project arms race

Project Arms RaceProject Arms Race

Jason Jason

&&

MichaelleMichaelle

Armaments in the PastArmaments in the Past

1939 - Richardson Model (explained later) 1939 - Richardson Model (explained later) of Arms Race developed for combatants of of Arms Race developed for combatants of WWI.WWI.• Defenses and armaments dealt with at that Defenses and armaments dealt with at that

time were guns (pistols,machine guns), time were guns (pistols,machine guns), flamethrowers, grenades, tanks.flamethrowers, grenades, tanks.

Arms in the present dayArms in the present day

RichardsonRichardson model can still be used todaymodel can still be used today• Defences and armaments dealt with at Defences and armaments dealt with at

the present day are nuclear bombs, jet the present day are nuclear bombs, jet fighters, etc.fighters, etc.

Problems to DiscussProblems to Discuss Why are nations spurred to arm when other Why are nations spurred to arm when other

nations build defenses?nations build defenses?

How do we develop a model that depicts how a How do we develop a model that depicts how a nation determines how much to spend on arms?nation determines how much to spend on arms?

How does a nation’s ill or good feelings towards How does a nation’s ill or good feelings towards other nations affect the time it takes for nations other nations affect the time it takes for nations to be at peace(steady state)?to be at peace(steady state)?

How do steady states differ in regards to How do steady states differ in regards to maximum arms expenditure in a nation?maximum arms expenditure in a nation?

Why are nations spurred to arm Why are nations spurred to arm when other nations build defenses?when other nations build defenses? Nations are defensive and fearful in nature and Nations are defensive and fearful in nature and

will build defences for own protectionwill build defences for own protection..• (eg. Nation ‘X’ builds a tank for protection on time-step of the (eg. Nation ‘X’ builds a tank for protection on time-step of the

first year(t=1))first year(t=1))

This might seem offensive in the eyes of other This might seem offensive in the eyes of other nation(s)nation(s)..

……

X(1) = 1X(1) = 1

Cont…Cont… Nation ‘Y’ feels threatened and builds twice as many tanks Nation ‘Y’ feels threatened and builds twice as many tanks

as ‘X’ did at time t=1.as ‘X’ did at time t=1.• Introduce constant ‘b’. This determines how fearful nation ‘Y’ Introduce constant ‘b’. This determines how fearful nation ‘Y’

is of ‘X’.is of ‘X’. In this case, b=2.In this case, b=2.

This is a simplified formulation of how the Richardson Model This is a simplified formulation of how the Richardson Model works. This simplified version is the Mutual Fear Model.works. This simplified version is the Mutual Fear Model.• Note: new variables will be added on later slidesNote: new variables will be added on later slides

++X(1) = 1X(1) = 1

b*X(1)b*X(1)

Y(1) = b*X(1) = 2Y(1) = b*X(1) = 2

How do we develop a model that depicts How do we develop a model that depicts how a nation determines how much to how a nation determines how much to

spend on arms?spend on arms?• We will end up solving a system of continuous-time differential We will end up solving a system of continuous-time differential

equations which depicts how a nation reacts to an arms equations which depicts how a nation reacts to an arms increase of another nationincrease of another nation• For ease of explanation, we will use a system of 2 equations while For ease of explanation, we will use a system of 2 equations while

formulating the model, and add one more nation to the model later.formulating the model, and add one more nation to the model later.

• From previous, it is easy to see we can start with simple model From previous, it is easy to see we can start with simple model of of dx/dt=ay, dy/dt =bxdx/dt=ay, dy/dt =bx..• Where ‘a’ and ‘b’ are “fear” constants.Where ‘a’ and ‘b’ are “fear” constants.

• i.e. if ‘a’ is large(>1), nation ‘X’ fears nation ‘Y’ more, (=1), nation ‘X’ i.e. if ‘a’ is large(>1), nation ‘X’ fears nation ‘Y’ more, (=1), nation ‘X’ wants to match ‘Y’.wants to match ‘Y’.

• This mutual fear model doesn’t account for a maximum This mutual fear model doesn’t account for a maximum expenditure of a nation on arms. Therefore we end up with a expenditure of a nation on arms. Therefore we end up with a runaway model that increases infinitely.runaway model that increases infinitely.

• ……

Hyperbolic functionHyperbolic function Recognize that solving dy/dx in the Mutual fear system and Recognize that solving dy/dx in the Mutual fear system and

plotting the solutions, we have a hyperbolic function:plotting the solutions, we have a hyperbolic function:

Later, modified equations will Later, modified equations will

also behave in this manner.also behave in this manner.

