E ﬀect of Force to Space Ratios on
Conventional Combat Dynamics (Alexandria, VA: Institute for Defense Analyses 1991).744 B. GARFINKEL AND A. DAFOE
To isolate the core mechanism behind this prediction, we now present a
signiﬁcantly simpler model of a ground invasion.
An idealized model
We consider a contest success function in which the output is the expected
size of the ground force that is able to penetrate the defensive line. The
inputs are the total sizes of the attacking and defending ground forces.
Possible units include simple headcounts or Armored Division Equivalents
(ADEs), which adjust for the quality of each force.
S≡Size of force that breaks through the defensive line
A≡Initial attacking force size
D≡Initial defensive force size
Our choice to use the size of the force that penetrates the defensive line
as a measure of success re ﬂects the view that, across a wide range of
scenarios, it is a reasonable proxy for the attacker ’s ability to achieve its
objectives. Its ability to capture territory or to attack remaining defenders
from the ﬂanks and rear, for instance, will typically increase with S. In this
case we treat the attacker as attempting to maximize the expected value of
Sand the defender as trying to minimize it.
We accept the standard assumption, discussed above, that the attacker
must establish a greater local force concentration to break through. We also
accept the standard assumption that there is a limit to how densely a force
can fruitfully concentrate at a particular point. Beyond a certain concentra-
tion, combatants and vehicles will face risks from collateral damage to one
another, increase the expected e ﬀectiveness of enemy artillery, and limit
each other ’s mobility.
Then, to produce an idealized model, we add four simplifying assump-
tions. First, we assume that breaking through at a given point requires
complete attrition of the defender ’s forces at that point and that each
side ’s losses grow linearly with the initial size of the other ’s force. Second,
we assume that points on the front are homogeneous: they do not vary
in limits on force concentration, defensive advantage, or other such
factors. Third, we assume that reinforcements do not arrive during bat-
tles. Fourth, we assume that the defender cannot anticipate where the
attacker will strike and that, given this fact, both sides distribute their
forces optimally.22
We show in our appendix that a Nash equilibrium is for the defender to
distribute its forces evenly across the points, while the attacker distributes
22This model resembles but is distinct from ‘Colonel Blotto ’models, a well-studied class of game-
theoretic models that also concern the optimal allocation of resources across battle ﬁelds. Brian
Roberson, ‘The Colonel Blotto Game ’,Economic Theory 29/1 (2006).THE JOURNAL OF STRATEGIC STUDIES 745
its forces to achieve the maximum practical concentration at as many points
as possible.
LetNbe the number of points along the front where the attacker might
strike. Let Bbe the force concentration ratio required to break through and
Mthe maximum practical force concentration. Then, as shown in our
appendix, the contest success function is given by the following expression:
E½S/C138¼XN
i¼1max 0;max f0;minfM;A/C0M/C3ði/C01Þgg /C0 B/C3minfM;D
Ng/C26/C27
Figure 1 presents a graphical depiction of the function for a representa-
tive set of parameters. Figures 2 and 3present the relevant scaling e ﬀects
more clearly, by depicting success-from-a-ratio for an equal investment ratio
and ratio-for-success for a threshold of one unit of penetration. Clearly, OD-
scaling is observed.
We can explain the basic mechanism responsible for OD-scaling in this
case by focusing on success-from-a-ratio and considering what happens as
investments increase. Initially, the attacker can use surprise to achieve much
higher degrees of force concentration at a small number of points. These
diﬀerences in concentration allow the attacker to break through the defen-
der ’s lines. For some time, higher force levels will also enable larger break-
throughs. However, given that space is ﬁnite, the defender will eventually
Figure 1. The CSF for our idealized model. The chosen parameters correspond to 10
possible points of attack, a 2:1 force ratio requirement for achieving breakthrough, and
force size units normalized to the maximum concentration level at each point.746 B. GARFINKEL AND A. DAFOE
begin to saturate the front by achieving close to the maximum concentra-
tion at a large number of points. This will place increasingly strict limits on
how di ﬀerent the two actors ’patterns of concentration along the front can
be. Since defense is also locally superior –in the sense that the defender has
a natural edge in battles where concentrations are equal –the attacker ’s
capacity to break through must decline too.
