Tragedy of Great Power Politics? Modeling International War
Recently I finished reading John Mearsheimer’s excellent political science book The Tragedy of Great Power Politics. In this book, Mearscheimer lays out his “offensive realism” theory of how countries interact with each other in the world. The book is quite readable and wellthoughtout — I’d recommend it to anyone who has an inkling for political history and geopolitics. However, as I was reading this book, I decided that there was a point of Mearsheimer’s argument which could be improved by a little mathematical analysis.
The main tenant of the book is that states are rational actors who act to to maximize their standing in the international system. However, states don’t seek to maximize their absolute power, but instead their relative power as compared to the other states in the system. In other words, according to this logic the United Kingdom position in the early 19th century — when its army and navy could trounce most of the other countries on the globe — was better than it is now — when many other countries’ armies and navies are comparable to that of the UK, despite the UK current army and navy being much better now than they were in the early 19th century. According to Mearsheimer, the main determinant of state’s international actions is simply maximizing its relative power in its region. All other considerations — capitalist or communist economy, democratic or totalitarian government, even desire for economic growth — matter little in a state’s choice of what actions it will take. (Perhaps it was this simplification of the problem which made the book really appeal to me as a physicist.)
Most of Mearsheimer’s book is spent exploring the logical corollaries of his
main tenant, along with some historical examples. He claims that his idea has
three different predictions for three different possible systems. 1) A balanced
bipolar system (one where two states have roughly the same amount of power and
no other state has much to speak of) is the most stable. War will probably not
break out since, according to Mearsheimer, each state has little to gain from a
war. (His example is the Cold War, which didn’t see any actual conflict between
the US and the USSR.) 2) A balanced multipolar system (
While I liked Mearsheimer’s argument in general, something irked me about the statement about bipolarity being stable. I didn’t think that the stability of bipolarity (corollary 1 above) actually followed from his main hypothesis. After spending some extra time thinking in the shower, I decided how I could model Mearsheimer’s main tenant quantitatively, and that it actually suggested that bipolarity was actually unstable!!
Let’s see if we can’t quantify Mearsheimer’s ideas with a model. Each state in
the system has some power, which we’ll call
where we take the sum over the relevant players in the system. If there was
some action that changed the power of some of the players in the system (say a
war), then the relative power would also change with time
A state will pursue an action that increases its relative power
 War always reduces a state’s absolute power. This is simply a statement that
in general, war is destructive. Many people die and buildings are bombed,
neither of which is good for a state. Mathematically, this statement is that in
wartime,
$dP_i/dt < 0$ always. Note that this doesn’t imply that that$dR_i/dt$ is always negative.

The change in power of two states A & B in a war should depend only on how much power A & B have. In addition, it should be independent of the labeling of states. Mathematically,
$dP_a / dt = f(P_a, P_b)$ , and$dP_b/dt = f(P_b, P_a)$ with the same function$f$ ^{[2]}. 
If State A has more absolute power than State B, and both states are in a war, then State B will lose power more rapidly than State A. This is almost a restatement of our definition of power. We defined power such that if State A has more absolute power than State B, then State A will win a war against State B. So we’d expect that power translates to the ability to reduce another state’s power, and more power means the ability to reduce another state’s power more rapidly.

For simplicity, we’ll also notice that the decrease of a State A’s absolute power in wartime is largely dependent on the power of State B attacking it, and is not so much dependent on how much power State A has.
In general, I think that assumptions 13 are usually true, and assumption 4 is pretty reasonable. But to simplify the math a little more, I’m going to pick a definite form for the change of power. The simplest possible behavior that capture all 4 of the above assumptions is:
where
Let’s examine the case that was bothering me most — a balanced bipolar system.
Now we have only two states in the system, X and Y. For starters, let’s address
the case where both states start out with equal power
But it gets worse. For a real balanced bipolar system, both states won’t have
exactly the same power, but will instead be approximately equal. Let’s say that
the relative power between the two states differs by some small (positive)
number
In other words, if the power balance is slightly upset, even by an infinitesimal amount, then the more powerful state should go to war! For a balanced bipolar system, peace is unstable, and the two countries should always go to war according to this simple model of Mearsheimer’s realist world.
Of course, we’ve just considered the simplest possible case — only two states in the system (whereas even in a bipolar world there are other, smaller states around) who act with perfect information (i.e. both know the power of the other state) and can control when they go to war. Also, we’ve assumed that relative power can change only through a decrease of absolute power, and in a deterministic way (as opposed to something like economic growth). To really say whether bipolarity is stable against war, we’d need to address all of these in our model. A little thought should convince you which of these either a) makes a bipolar system stable against war, and b) makes a bipolar system more or less stable compared to a multipolar system. Maybe I’ll address these, as well as balanced and unbalanced multipolar systems, in another blog post if people are interested.
1. ^
for all
2. ^ This implicity assumes that it doesn’t matter which state attacked the other, or where the war is taking place, or other things like that.
3. ^ Incidentally this form for the rateofchange of the power also has the
advantage that it is scalefree, which we might expect since there is no
intrinsic “power scale” to the problem. Of course there are other forms with
this property that follow some or all of the assumptions above. For instance,
something of the form
4. ^ Homework for those who are not satisfied with my assumptions: Show that any
functional form for
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