Phaseplane (Mutual Fear)

0

1E+100

2E+100

3E+100

4E+100

5E+100

6E+100

7E+100

0 1E+100 2E+100 3E+100 4E+100 5E+100 6E+100

Purple

Gre

en Series1

Cont…(adding a maximum Cont…(adding a maximum constantconstant

Unless nation has unlimited budget, we will need Unless nation has unlimited budget, we will need to introduce a new constantto introduce a new constant

Let Kx and Ky be maximum expenditure on arms Let Kx and Ky be maximum expenditure on arms of nations X and Y respectively.of nations X and Y respectively.• Introduce new term (1-x/Kx) into equationIntroduce new term (1-x/Kx) into equation• This term =0 if the expenditure of nation X (denoted by This term =0 if the expenditure of nation X (denoted by

‘x’) at time t equals the maximum expenditure‘x’) at time t equals the maximum expenditure.. Similar for ‘Y’Similar for ‘Y’

Modified model is now Modified model is now • dx/dt = ay(1-x/Kx)dx/dt = ay(1-x/Kx)• dy/dt = bx(1-y/Ky)dy/dt = bx(1-y/Ky)

Cont…(adding an economic constant)Cont…(adding an economic constant)

Nations while competing sometimes forget Nations while competing sometimes forget they need to take care of their they need to take care of their country(economically).country(economically).

Introduce economic constants ‘m’ and ‘n’ Introduce economic constants ‘m’ and ‘n’ for nations X and Y respectively.for nations X and Y respectively.• e.g. if nation ‘X’ wants to spend ‘m’ times e.g. if nation ‘X’ wants to spend ‘m’ times

expenditures of arms on production of cars, we expenditures of arms on production of cars, we minus the term (mx) from the model.minus the term (mx) from the model.

Result model is :Result model is :dx/dt = ay(1-x/Kx) - mxdx/dt = ay(1-x/Kx) - mx Similar for ‘Y’ : Similar for ‘Y’ : dy/dt = bx(1-y/Ky) – nydy/dt = bx(1-y/Ky) – ny

Cont…(adding love/hate constant)Cont…(adding love/hate constant)

Nations can have underlying grievances or good Nations can have underlying grievances or good will towards other nations.will towards other nations.

New constants introduced will simply be added to New constants introduced will simply be added to the existing modelthe existing model• e.g. ‘r’ and ‘s’ are the love/hate constant for e.g. ‘r’ and ‘s’ are the love/hate constant for

nations ‘y’ and ‘x’ respectivelynations ‘y’ and ‘x’ respectively

If constant is:If constant is:• >0, adds to expenditure since there is an underlying >0, adds to expenditure since there is an underlying

grievance to corresponding country.grievance to corresponding country.• <0, underlying good will<0, underlying good will• =0 neutral=0 neutral

Finalized ModelFinalized Model

Finalized model for 2 nations X,Y:Finalized model for 2 nations X,Y:

• dx/dt = ay(1-x/Kx) – mx + rdx/dt = ay(1-x/Kx) – mx + r

• dy/dt = bx(1-y/Ky) – ny + sdy/dt = bx(1-y/Ky) – ny + s

AssumptionsAssumptions

• Fear constants>= 1 because nations want to Fear constants>= 1 because nations want to either match or better arms expenditures of their either match or better arms expenditures of their competing nation.competing nation.

Assume a nation’s fear is constant towards all other Assume a nation’s fear is constant towards all other nations.nations.

• Economic constants >=0. Economic constants >=0. Assume these constants are for the production of the Assume these constants are for the production of the

corresponding nation’s highest grossing product in order corresponding nation’s highest grossing product in order to boost economy.to boost economy.

m,n increases as love doesm,n increases as love does

• Love/Hate constant can be any real number.Love/Hate constant can be any real number. Assume nations either loves or hate all other nations, that Assume nations either loves or hate all other nations, that

is a nation either hates all other nations or loves all other is a nation either hates all other nations or loves all other nations. nations.

Fear reduces as love increases.Fear reduces as love increases.