In our appendix, we consider each of the assumptions used in this simple
model in turn. We ﬁnd that although they are highly idealized, it appears to be
the case that reasonable adjustments to the assumptions, for instance to allow
for reinforcement dynamics and a more realistic attrition law, do not challenge
the basic mechanism described here. The more complex Biddle et al. model also
arrives at the same conclusion.
Case study: Normandy landings
The Allied invasion of France in the Second World War provides a useful
illustration of the dynamics discussed above. In this case, the Allied forces
entered France from the sea rather than over land. However, this distinction
does not imply any signi ﬁcant change to the abstract model.
Figure 2. Success-from-a-ratio for our idealized model, for an equal investment ratio.
Parameters same as above.THE JOURNAL OF STRATEGIC STUDIES 747
First, it was the case that the Normandy landings depended heavily on the
Allies ’ability to exploit superior force concentrations at the points where they
landed.23They were able to achieve the necessary concentrations by creating
uncertainty about their chosen landing points, including by running a pro-
tracted disinformation campaign, which forced the German military to spread
its forces out between a larger number of plausible landing points. However,
even with the bene ﬁt of a lopsided local force ratio, the inherent di ﬃculty of
amphibious landings caused the Allies to struggle and su ﬀer several times
greater casualties. In addition, despite being spread between ﬁve beaches
and arriving in multiple waves, the Allies also appeared to be approaching
the maximum useful levels of concentration –suﬀering from congestion issues,
particularly, on Sword Beach.
A counterfactual version of the Normandy landings with much higher
force levels on both sides could have allowed the German forces to more
comfortably defend each of the landing sites they considered plausible. On
the other hand, a counterfactual version with much lower force levels could
Figure 3. Ratio-for-success for our idealized model, for a threshold of one unit.
Parameters same as above.
23Olivier Wieviorka, Normandy: The Landings to the Liberation of Paris (Cambridge, MA: Harvard
University Press 2008).748 B. GARFINKEL AND A. DAFOE
have prevented the Allies from landing a large enough force to achieve their
objectives on the mainland.
Cyberattacks
Background
A cyberattack is an attempt to exploit another actor ’s computer systems. A
cyberattack may be associated with a number of possible objectives, includ-
ing stealing con ﬁdential information, disrupting the availability of a service,
or damaging connected physical objects and infrastructure.
One particularly noteworthy class of cyberattacks involves the exploi-
tation of zero days , or software vulnerabilities that are unknown to
developers and users of the softwar e. A wide range of political actors,
including national governments and c riminal organizations, are believed
to stockpile knowledge of zero days for potential use in future o ﬀensive
operations. For instance, the American/Israeli Stuxnet attack on the
Iranian Natanz nuclear facility exploited four distinct zero days for the
Windows operating system.24
We can regard any party attempting to acquire zero days for a particular
system as an attacker and the set of parties with an interest in patching
vulnerabilities, collectively, as the defender. An attacker acquires a zero day,
then, when it discovers a vulnerability that the defender has not.
Since both attackers and defenders bene ﬁt from discovering vulnerabil-
ities, we can ask about the e ﬀect of scaling up their discovery capabilities. In
fact, this question is far from hypothetical. We should want to know, for
instance, whether ongoing projects to develop more e ﬀective vulnerability
discovery tools are more likely to empower attackers or defenders.
Only a small literature has examined scaling e ﬀects. One relevant but
inconclusive empirical debate considers whether publishing software source
code –which is a useful resource for anyone hoping to discover vulnerabilities
–tends to overall bene ﬁt attackers or defenders more.25A related debate
considers the returns to defensive vulnerability discovery. Rescorla claims that
any vulnerability discovered by a defender is unlikely to have been rediscov-
ered by an attacker.26(The most sophisticated analysis of vulnerability redis-
covery to date, although also not conclusive, supports this claim.)27From this,
he argues that defensive investments have only a minimal e ﬀect on an
attacker ’s ability to acquire zero days; scaling up investments would strongly
favor the attacker.
24Thomas M. Chen and Saeed Abu-Nimeh, ‘Lessons from Stuxnet ’,Computer 44/4 (2011).
