Graph Algebra and Richardson’s Arms Race Model Revisited

Graph Algebra and Richardson’s Arms Race Model Revisited Courtney Brown, Ph.D. Emory University

An exact representation of Richardson’s arms race model

Alternative specifications can easily reproduce the model in reduced form. • Alternative #1: True feedback with grievances and ambitions added directly to the dependent variable. • In this case, grievances and ambitions are truly independent of the effects of the opponent’s spending.

Alternative specification #1

Alternative #2: True feedback with grievances and ambition added before processing the input of opponent’s spending • In this case, the grievances and ambitions of leaders are considered in combination with the spending of an opponent prior to processing. • This would be the case if leaders consider the inputs as a lump sum.

Alternative #3: True feedback with grievances and ambition added before processing the input of opponent’s spending • This model focuses on the difference between each country’s spending on arms. • This would be the case in which country X reacts not to the total spending of country Y, but to how much more country Y is spending relative to country X.

Alternative specification #3 Δxt+1 = ayt - (1 + m)axt + ag Δyt+1 = bxt - (1 + n)byt + bh

And on the wild side • It is easy to use graph algebra to include new ideas, such as to use a forced oscillator instead of a constant parameter. • This would make sense if a country’s spending on arms changes with respect to an electoral calendar. • Near an election, leaders of a democratic country may find it useful to encourage fear of the opposing country among the populace, and thus increase arms spending.

Nonautonomous functions of this sort can produce chaos if we add a touch of nonlinearity.

Graph Algebra and Richardson’s Arms Race Model Revisited