Mathematical MethodsMathematical Methods Euler’s method, to solve for our continuous Euler’s method, to solve for our continuous

time model of 2 D.E. equations. Using the time model of 2 D.E. equations. Using the loop:loop:

While (t <= tend)While (t <= tend) RHS1 = a * y * (1 - (x / Kp)) - (m * x) + rRHS1 = a * y * (1 - (x / Kp)) - (m * x) + r RHS2 = b * x * (1 - (y / Kg)) - (n * y) + sRHS2 = b * x * (1 - (y / Kg)) - (n * y) + s

x = x + deltat * RHS1x = x + deltat * RHS1 y = y + deltat * RHS2y = y + deltat * RHS2

t = t + deltatt = t + deltat Row = Row + 1Row = Row + 1

Steady StateSteady State

Plotting nullclines – shows where Plotting nullclines – shows where steady state is:steady state is:

• a=2, b=1.5a=2, b=1.5

• Kx=7, Ky=5Kx=7, Ky=5

• m=0.5, n=0.5m=0.5, n=0.5

• r=0.75, s=1.5r=0.75, s=1.5

• xint=2, yint=3xint=2, yint=3

Steady State (cont…)Steady State (cont…)Expenditures vs Time (Mutual Grievance_withMax)

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12.6

25.2

37.8

50.4 63

75.6

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126

139

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189

Time

Exp

end

itu

res

Expenditures of X

Expenditures of Y

• Steady state confirmed, (x = 5.5, y= 4.5)Steady state confirmed, (x = 5.5, y= 4.5)

Stability AnalysisStability Analysis

Jacobian of our 2-system model is:Jacobian of our 2-system model is:

•Eigenvalues : -2.16, -0.29Eigenvalues : -2.16, -0.29

•Therefore the steady states are Therefore the steady states are stable.stable.

3 Nation Model3 Nation Model We will now add a third nation to the modelWe will now add a third nation to the model

Nation Z will be added to the model Nation Z will be added to the model • Constants Constants c, Kz, l, uc, Kz, l, u will be added as fear, maximum, will be added as fear, maximum,

economic, and love/hate constants respectively.economic, and love/hate constants respectively.

Now nations are fearful of 2 nations at Now nations are fearful of 2 nations at once.once.

Final equations of 3 nations become:Final equations of 3 nations become:• dx/dt = 2a(y+z)(1-x/Kx) – mx + rdx/dt = 2a(y+z)(1-x/Kx) – mx + r• dy/dt = 2b(x+z)(1-y/Ky) – ny + sdy/dt = 2b(x+z)(1-y/Ky) – ny + s• dz/dt = 2c(x+y)(1-z/Kz) – lz + udz/dt = 2c(x+y)(1-z/Kz) – lz + u

Effect of love/hate constantEffect of love/hate constantScenario 1Scenario 1

Scenario 1 (nations all hate each other with Z Scenario 1 (nations all hate each other with Z nation being the biggest hater.nation being the biggest hater.

Expenditures vs Time (3 Nations)

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Expenditures of X

Expenditures of Y

Expenditures of Z

• r=0.75,s=1.5,r=0.75,s=1.5,u=3.5u=3.5

• Peace of arms Peace of arms race at t= 4.9 yearsrace at t= 4.9 years


Scenario 2 (nations all love each other with X Scenario 2 (nations all love each other with X nation being the biggest lover.nation being the biggest lover.

• r=-5r=-5,s=-2.5,u=-1.75,s=-2.5,u=-1.75

• Nations all disarm at Nations all disarm at t=3.2 yearst=3.2 years


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Expenditures of X

Expenditures of Y

Expenditures of Z


Scenario 3 (there is some love as nation Y is the Scenario 3 (there is some love as nation Y is the only nation who loves.only nation who loves.

• r=3,r=3,s=-2.5s=-2.5,u=1.75,u=1.75

• Peace of arms race Peace of arms race at t=3.7 yearsat t=3.7 years

•Large reason Large reason expenditures of expenditures of nation Y is lownation Y is low

•Only country with no Only country with no grievances.grievances.


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Exp

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Expenditures of X

Expenditures of Y

Expenditures of Z

How do steady states differ in regards to How do steady states differ in regards to maximum arms expenditure in a nation?maximum arms expenditure in a nation?


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Time

Exp

end

itu

res

Expenditures of X

Expenditures of Y

Expenditures of Z

• With Z being richest nation, Y being poorest, With Z being richest nation, Y being poorest, steady states are higher and lower, respectively.steady states are higher and lower, respectively.

• How rich or poor a country is reflects position of How rich or poor a country is reflects position of steady state.steady state.

Model CritiqueModel Critique

Variables were very interactive, therefore Variables were very interactive, therefore some of them were confounding if not some of them were confounding if not changed appropriately.changed appropriately.• Could change some of these variables to be Could change some of these variables to be

proportional to say the love variable.proportional to say the love variable.

With D.E. of 3 nations, could have allowed With D.E. of 3 nations, could have allowed the nations to hate one and love one, the nations to hate one and love one, instead of having to make them hate all or instead of having to make them hate all or love all.love all.• That is to say we could introduce a different fear That is to say we could introduce a different fear

constant for each nation involved.constant for each nation involved.

NO NUKES! NO NUKES!NO NUKES! NO NUKES!

project arms race

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