25Guido Schryen, ‘Is Open Source Security a Myth? ’,Communications of the ACM 54/5 (2011).
26Eric Rescorla, ‘Is Finding Security Holes a Good Idea? ’,IEEE Security & Privacy 3/1 (2005).
27Zero Days, Thousands of Nights: The Life and Times of Zero-Day Vulnerabilities and Their Exploits
(Santa Monica, CA: RAND Corporation 2017).THE JOURNAL OF STRATEGIC STUDIES 749
We now present a simple model that, while consistent with Rescorla ’s
analysis for su ﬃciently low investment levels, ultimately predicts OD-scaling.
An idealized model
We consider a contest success function in which the output is the expected
number of zero days that the attacker acquires over a given window of time.
The inputs are quantities of e ﬀort devoted to vulnerability discovery over
this time.
S≡Zero days acquired by attacker
A≡Attacker e ﬀort
D≡Defender e ﬀort
For our idealized model, we make three simplifying assumptions. First, we
assume that there is no correlation between the vulnerabilities the attacker
discovers and the vulnerabilities the defender discovers. Second, we assume
that no new vulnerabilities are introduced over the period of time under
consideration. Third, we assume that, until all vulnerabilities are discovered,
the number of vulnerabilities each actor discovers grows linearly with e ﬀort.
Now, let Nrepresent total number of vulnerabilities in the defenders ’
systems. In addition, let the units of e ﬀort be normalized so that one unit of
eﬀort corresponds to one expected discovery. Then, as shown in our
appendix, it follows that the contest success function takes the form:
E½S/C138¼minfN;Ag/C1N/C0minfN;Dg
N
Figure 4 presents a graphical depiction of the function. One can see that
increasing investments tends to bene ﬁt the attacker when investment levels
are low and tends to bene ﬁt the defender when they are su ﬃciently high.
Figures 5 and 6show this phenomenon more clearly, depicting the scaling
of success-from-a-ratio, for an even investment ratio, and ratio-for-success,
for a threshold of one zero day. For both metrics, we observe OD-scaling.
Why, intuitively, does the model predict OD-scaling? The underlying
mechanism is essentially the same as in the ground invasion case. We
again focus on success-from-a-ratio and consider what happens as D
increases.
Initially, due to randomness, the two actors are unlikely to discover
precisely the same vulnerabilities. When their investments begin to increase
from nothing, the attacker will therefore tend to ﬁnd some vulnerabilities
that the defender does not. However, given that the number of vulnerabil-
ities is ﬁnite, the defender will eventually begin to achieve saturation by
ﬁnding very large portions of them. This will place increasingly strict limits
on how di ﬀerent the sets of vulnerabilities discovered by the two actors can
be. Since defense is also locally superior –in the sense that any vulnerability750 B. GARFINKEL AND A. DAFOE
the attacker and defender both discover does not become a zero day –the
expected number of zero days must eventually begin to shrink too.
In our appendix, we consider each of the assumptions used in this simple
model in turn. We again ﬁnd that although they are highly idealized, it
appears to be the case that reasonable adjustments to the assumptions, for
instance to allow for correlations between discoveries and diminishing
returns on e ﬀort, do not challenge the basic mechanism described here.
Case study: Automated vulnerability discovery
The above analysis can also help us to understand the signi ﬁcance of
ongoing work on automated vulnerability discovery.
Currently, manual testing and inspection still consistently allow actors to
discover vulnerabilities that automated tools cannot.28However, there has
recently been some noteworthy progress in adapting techniques from the
ﬁeld of arti ﬁcial intelligence to the problem of vulnerability discovery.29
Figure 4. The CSF for our idealized model, for a case with 10 vulnerabilities.
28Andrew Austin and Laurie Williams, ‘One Technique is Not Enough: A Comparison of Vulnerability
Discovery Techniques ’,2011 International Symposium on Empirical Software Engineering and
Measurement (2011).
