Concentration and Asymmetry in Air Combat
Lessons for the defensive employment of air power
Ian Horwood, Niall MacKay & Christopher Price
Naval Postgraduate School, Monterey, 28th January 2016
Based on RAF Air Power Review 17 no.2 (2014) 68-91
What do the historical data tell us about the
nature of concentration in air combat?
What do the historical data tell us about the
nature of concentration in air combat?
What can a historical analysis tell us about the implications
for tactical and operational principles?
A natural assumption:
Fighting strength = number of units × unit effectiveness
A natural assumption:
Fighting strength = number of units × unit effectiveness
A thought-provoking alternative:
Lanchester’s Square Law
Frederick Lanchester (1868-1946)
Automotive and aeronautical engineer
who built the first British motor car
Frederick Lanchester (1868-1946)
Automotive and aeronautical engineer
who built the first British motor car
Invented the carburettor, disc brakes, the accelerator pedal;
theory of lift and drag.
Lanchester’s Square Law (1914)
The aimed-fire model: G (t) Green units fight R(t) Red units.
dG
= −r R
dt
Green’s instantaneous loss-rate is proportional to Red numbers
dR
= −g G
dt
and vice versa.
Lanchester’s Square Law (1914)
The aimed-fire model: G (t) Green units fight R(t) Red units.
dG
= −r R
dt
Green’s instantaneous loss-rate is proportional to Red numbers
dR
= −g G
dt
and vice versa. Divide:
dR
gG
=
dG
rR
and integrate:
or
r R dR = g G dG
1 2 1
r R = g G 2 + constant
2
2
throughout the battle, the Square Law.
Lanchester’s Square Law (1914)
The constancy of
r R2 − gG 2
tells us how to combine numbers (R, G ) and effectiveness (r , g ).
Lanchester’s Square Law (1914)
The constancy of
r R2 − gG 2
tells us how to combine numbers (R, G ) and effectiveness (r , g ).
Numbers win:
suppose we begin with twice as many Reds as Greens, R 0 = 2G 0 ,
but that Greens are three times more effective,
g = 3r .
Then
r R 2 − g G 2 = r (2G 0 )2 − 3r G 20 = r G 20 > 0,
and Red wins: the battle ends with G = 0, R = G 0 .
Lanchester’s Square Law (1914)
The constancy of
r R2 − gG 2
tells us how to combine numbers (R, G ) and effectiveness (r , g ).
Numbers win:
suppose we begin with twice as many Reds as Greens, R 0 = 2G 0 ,
but that Greens are three times more effective,
g = 3r .
Then
r R 2 − g G 2 = r (2G 0 )2 − 3r G 20 = r G 20 > 0,
and Red wins: the battle ends with G = 0, R = G 0 .
Concentration is good:
If Red divides its forces, and Green
p fights each half in turn,
Green wins the first battle, with 2/3
p ≃ 80% of G 0 remaining,
Green wins the second battle, with 1/3 ≃ 60% of G 0 remaining.
Lanchester’s Linear Law
Ancient warfare, along a fixed, narrow battle-line with N(t)
fighting on each side:
dG
= −r N
dt
dR
= −g N
dt
Modern warfare, but with hidden targets (the unaimed-fire
model):
dR
dG
= −r RG
= −g G R
dt
dt
Either way, dR/d G is now fixed, and the constant quantity is
rR − gG ,
the more intuitive linear law: fighting strength is just
numbers × effectiveness.
Asymmetric warfare
Green attacks, Red defends:
dG
= −rR,
dt
dR
R
= −g G
dt
K
where K > R 0 is a constant parameter. Then
1
r RK − g G 2
2
is conserved, so that
– Green benefits more from numbers and concentration, but
√
– needs g to be at least twice as great, or G 0 to be 2 times as
great, as in a symmetric aimed-fire battle.
Asymmetric warfare
Green attacks, Red defends:
dG
= −rR,
dt
dR
R
= −g G
dt
K
where K > R 0 is a constant parameter. Then
1
r RK − g G 2
2
is conserved, so that
– Green benefits more from numbers and concentration, but
√
– needs g to be at least twice as great, or G 0 to be 2 times as
great, as in a symmetric aimed-fire battle.
