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Project Arms Race

Project Arms Race. Jason & Michaelle. Armaments in the Past. 1939 - Richardson Model (explained later) of Arms Race developed for combatants of WWI. Defenses and armaments dealt with at that time were guns (pistols,machine guns), flamethrowers, grenades, tanks. Arms in the present day.

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Project Arms Race

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  1. Project Arms Race Jason & Michaelle

  2. Armaments in the Past • 1939 - Richardson Model (explained later) of Arms Race developed for combatants of WWI. • Defenses and armaments dealt with at that time were guns (pistols,machine guns), flamethrowers, grenades, tanks.

  3. Arms in the present day • Richardsonmodel can still be used today • Defences and armaments dealt with at the present day are nuclear bombs, jet fighters, etc.

  4. Problems to Discuss • Why are nations spurred to arm when other nations build defenses? • How do we develop a model that depicts how a nation determines how much to spend on arms? • How does a nation’s ill or good feelings towards other nations affect the time it takes for nations to be at peace(steady state)? • How do steady states differ in regards to maximum arms expenditure in a nation?

  5. Why are nations spurred to arm when other nations build defenses? • Nations are defensive and fearful in nature and will build defences for own protection. • (eg. Nation ‘X’ builds a tank for protection on time-step of the first year(t=1)) • This might seem offensive in the eyes of other nation(s). • … X(1) = 1

  6. Cont… • Nation ‘Y’ feels threatened and builds twice as many tanks as ‘X’ did at time t=1. • Introduce constant ‘b’. This determines how fearful nation ‘Y’ is of ‘X’. • In this case, b=2. • This is a simplified formulation of how the Richardson Model works. This simplified version is the Mutual Fear Model. • Note: new variables will be added on later slides + b*X(1) X(1) = 1 Y(1) = b*X(1) = 2

  7. How do we develop a model that depicts how a nation determines how much to spend on arms? • We will end up solving a system of continuous-time differential equations which depicts how a nation reacts to an arms increase of another nation • 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. • From previous, it is easy to see we can start with simple model of dx/dt=ay, dy/dt =bx. • Where ‘a’ and ‘b’ are “fear” constants. • i.e. if ‘a’ is large(>1), nation ‘X’ fears nation ‘Y’ more, (=1), nation ‘X’ wants to match ‘Y’. • This mutual fear model doesn’t account for a maximum expenditure of a nation on arms. Therefore we end up with a runaway model that increases infinitely. • …

  8. Hyperbolic function • Recognize that solving dy/dx in the Mutual fear system and plotting the solutions, we have a hyperbolic function: • Later, modified equations will also behave in this manner.

  9. Cont…(adding a maximum constant • Unless nation has unlimited budget, we will need to introduce a new constant • Let Kx and Ky be maximum expenditure on arms of nations X and Y respectively. • Introduce new term (1-x/Kx) into equation • This term =0 if the expenditure of nation X (denoted by ‘x’) at time t equals the maximum expenditure. • Similar for ‘Y’ • Modified model is now • dx/dt = ay(1-x/Kx) • dy/dt = bx(1-y/Ky)

  10. Cont…(adding an economic constant) • Nations while competing sometimes forget they need to take care of their country(economically). • Introduce economic constants ‘m’ and ‘n’ for nations X and Y respectively. • e.g. if nation ‘X’ wants to spend ‘m’ times expenditures of arms on production of cars, we minus the term (mx) from the model. • Result model is :dx/dt = ay(1-x/Kx) - mx • Similar for ‘Y’ : dy/dt = bx(1-y/Ky) – ny

  11. Cont…(adding love/hate constant) • Nations can have underlying grievances or good will towards other nations. • New constants introduced will simply be added to the existing model • e.g. ‘r’ and ‘s’ are the love/hate constant for nations ‘y’ and ‘x’ respectively • If constant is: • >0, adds to expenditure since there is an underlying grievance to corresponding country. • <0, underlying good will • =0 neutral

  12. Finalized Model • Finalized model for 2 nations X,Y: • dx/dt = ay(1-x/Kx) – mx + r • dy/dt = bx(1-y/Ky) – ny + s

  13. Assumptions • Fear constants>= 1 because nations want to either match or better arms expenditures of their competing nation. • Assume a nation’s fear is constant towards all other nations. • Economic constants >=0. • Assume these constants are for the production of the corresponding nation’s highest grossing product in order to boost economy. • m,n increases as love does • Love/Hate constant can be any real number. • Assume nations either loves or hate all other nations, that is a nation either hates all other nations or loves all other nations. • Fear reduces as love increases.

  14. Mathematical Methods • Euler’s method, to solve for our continuous time model of 2 D.E. equations. Using the loop: While (t <= tend) RHS1 = a * y * (1 - (x / Kp)) - (m * x) + r RHS2 = b * x * (1 - (y / Kg)) - (n * y) + s x = x + deltat * RHS1 y = y + deltat * RHS2 t = t + deltat Row = Row + 1

  15. Steady State • Plotting nullclines – shows where steady state is: • a=2, b=1.5 • Kx=7, Ky=5 • m=0.5, n=0.5 • r=0.75, s=1.5 • xint=2, yint=3

  16. Steady State (cont…) • Steady state confirmed, (x = 5.5, y= 4.5)

  17. Stability Analysis • Jacobian of our 2-system model is: • Eigenvalues : -2.16, -0.29 • Therefore the steady states are stable.

  18. 3 Nation Model • We will now add a third nation to the model • Nation Z will be added to the model • Constants c, Kz, l, u will be added as fear, maximum, economic, and love/hate constants respectively. • Now nations are fearful of 2 nations at once. • Final equations of 3 nations become: • dx/dt = 2a(y+z)(1-x/Kx) – mx + r • dy/dt = 2b(x+z)(1-y/Ky) – ny + s • dz/dt = 2c(x+y)(1-z/Kz) – lz + u

  19. Effect of love/hate constantScenario 1 • Scenario 1 (nations all hate each other with Z nation being the biggest hater. • r=0.75,s=1.5,u=3.5 • Peace of arms race at t= 4.9 years

  20. Effect of love/hate constantScenario 2 • Scenario 2 (nations all love each other with X nation being the biggest lover. • r=-5,s=-2.5,u=-1.75 • Nations all disarm at t=3.2 years

  21. Effect of love/hate constantScenario 3 • Scenario 3 (there is some love as nation Y is the only nation who loves. • r=3,s=-2.5,u=1.75 • Peace of arms race at t=3.7 years • Large reason expenditures of nation Y is low • Only country with no grievances.

  22. How do steady states differ in regards to maximum arms expenditure in a nation? • With Z being richest nation, Y being poorest, steady states are higher and lower, respectively. • How rich or poor a country is reflects position of steady state.

  23. Model Critique • Variables were very interactive, therefore some of them were confounding if not changed appropriately. • Could change some of these variables to be proportional to say the love variable. • With D.E. of 3 nations, could have allowed the nations to hate one and love one, instead of having to make them hate all or love all. • That is to say we could introduce a different fear constant for each nation involved.

  24. NO NUKES! NO NUKES!

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