29Michael Kan, ‘Mayhem Supercomputer Takes on Humans at Def Con ’,Computer World , 6 August 2016;
Teresa N. Brooks, ‘Survey of Automated Vulnerability Detection and Exploit Generation Techniques in
Cyber Reasoning Systems ’,arXiv preprint arXiv:1702.06162 (2017); Konstantin Böttinger, Patrice
Godefroid, and Rishabh Singh, ‘Deep Reinforcement Fuzzing ’,arXiv preprint arXiv:1801.04589 (2018).THE JOURNAL OF STRATEGIC STUDIES 751
One obvious question is whether we should expect this progress, if it
continues, to favor o ﬀense or defense. To answer this question, let us consider
again our simpli ﬁed model. If we take investments to be in units of vulnerability
discovery capability, then we can interpret improvements in the e ﬃciency of
discovery tools as increases in the two actors ’investments. Our model suggests
that, all else being equal, these increases will tend to bene ﬁt attackers up until
some point where they become large enough to bene ﬁt defenders.
However, all else might not be equal. One especially relevant possibility to
note is that the vulnerabilities discov ered by highly optimized tools might be
more correlated than the vulnerabilities that are discovered in a more ad-hoc
manner by humans today. Our robustnes s analysis, included in the appendix,
shows that an increase in the correlation between discoveries would favor the
defender.
In all, the short-term impact of increased automation is ambiguous. There
appears to be at least one factor favoring the attacker and at least one other
factor plausibly favoring the defender. A more thorough analysis would
attempt to weigh these factors against each other, perhaps on the basis of
data about discovery correlations between automated tools. The long-run
impact will also depend on just how e ﬀective these tools can become. If
they ever become e ﬃcient enough to allow defenders to discover
Figure 5. Success-from-a-ratio for our idealized model, for a case with 10 vulnerabil-
ities and an even investment ratio.752 B. GARFINKEL AND A. DAFOE
signiﬁcant majorities of the vulnerabilities in their systems, outpacing the
rate at which new vulnerabilities are introduced, then the picture would
seem to decisively favor defense.30
When should we expect OD-scaling?
A general model of con ﬂict
We now present a general model that can be applied to many di ﬀerent forms of
conﬂict, with the goal of explaining the co nditions under which OD-scaling is
likely to occur. For simplicity, we focus speci ﬁcally on the success-from-a-ratio
metric.
The model
In this model, there is some attack surface that can be thought to consist of
discrete attack vectors . By making investments the attacker and defender
can increase their coverage of these vectors, each up to some maximum
coverage level . The manner in which the actors ’investments produce pat-
terns of coverage is determined by their respective investment functions .
Figure 6. Ratio-for-success for our idealized model, for a case with 10 vulnerabilities
and a threshold of one expected zero day.
30For a recent comment on the plausibility of this long-run scenario, see Bruce Schneier, ‘Artiﬁcial
Intelligence and the Attack/Defense Balance ’,IEEE Security & Privacy 2 (2018).THE JOURNAL OF STRATEGIC STUDIES 753
Each attack vector is furthermore associated with a local contest success
function , which maps the two actors ’coverage levels to o ﬀensive success
at that point. Finally, there is an aggregation function that maps the values of
all of the local contest success functions to the overall success of the attack.
The invasion and cyberattack models as special cases
The idealized models introduced above to study ground invasions and
cyberattacks can be understood as special cases of this more general model.
For our ground invasion model, the attack surface is the set of points that
make up the defensive line. Local coverage levels AiandDirepresents how large
a force each actor has concentrated at each point i. For all points, and both actors,
the maximum coverage level is some constant N.T h ed e f e n d e r ’s investment
function converts investments into equ al levels of coverage across the points.
The attacker ’s investment function converts i nvestments into maximum cover-
a g eo fa sm a n yp o i n t sa sp o s s i b l e .T h el o c a lC S F Fi(Ai,Di) indicates how large of an
attacking force breaks thro u g ha tag i v e np o i n t .I ti sb a s e do nal i n e a ra t t r i t i o n
law, with Bbeing the local coverage ratio n ecessary to break through.
FiðAi;DiÞ¼Ai/C0B/C3Di;Ai>B/C3Di
0; otherwise/C26
The aggregation function adds up the values of each of the local CSFs to
ﬁndS, the total size of the attacking force that breaks through. This then
implies the global contest success presented above.