Red has a defender’s advantage
Scaling laws for air combat
Fit loss-rates to powers of own and enemy numbers:
dG
= −rR r 1 G g 2
dt
dR
= −g G g 1 R r 2
dt
Scaling laws for air combat
Fit loss-rates to powers of own and enemy numbers:
dG
= −rR r 1 G g 2
dt
dR
= −g G g 1 R r 2
dt
Divide, re-arrange, integrate: we find that
r ρ g γ
R − G
ρ
γ
is constant, where ρ = 1 + r 1 − r 2 and γ = 1 + g 1 − g 2
Scaling laws for air combat
Fit loss-rates to powers of own and enemy numbers:
dG
= −rR r 1 G g 2
dt
dR
= −g G g 1 R r 2
dt
Divide, re-arrange, integrate: we find that
r ρ g γ
R − G
ρ
γ
is constant, where ρ = 1 + r 1 − r 2 and γ = 1 + g 1 − g 2 ,
the exponents, capture the conditions of battle:
– Green should concentrate its force if γ > 1, divide if γ < 1.
– if ρ > γ then Green has a defender’s advantage, by a factor ρ/γ
Scaling laws for air combat
The crucial tactical dependence is of the loss ratio on force
numbers:
√
ρ−γ
dG
r R ρ−1
=
=
RG
dR
g G γ−1
Linear Law:
ρ = γ = 1, and
dG
dR
Square Law:
ρ = γ = 2, and
Asymmetric:
ρ = 1, γ = 2, and
r
R
G
ρ+γ−2
doesn’t depend on R or G .
dG
R
∝ .
dR
G
dG
1
∝ .
dR
G
.
Scaling laws for air combat
‘The dependence of the casualty exchange ratio on the force ratio
is not linear; it is exponential’
– Col. John Warden, USAF, The Air Campaign
Cites a 1970 study of Korea and WW2.
Scaling laws for air combat
‘The dependence of the casualty exchange ratio on the force ratio
is not linear; it is exponential’
– Col. John Warden, USAF, The Air Campaign
Cites a 1970 study of Korea and WW2.
Well, no.
NJM, Is air combat Lanchestrian?, MORS Phalanx 44, no. 4 (2011) 12-14
0.0
0.5
log(CER)
1.0
1.5
Loss ratio vs Force ratio: Battle of Britain
−0.5
0.0
0.5
1.0
log(FR)
log dG /dR vs log R/G
G =Luftwaffe, R=Royal Air Force
−4
−3
log(CER)
−2
−1
Loss ratio vs Force ratio: Pacific air war
−6
−5
−4
−3
−2
−1
0
log(FR)
log dG /dR vs log R/G
G =Americans, R=Japanese
−2
−3
−4
log(CER)
−1
Loss ratio vs Force ratio: Korea
−2.0
−1.5
−1.0
−0.5
0.0
0.5
log(FR)
log dG /dR vs log R/G
G =Americans, R=KPAF/Chinese
0.0
0.5
log(CER)
1.0
1.5
Loss ratio vs Sorties: Battle of Britain
5.0
5.5
6.0
6.5
7.0
log(sorties)
log dG /dR vs
1
2
log(RG ),
G =Luftwaffe, R=Royal Air Force
P 2
R = 0.17, p = 0.003
−4
−3
log(CER)
−2
−1
Loss ratio vs Sorties: Pacific air war
3
4
5
6
7
8
log(sorties)
log dG /dR vs
1
2
log(RG ),
G =Americans, R=Japanese
P 2
R = 0.30, p = 10−5
−2
−4
−3
log(CER)
−1
Loss ratio vs Sorties: Korea
6.5
7.0
7.5
8.0
log(sorties)
log dG /dR vs
1
2
log(RG ),
P
G =Americans, R=North Korean/Chinese
R 2 = 0.11, p = 0.11
Subtleties I
5.5
6.0
log R
6.5
7.0
G and R are highly correlated (0.74 for the Battle of Britain):
4
5
6
log G
log R vs log G
7
Subtleties I
In
dG
= −r R r 1 G g 2
dt
◮
dR
= −g G g 1 R r 2 ,
dt
the overall powers in the loss-rates, g 1 + r 2 and r 1 + g 2 , are
better-determined than their constituents
Subtleties I
In
dG
= −r R r 1 G g 2
dt
dR
= −g G g 1 R r 2 ,
dt
◮
the overall powers in the loss-rates, g 1 + r 2 and r 1 + g 2 , are
