Thursday, January 30, 2014

War is Mostly in the Error Term, But That's Okay



Erik Gartzke's 1999 article "War is in the Error Term" is often described as a critique of Fearon 1995's "Rationalist Explanations for War". I'm not sure where this interpretation came from, because the article itself argues (appropriately) that we must either accept the limitations implied by Fearon's work or hold logic in contempt. Gartzke is not claiming to be engaged in reductio ad absurdum.\(^1\) In no way is he suggesting that we should or can dismiss Fearon's argument because we don't like its conclusions. But nonetheless, the idea is out there, and people sometimes ask me how I'd respond to it, so I figured I'd write this post.

Brief Synopsis

There are two points I want to make. The first is that Gartzke overstates the implications of Fearon's work.\(^2\) Gartzke suggests, as many believe, that Fearon's argument implies that observable factors have no bearing on the equilibrium probability of war among those states that are actually engaged in meaningful crisis bargaining (which Gartzke rightly argues most states probably are not). That's potentially true within certain model specifications, but it is not a general implication of Fearon's work or bargaining models more generally, as I've discussed before.

Second, even if all we could hope to achieve through empirical analysis was gaining a better sense of which states are truly at no risk of war as compared to those that are taking calculated risks but getting lucky, that's no reason to pack up and go home. We need a better understanding of which 0s represent cases where one side won the hand without having to go to the flop and which are cases where no one was even playing poker.

I have heard some big names in the field express the view that the reason why there is so little new empirical work on interstate conflict is because Fearon proved that there was nothing left to learn from the endeavor. I think this is wrong for at least two reasons, one of which is of little relevance to this post---that is, I think most empirically minded conflict scholars no longer have much interest in interstate conflict, believing that civil conflict is the only real game in town\(^3\)---but insofar as anyone truly believes that it's no longer possible to do meaningful empirical work on the study of interstate conflict because of Fearon, they are mistaken. I hope to convince a few of you of that below.\(^4\)

Details of the Argument

The following is but one way of illustrating the above claims. Various different assumptions could be made along the way to proving the same point, and the parameter values chosen for the simulations are admittedly arbitrary. However, those of you who like to see things worked out in detail may start to see what is ultimately a broader point after going through this. If not, I encourage you to play around with some models and simulations of your own.

Suppose we have two states, \(A\) and \(B\). They bargain over some good or bundle of goods normalized to be worth 1 to each side. Let \(x\) be the share that would be allocated to \(A\) under a negotiated agreement and \(1-x\) the share that would be allocated to \(B\), with \(u_i(x) = x\) for all \(i \in \{A,B\}\).

Assume that \(A\) is expected to acquire \(w\) of the good in war, leaving \(B\) with \(1-w\). Naturally, each side will incur some loss of utility associated with incurring the costs of war. Let \(c_A\) and \(c_B\) denote the amount of utility to be lost by \(A\) and \(B\), respectively.

Thus, we have
\begin{array}{ccc}
& u_A & u_B\\
\mbox{Negotiation} & x & 1-x\\
\mbox{War} & w-c_A & 1-w-c_B.\\
\end{array}

Now let's put some structure on \(w\). Assume that \(w = \frac{e_Am_A}{e_Am_A + e_Bm_B}\), where \(e_i>0\) is \(i\)'s martial effectiveness, by which I mean all the intangible factors like morale and courage and battlefield acumen related to how good \(i\)'s forces are at the actual conduct of war, and \(m_i>0\) is \(i\)'s military capabilities, by which I mean all the physical resources such as personnel, tanks, bombs, and planes, that can be brought to bear in a conflict. This formulation ensures that \(A\)'s expected share is increasing in both \(A\)'s effectiveness and capabilities, decreasing in \(B\)'s effectiveness and capabilities, and that the importance of each factor depends directly on the other. That is, with apologies to Sylvester Stallone, a single soldier isn't going to defeat an opposing army, no matter how well-trained he is. Similarly, having lots of shiny toys with which to blow stuff up doesn't help if your army is full of inexperienced, incompetent cowards.