For our vulnerability discovery model, the attack surface is the set of
software vulnerabilities in the defender ’s system. Coverage is a binary vari-
able that represents whether or not each actor has discovered the given
vulnerability. For all points, and for both actors, the maximum coverage
level is 1. The actors ’investment functions convert investments into cover-
age of random and uncorrelated sets of vulnerabilities. The local CSF indi-
cates whether or not a given vulnerability becomes a zero day. For a
particular point iand local coverage levels Aiand Di, it is given by:
FiðAi;DiÞ¼1;Ai>Di
0;otherwise/C26
The aggregation function adds up the values of the local CSFs to ﬁndS, the
total number of zero days the attacker acquires. This then implies the global
contest success function presented above.31
31One limitation of the general model is that it represents con ﬂict as consisting of only a single round,
in which all investments are applied. It is therefore unable to represent changes in coverage levels
over time. For example, it cannot represent the arrival of reinforcements or multiple waves of
attackers in the ground invasion case or represent multiple salvos in the missile defense case
discussed below. We hypothesize that extending the relevant models to account for multiple rounds
will not typically change the basic scaling phenomena predicted in each case, but we have not yet
explored this hypothesis in depth.754 B. GARFINKEL AND A. DAFOE
A common mechanism for OD-scaling
Both of the special cases just described have a number of common proper-
ties. These properties are jointly su ﬃcient (although not necessary) to
produce OD-scaling.
●Local defense superiority : When the defender provides maximum
coverage of any given attack vector, the output of the associated
local CSF is approximately zero.
●Defensive saturation : Above some level of investment, the defender
provides maximum coverage of all attack vectors.
●Gap exploitation : For some lower level of investment, the output of at
least one local CSF is likely to be non-negligible. This is the result of the
attacker ’s coverage of the associated attack vector substantially
exceeding the defender ’s.
●Aggregative success : The output of the global CSF is approximately
zero if the outputs of all local CSFs are approximately zero. It is also
non-negligible if the output of at least one local CSF is non-negligible.
Together local defense superiority, defensive saturation, and aggregative
success imply that the output of the global CSF will be approximately zero
when investments are su ﬃciently high. Before saturation occurs, however,
gap exploitation and aggregative success imply that the output of the
global CSF will be non-negligible.
Success-from-a-ratio begins at zero. As the investment level grows, it rises
to some non-negligible value. Ultimately, it must decrease toward zero
again. OD-scaling occurs.
Additional special cases
Brieﬂy, we now consider two additional cases: missile attacks and drone
swarm attacks. We use the general model just described as a tool for
assessing what scaling behavior these two cases are likely to exhibit. In
particular, we ask whether the four properties su ﬃcient for OD-scaling hold.
Other cases that it may be worth considering, in future work, include
cyberattacks based on social engineering (such as phishing attacks), blockades,
aerial bombings, attacks with pathogens, and various forms of terrorism
Missile attacks
In this case, the attacker attempts to destroy a target by striking it with
missiles. In turn, the defender attempts to protect the target by destroying
the attacker ’s missiles. To make the case more concrete, we will take the
target to be a warship and the defender to be relying exclusively on surface-
to-air missiles.THE JOURNAL OF STRATEGIC STUDIES 755
We can conceptualize the attack vectors as non-overlapping trajectories by
which the attacker ’s missiles can hit the ship. Furthermore, we can conceptua-
lize the attacker and the defender as covering the same vector if a projectile
launched by the defender follows a trajectory that leads it to intercept one of
the attacker ’s missiles. Investments are measured in units of missiles.
The local CSFs indicate how many missiles associated with a given attack
vector hit the ship, over a ﬁxed period of time. They exhibit local defense
superiority, since a missile that is intercepted will not reach its target. In
addition, consistent with aggregative success, the global CSF indicates the
total number of missiles that hit the ship.
In this case, gap exploitation occurs due to the inaccuracy of defensive
systems. In particular, each surface-to-air missile has some non-zero chance
of following a trajectory that does not lead it to intercept the intended
oﬀensive missile. This means that, as total investments scale up, the
expected number of o ﬀensive missiles that fail to be intercepted will initially
increase.