better-determined than their constituents
◮
variation is less significant along the lines of constant g 1 + r 2
and r 1 + g 2 than orthogonal to them
Subtleties I
In
dG
= −r R r 1 G g 2
dt
dR
= −g G g 1 R r 2 ,
dt
◮
the overall powers in the loss-rates, g 1 + r 2 and r 1 + g 2 , are
better-determined than their constituents
◮
variation is less significant along the lines of constant g 1 + r 2
and r 1 + g 2 than orthogonal to them
So in
√
(r 1 +g 2 )−(g 1 +r 2 )
dG
r R ρ−1
= RG
=
γ−1
dR
gG
r
R
G
(r 1 −g 2 )+(g 1 −r 2 )
,
Subtleties I
In
dG
= −r R r 1 G g 2
dt
dR
= −g G g 1 R r 2 ,
dt
◮
the overall powers in the loss-rates, g 1 + r 2 and r 1 + g 2 , are
better-determined than their constituents
◮
variation is less significant along the lines of constant g 1 + r 2
and r 1 + g 2 than orthogonal to them
So in
√
(r 1 +g 2 )−(g 1 +r 2 )
dG
r R ρ−1
= RG
=
γ−1
dR
gG
◮
r
R
G
(r 1 −g 2 )+(g 1 −r 2 )
,
it’s easier to spot dependence on sortie numbers than on
force ratio
Subtleties II
◮
When g 1 + r 2 6= 1 or r 1 + g 2 6= 1, autonomous battles
(‘raids’) should not be aggregated into daily data.
Subtleties II
◮
When g 1 + r 2 6= 1 or r 1 + g 2 6= 1, autonomous battles
(‘raids’) should not be aggregated into daily data.
◮
If they are, the effect is to push the overall powers g 1 + r 2
and r 1 + g 2 away from their true values and towards one, and
to reduce the quality of the fit.
Subtleties II
◮
When g 1 + r 2 6= 1 or r 1 + g 2 6= 1, autonomous battles
(‘raids’) should not be aggregated into daily data.
◮
If they are, the effect is to push the overall powers g 1 + r 2
and r 1 + g 2 away from their true values and towards one, and
to reduce the quality of the fit.
◮
For the values we find, the difference
ρ − γ = (r 1 + g 2 ) − (g 1 + r 2 ),
and thus the asymmetry, is underestimated.
Subtleties II
0
10
20
y
30
40
50
Example: y = x 2
1
2
3
4
5
6
7
x
has log y = 2 log x, of course.
Subtleties II
0
10
20
y
30
40
50
Example: y = x 2 and sums of these: e.g. not only (3, 9) but also
(1 + 2, 1 + 4) = (3, 5) and (1 + 1 + 1, 1 + 1 + 1) = (3, 3).
1
2
3
4
5
6
7
x
and the best fit is now log y = 1.5 log x, with ΣR 2 = 0.6.
Asymmetry in air combat
What are the exponents for air combat?
Asymmetry in air combat
What are the exponents for air combat?
Battle of Britain: Germans 1.3 , British 0.8
Asymmetry in air combat
What are the exponents for air combat?
Battle of Britain: Germans 1.3 , British 0.8
Pacific air war: Americans 1.3, Japanese 0.9
Asymmetry in air combat
What are the exponents for air combat?
Battle of Britain: Germans 1.3 , British 0.8
Pacific air war: Americans 1.3, Japanese 0.9
Korea: Americans 1.2, North Koreans 0.1
– and these differences are understated.
The Battle of Britain: The Big Wing
Should the RAF’s squadrons mass into wings (3 squadrons)
or ‘Big Wings’ (5 or more) before engaging?
The Battle of Britain: The Big Wing
Should the RAF’s squadrons mass into wings (3 squadrons)
or ‘Big Wings’ (5 or more) before engaging?
Is mere concentration of numbers advantageous for the RAF?
