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Math 3MB3 Final Project Report

Project 5: Arms Race

Imagine two economically competing countries: Purple and Green. Both countries desirepeace and hope to avoid war.
While neither country will go out of its way to launch an aggression, it also would not sitidly by if their country was attacked. Each country believes in self-defence and will prepareto fight to protect itself. Both nations feel that the maintenance of certain amount of armyand the stockpiling of weapons is "defensive" when they do it but "offensive" when the othernation does it.

In this project, you will investigate with a simple model about how arms or militaryspending varies over time in each nation, based on various assumptions about how eachcountry's arming and military  behaviour influences the other.

Base model

Let p(t) and g(t) represent the total arms and military spending in year t by Purple andGreen, respectively. The two nations are in competition, so there is an underlying sense ofmutual fear: the more one nation arms, the more the other nation is incentivized to arm.To start, we could assume that each nation's armament rate is directly proportional to theother's arms spending (these should be linear terms in your model). We may also want toaccount for the fact that excessive armament expenditures come at a cost to each nation'seconomy, i.e., the rate of change of each nation's expenditures is directly and negativelyproportional to its own expenditure (again, these should be linear terms in your model).The sign of these constant parameters determines whether the term models grievances, whichincrease armament spending, or feelings of good will, which decrease it.

Construct a continuous model to describe this scenario. What does your model predictfor the long-term arms spending of each country? Notice that analytical discussions will berequired to get full score of the this part of the base model for this topic.Finally, we can additionally model any underlying grievances or feelings of goodwill ofeach country toward the other using constant terms, which should be invariant to spend-Try to discuss the impact of such goodwills to the model predictions usingings or time.simulations.

Possible extensions

Logistic mutual fear

For the mutual fear terms, assume that there is some inherent limit to the amount a nationcan spend on armaments each year (call it K, and K, for Purple and Green, respectively).Then, you could replace the linear terms with logistic terms (e.g. a(1 - p(t)/Kp)9(t) forPurple). What could be changed in your model prediction after the replacement?