This somewhat abstract argument is consistent with models that have
been developed in the missile defense literature. For instance, Washburn
and Kress present a simple single-round model in which the defender
distributes its missiles evenly among the o ﬀensive missiles, each defensive
missiles has some probability Pof hitting the intended o ﬀensive missile (and
no probability of hitting any other), and each o ﬀensive missile has some
probability Qof hitting the target if it is not destroyed ﬁrst.32LetAand Dbe
the number of missiles launched by the two actors. Then, making the
mathematically convenient assumption thatD
Ais an integer, we show in
our appendix that the expected number of missiles that hit the target is
given by:
ES½/C138 ¼ Q/C3A/C31/C0P ðÞD=A
This expression implies o ﬀensive scaling. The associated function for
success-from-a-ratio, FR∗(D)=Q∗D∗(1–P)R∗, increases linearly with D.
The key remaining question, then, is whether the property of defensive
saturation holds. This property would seem to imply that, for su ﬃciently
high investment levels, the defender could create a virtual wall of missiles
around the ship. In this case, the model just described would cease to be
appropriate, as all trajectories that an o ﬀensive missile could take would
lead to interception.
However, the plausibility of defensive saturation is questionable. Even if
the defender possessed an essentially unlimited supply of surface-to-air
missiles, it seems that the need to prevent these missiles from interfering
32Alan R. Washburn and Moshe Kress, Combat Modeling (New York: Springer 2009), 65 –67.756 B. GARFINKEL AND A. DAFOE
with one another and the need to avoid fratricide would be key limiting
factors.33
In short, in contrast with the primary two cases considered above, it
seems likely that missile defense presents a case of purely o ﬀensive scaling
with regard to the number of missiles used by both sides.
Drone swarm attacks
In this case, a large number of drones launched by an attacker attempt to
navigate close enough to a target to damage it with short-range munitions.
The defender attempts to destroy the drones before they can get in range of
the target. For instance, Allen and Chan consider a scenario in which an
attacker directs millions of cheap kamikaze drones at an aircraft carrier
battle group.34
This variety of attack is speculative. The only military drone swarms
known to be in development, such as the US Department of Defense ’s
Perdix systems, are not designed to carry munitions.35However, some
analysts predict that, as the technology progresses, the possibility of using
large swarms to destroy targets will become irresistible.
Individually, drones o ﬀer a number of advantages over human comba-
tants or manned vehicles.36Since they do not need to carry or protect a
human passenger, they can be especially long-ranged, fast, small, cheap to
produce, and expendable. To the extent that they exhibit autonomy, they
can also process information to make decisions much more quickly. Acting
together, though, their advantages can multiply. For orders-of-magnitude
less than the cost of, for instance, a modern ﬁghter jet or warship, a well-
coordinated swarm of drones may be able to overwhelm the defenses of a
target, dividing its attention and forcing it to expend its ammunition. Even if
individual drones are not especially sophisticated or deadly, a large swarm
may still have a very high probability of destroying its target.
Although a number of possible defenses against swarming attacks have
been proposed, we focus on the case where the counter-swarm is the
primary line of defense and ask what the impact of growing swarm sizes
would likely be.37
33Note that our analysis here considers what would happen if the number of missiles used by both
actors increased enormously. Once we also take into account practical limits on the number of
missiles that could plausibly be stored and launched by an individual ship, it becomes more doubtful
that defensive saturation could ultimately be observed in practice.
34Greg Allen and Taniel Chan, Arti ﬁcial Intelligence and National Security (Cambridge, MA: Belfer Center
for Science and International A ﬀairs 2017).
35Paul Scharre, Army of None (New York: Norton 2018).
36Paul Scharre, Robotics on the Battle ﬁeld: The Coming Swarm (Washington DC: Center for a New
American Security 2014).
37Paul Scharre, ‘Counter-Swarm: A Guide to Defeating Robotic Swarms ’,War on the Rocks , 31 March
2015.THE JOURNAL OF STRATEGIC STUDIES 757
Atﬁrst glance, this case is structurally quite similar to missile defense,
with individual defensive drones attempting to destroy individual o ﬀensive
drones before they can reach the target. However, the potential ability of
drones within a swarm to coordinate their movements closely with one
another, hover within a ﬁxed region of space, and defend against multiple
attacking drones sequentially suggests that defensive saturation is more
plausible than it is in the missile defense case.