The Battle of Britain: The Big Wing
Should the RAF’s squadrons mass into wings (3 squadrons)
or ‘Big Wings’ (5 or more) before engaging?
Is mere concentration of numbers advantageous for the RAF?
Is the RAF exponent ρ > 1?
The Battle of Britain: The Big Wing
Should the RAF’s squadrons mass into wings (3 squadrons)
or ‘Big Wings’ (5 or more) before engaging?
Is mere concentration of numbers advantageous for the RAF?
Is the RAF exponent ρ > 1?
No, ρ ≤ 0.8.
Ian Johnson & NJM, Lanchester models and the Battle of Britain, Naval
Research Logistics 58 (2011) 210-222.
NJM & Chris Price, Safety in Numbers: Ideas of concentration in Royal Air
Force fighter defence from Lanchester to the Battle of Britain,
History 96 (2011) 304-325.
J. F. C. Fuller (1878-1966) and the Principles of War
‘The Principles of War with Reference to the Campaigns of 1914-15’, JRUSI 61 (1916)
J. F. C. Fuller (1878-1966) and the Principles of War
‘The Principles of War with Reference to the Campaigns of 1914-15’, JRUSI 61 (1916)
‘The principle of concentration of force’
The Principle of Concentration of Force
‘The decisive, synchronized application of superior fighting power’
– UK, 2011
The Principle of Concentration of Force
‘The decisive, synchronized application of superior fighting power’
– UK, 2011
‘Mass is one of the principles of war. The purpose of mass is to
concentrate the effects of combat power at the most advantageous
place and time to achieve decisive results.’ – USAF, 2015
The Principle of Concentration of Force
‘The decisive, synchronized application of superior fighting power’
– UK, 2011
‘Mass is one of the principles of war. The purpose of mass is to
concentrate the effects of combat power at the most advantageous
place and time to achieve decisive results.’ – USAF, 2015
But how should a defensive air force (perhaps small, technically
inferior, or outnumbered) prevent the application of this principle?
The Principle of Concentration of Force
‘The decisive, synchronized application of superior fighting power’
– UK, 2011
‘Mass is one of the principles of war. The purpose of mass is to
concentrate the effects of combat power at the most advantageous
place and time to achieve decisive results.’ – USAF, 2015
But how should a defensive air force (perhaps small, technically
inferior, or outnumbered) prevent the application of this principle?
Fuller (1925): Lanchester’s laws ‘an act of mathematical
certainty’,
The Principle of Concentration of Force
‘The decisive, synchronized application of superior fighting power’
– UK, 2011
‘Mass is one of the principles of war. The purpose of mass is to
concentrate the effects of combat power at the most advantageous
place and time to achieve decisive results.’ – USAF, 2015
But how should a defensive air force (perhaps small, technically
inferior, or outnumbered) prevent the application of this principle?
Fuller (1925): Lanchester’s laws ‘an act of mathematical
certainty’, but ‘whilst formerly the application of the principle of
concentration aimed at massing numbers of men, it should now
aim at accentuating weapon-power.’
Tactics in the RAF approaching the Battle of Britain
‘British air doctrine was based upon Lanchester’ – Higham
Tactics in the RAF approaching the Battle of Britain
‘British air doctrine was based upon Lanchester’ – Higham
Tactics in the RAF approaching the Battle of Britain
‘British air doctrine was based upon Lanchester’ – Higham
Wing Cdr A.B. Ellwood, RAF Staff College, 1939: ‘The ultimate aim of
fighters is to stop enemy attacks. To do this they must obviously intercept and
engage enemy bombers, if possible before they reach their objectives’.
Tactics in the RAF approaching the Battle of Britain
‘British air doctrine was based upon Lanchester’ – Higham
Wing Cdr A.B. Ellwood, RAF Staff College, 1939: ‘The ultimate aim of
fighters is to stop enemy attacks. To do this they must obviously intercept and
engage enemy bombers, if possible before they reach their objectives’.
But defenders might face multiple incoming raids against different targets.
How to respond? Fighters should avoid ‘nibbling at every enemy formation as
opposed to bringing maximum force to bear on certain raids, with the objective
of destroying them utterly’.