Ultimately, it is plausible but far from certain that swarm defense will
eventually present another case of OD-scaling with regard to the number of
drones used by both sides.
Three e ﬀects of technological progress
It is well-understood that technological progress can impact o ﬀense-defense
balances. In fact, perhaps the primary motivation for developing the con-
cept has been to understand the distinctions between di ﬀerent eras of
military technology.
For instance, European powers ’failure to predict the grueling attrition
warfare that would characterize much of the First World War is often
attributed to their failure to recognize that new technologies, such as
machine guns and barbed wire, had shifted the European o ﬀense-defense
balance for conquest signi ﬁcantly toward defense.38
We see three primary ways in which technological progress can in ﬂuence
oﬀense-defense balances.
Pathway 1 is to introduce a new form of con ﬂict with distinct o ﬀense-
defense dynamics. For instance, the emergence of cyberattacks as a form of
conﬂict has made it possible to discuss the ‘cyber o ﬀense-defense balance. ’
Pathway 2 is to change the character of an existing form of con ﬂict in a
manner that implies an updated contest success function. A CSF that maps
relative force sizes into the expected success of a ground invasion, for example,
will be di ﬀerent if they are equipped with mid-nineteenth century weaponry
than if they are equipped with early twentieth century weaponry.39
Pathway 3 is to change the quantity of investments applied to the
associated CSF. Technological progress can support an increase in invest-
ments –whether measured in units of ﬁnancial cost (such as dollars) or in
units of capability (such as Armored Division Equivalents) –by generating
wealth through economic growth or by increasing the power and cost-
38Stephen Van Evera, ‘The Cult of the O ﬀensive and the Origins of the First World War ’,International
Security 9/1 (1984).
39Concretely, holding force sizes ﬁxed, the conventional wisdom holds that a con ﬂict with mid-
nineteenth century technology could be expected to produce a better outcome for the attacker
than a con ﬂict with early twentieth century technology. See, for instance, Van Evera, ‘Oﬀense,
Defense, and the Causes of War ’.758 B. GARFINKEL AND A. DAFOE
eﬀectiveness of technologies used in con ﬂict. We have seen, in turn, that
rising investments can signi ﬁcantly in ﬂuence the o ﬀense-defense balance.
Pathway 1 and Pathway 2 are widely recognized. The third pathway, how-
ever, appears to have received rather less attention.40Arguably, this relative
neglect has been understandable. The concept of an o ﬀense-defense balance
has been discussed primarily in the context of ground invasions, often with a
focus on head counts as inputs. Technological progress does not support the
‘scaling up ’of ground force levels nearly as directly as it supports the scaling up
of, for example, the number of software vulnerabilities an actor discovers. In
addition, the e ﬀects of changing investment levels may also be obscured by the
fact that technological progress is continually changing the associated con ﬂict
domains and CSFs as well.
However, we argue that this third pathway will become increasingly di ﬃcult
to ignore. Brundage et al. argue that scalability is one of the key distinguishing
features of digital and arti ﬁcially intelligent systems.41The marginal cost of
increasing the e ﬀectiveness of a given piece of software or increasing the
number of copies in operation will often be quite low; the marginal cost can
also often be expected to decline exponentially with the cost of computing
power over time. The manufacture and operation of certain robots may also
lend itself to unusually rapid scaling. For instance, anticipating a continual
decline in manufacturing costs, Scharre suggests that it may be possible to
deploy swarms of ‘billions ’of drones in the future.42Overall, a trend toward
more highly automated and digitally-oriented con ﬂict suggests that scaling
phenomena may become much more salient.
Above, we discussed two concrete instances of this trend. First, we argued
that progress in developing and applying automated vulnerability tools may
dramatically increase actors ’abilities to discover software vulnerabilities. All
else being equal, a plausible e ﬀect of this trend would be to tilt the relevant
oﬀense-defense balance further toward o ﬀense in the short-run, then toward
defense in the long run.43
40Glaser and Kaufman brie ﬂy discuss the signi ﬁcance of force-to-space ratios in their classic article on
the oﬀense-defense balance. Elsewhere, without directly linking their result to the o ﬀense-defense
balance literature, Johnson and MacKay suggest that defense may be more di ﬃcult in the context of
conﬂicts between small-scale societies (such as hunter-gatherer groups) than in the context of
conﬂicts between large-scale societies (such as countries). Lindsay also suggests that the relative
costs of carrying out anonymous cyberattacks and attributing them may scale in a way that
ultimately favors defenders. Dominic D.P. Johnson and Niall J. MacKay, ‘Fight the Power:
Lanchester ’s Laws of Combat in Human Evolution ’,Evolution and Human Behavior 36/2 (2015); Jon
R. Lindsay, ‘Tipping the Scales: The Attribution Problem and the Feasibility of Deterrence against
Cyberattack ’,Journal of Cybersecurity 1/1 (2015).