Battle of Britain: protagonists
Dowding
Park
Battle of Britain: protagonists
Dowding
Park
‘It [is] better to have even one strong squadron of our fighters over
the enemy than a wing of three climbing up below them’ – Park,
memo to his subordinates, October 1940
Battle of Britain: protagonists
Leigh-Mallory
Bader
Battle of Britain: protagonists
Leigh-Mallory
Bader
‘It is much more economical to put up 100 against 100 than 12
against 100’ – Bader, Air Ministry meeting, 17th October 1940.
Battle of Britain: protagonists
Leigh-Mallory
Bader
‘It is much more economical to put up 100 against 100 than 12
against 100’ – Bader, Air Ministry meeting, 17th October 1940.
‘It was agreed that the more we could outnumber the enemy the
more we should shoot down’
Vietnam 1965-68; Rolling Thunder
Engagement-level data, and a simple linear regression of loss rates
against numbers.
Does a sortie lead to a kill, a loss, or neither?
Vietnam 1965-68; Rolling Thunder
Engagement-level data, and a simple linear regression of loss rates
against numbers.
Does a sortie lead to a kill, a loss, or neither?
F4 (US fighter) concentration tended to increase NVAF (but not
US) losses.
F105 (US bomber) numbers tended to cause neither.
Implication for US: F4s should sortie in numbers.
Vietnam 1965-68; Rolling Thunder
Engagement-level data, and a simple linear regression of loss rates
against numbers.
Does a sortie lead to a kill, a loss, or neither?
F4 (US fighter) concentration tended to increase NVAF (but not
US) losses.
F105 (US bomber) numbers tended to cause neither.
Implication for US: F4s should sortie in numbers.
NVAF (MiG 17,19,21) concentration tended to increase own
losses, whether against F4s or F105s.
Implication for NVAF: sortie sparingly, disrupt, avoid
engagement.
North Vietnamese Air Force
A small force in a ‘target-rich environment’, often in slow but
manoeuvrable aircraft, over their own territory with effective
ground fire and SAMs.
The Falklands War
‘The key to disrupting or dissuading a large number of aircraft from attacking
was to get in amongst them with a small number of easily controlled fighters.
One pair let loose against 10 or 15 bogeys could easily keep track of each
other, whereas the opposition would have great difficulty in sorting out friend
from foe.’ – Nigel Ward, CO 801 sqn, HMS Invincible
The Gulf War
– in which overwhelming force was applied, in operation Instant
Thunder, following the doctrine of John Warden...
The Gulf War
– in which overwhelming force was applied, in operation Instant
Thunder, following the doctrine of John Warden...
...who was an outspoken advocate of ‘Big Wing’ tactics.
The Gulf War
– in which overwhelming force was applied, in operation Instant
Thunder, following the doctrine of John Warden...
...who was an outspoken advocate of ‘Big Wing’ tactics.
Overwhelming material and technical advantage
should not be confused with tactical concentration.
Asymmetry in air combat
The starting hypothesis:
Air combat is a set of duels; random; linear-law.
To the extent to which it departs from this,
Asymmetry in air combat
The starting hypothesis:
Air combat is a set of duels; random; linear-law.
To the extent to which it departs from this,
Air combat does not obey the Square Law.
Asymmetry in air combat
The starting hypothesis:
Air combat is a set of duels; random; linear-law.
To the extent to which it departs from this,
Air combat does not obey the Square Law.
Rather, while there appears to be some Lanchestrian advantage in
numbers and concentration for the attacker (of surface targets),
there is none for the defender – the reverse, even:
Air combat is (somewhat) asymmetric.
Why the asymmetry?
Defender vs attacker of surface targets
Why the asymmetry?
Defender vs attacker of surface targets
Fighters vs bombers with escorts
Why the asymmetry?
Defender vs attacker of surface targets
Fighters vs bombers with escorts
Information asymmetry
Why the asymmetry?
Defender vs attacker of surface targets
Fighters vs bombers with escorts
Information asymmetry
Unmanned aircraft behave differently
Asymmetry in air combat
Take-home messages:
Asymmetry in air combat
Take-home messages:
Air combat is 80% linear-law, 20% asymmetric.
Asymmetry in air combat
Take-home messages:
Air combat is 80% linear-law, 20% asymmetric.
In defensive air power, at least,
Concentration 6= Mass.