41Miles Brundage, et al., ‘The Malicious Use of Arti ﬁcial Intelligence: Forecasting, Prevention, and
Mitigation ’,arXiv preprint arXiv:1802.07228 (2018).
42Paul Scharre, Robotics on the Battle ﬁeld: The Coming Swarm.
43However, as noted in the earlier section, automation may also change the CSF in a manner favorable
to the defender, by increasing the correlation between the vulnerabilities the actors discover. This
eﬀect could conceivably o ﬀset the initial bene ﬁt attackers derive from scaling e ﬀects.THE JOURNAL OF STRATEGIC STUDIES 759
Second, as Scharre argues, progres s in developing drone swarms for
use in combat may dramatically increase the number of weapons plat-
forms available to military actors. This raises the tactical possibility of
using large swarms of individually expendable drones to overwhelm the
defenses of targets, with the quantity of drones playing perhaps an even
greater role than their quality does in determining their chance of suc-
cess. Our analysis suggests that, for target defense, the net e ﬀect of a
trend of growing swarm sizes is likely to initially bene ﬁta t t a c k e r s .
Whether further increases eventually bene ﬁt defenders will depend on
whether it is feasible for defenders to essentially saturate the airspace
around a target with defensive drones.44
Conclusion
In this paper we have shown how investment levels can play an important
role in determining the o ﬀense-defense balance. For both ground invasions
and attempts to acquire zero days, our models predict that scaling up
investments will initially bene ﬁtoﬀense and ultimately bene ﬁt defense.
We argued that this pattern, which we call OD-scaling, is a useful baseline
assumption across a range of cases. A foothold for understanding the case
of a missile attack, for example, is asking whether or not the case displays
certain features that are su ﬃcient for OD-scaling.
Finally, we have suggested that scaling e ﬀects can be expected to play a
particularly large role in determining the o ﬀense-defense balance associated
with drone swarm warfare and automated vulnerability discovery. Scaling
eﬀects, therefore, are likely to become increasingly di ﬃcult to ignore and
essential to understand.
We have shied away from drawing speci ﬁc conclusions about the rela-
tionship between scaling e ﬀects and international security. The apparent
implications will vary with one ’s views on unresolved controversies in
oﬀense-defense theory, particular the balance ’s relationship with crisis and
peacetime stability. While some hold that shifts toward o ﬀense-dominance
obviously favor con ﬂict and arms racing, this position has been challenged
on a number of grounds. It has even been suggested that shifts toward
oﬀense-dominance can increase stability in a number of cases.
Providing a full account of these controversies is beyond the scope of this
paper. Nevertheless, further work to understand the security implications of
scaling e ﬀects will require direct engagement with o ﬀense-defense theory.
Given that o ﬀense-defense theory has also been developed primarily under
44Of course, it is not yet known how signi ﬁcant a role counter-swarms will play in defense against
oﬀensive drone swarms. If gun systems or area denial weapons play larger roles, for instance, then
this result has less direct signi ﬁcance.760 B. GARFINKEL AND A. DAFOE
the assumption that the o ﬀense-defense balance is a constant property of a
system and technological era, rather than a variable that changes with the
actors ’investments, our work may also suggest the need for substantial
theoretical revision.
Acknowledgements
We would also like to thank Al Brown, Ben Buchanan, Je ﬀDing, Jon Lindsay, David
Manheim, Anish Mohammed, Mike Montague, and Stefan Schubert for valuable
comments.
Disclosure statement
No potential con ﬂict of interest was reported by the authors.
Funding
This work was supported by the European Research Council.
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