Finally, let's add incomplete information, and do so in a very simple way. Let \(e_B\) be known to \(B\) but not \(A\). All \(A\) knows is that \(e_B = \color{blue}{\underline{e}_B}\) with probability \(\color{blue}{\phi}\) and \(e_B = \color{red}{\overline{e}_B}\) with probability \(\color{red}{1-\phi}\), where \(\color{blue}{\underline{e}_B} < \color{red}{\overline{e}_B}\). Everything else, including \(m_B\), is assumed to be common knowledge. That is, \(A\) knows how big a fighting for \(B\) has, and how well equipped that force is, but \(A\) does not know how well-trained they are, how brave they are, or whether the commanding officers will devise appropriate battlefield strategies. But \(A\) can make an educated guess. And as \(|\color{red}{\overline{e}_B} - \color{blue}{\underline{e}_B}| \rightarrow 0\), \(A\)'s uncertainty vanishes. That is, \(A\) is less and less uncertain about \(B\)'s martial effectiveness as the best and worst case scenarios grow more and more similar.

Note that this necessarily implies that \(w = \frac{e_Am_A}{e_Am_A + \color{blue}{\underline{e}_B}m_B}\), which we can denote \(\color{blue}{\overline{w}}\) for notational convenience, with probability \(\color{blue}{\phi}\), and  \(w = \frac{e_Am_A}{e_Am_A + \color{red}{\overline{e}_B}m_B}\), which I'll denote \(\color{red}{\underline{w}}\), with probability \(\color{red}{1-\phi}\), where \(\color{red}{\underline{w}} < \color{blue}{\overline{w}}\).

One of two things will happen in equilibrium. Either \(A\) will set \(x=\color{red}{\underline{w}} + c_B\), which \(B\) accepts for sure, or \(A\) sets  \(x=\color{blue}{\overline{w}} + c_B\), which \(B\) accepts if and only if \(B\)'s martial effectiveness is relatively low. That is, either \(A\) chooses a value of \(x\) that ensures peace or \(A\) makes a demand that provokes war with probability \(\color{red}{1-\phi}\). Let's call the smaller, safer version \(\color{red}{\underline{x}}\) and the larger, riskier one \(\color{blue}{\overline{x}}\).

Because I want to elucidate the implications for empirical work, let's now turn to simulations. As usual, I'll provide Stata code, but it's easy enough to reproduce what I'm doing here in R.
set obs 100000
gen eA = uniform()
gen eBh = uniform()
gen diff = uniform()
gen eBl = diff*eBh
What I've done here is created 100,000 observations and drawn random values for the martial effectiveness terms. \(e_A\) and \(\color{red}{\overline{e}_B}\) are uniformly distributed between 0 and 1, as is the relative size of \(\color{blue}{\underline{e}_B}\).

While \(B\)'s type is unknown to \(A\), and to us as analysts when we deal with real world data, to generate the simulated data, we need to assign a type.
gen phi = uniform()
gen draw = uniform()
gen low = draw < phi
Next, the rest of our basic parameters. I'll limit the range of the cost terms to make sure \(A\) finds gambling attractive in a non-trivial number of cases.
gen mA = uniform()
gen mB = uniform()
gen cA = uniform()
gen cB = uniform()
replace cA = 0.25*cA
replace cB = 0.25*cB
  
Let's create a standard balance of capabilities measure, the sort most people dump into their empirical models just because other studies have found that it correlates with dispute onset. This will be useful later.
gen balance = 1 - (mA/(mA+mB)) if mA>=mB
replace balance = 1 - (mB/(mA+mB)) if mB>mA
This variable ranges from 0 (complete preponderance) to 0.5 (capabilities perfectly balanced). You'll find it, or ones just like it, in lots of publications.

Now let's specify the \(w\)'s, per above.
gen wl = (eA*mA)/(eA*mA + eBh*mB)
gen wh = (eA*mA)/(eA*mA + eBl*mB)
Then the \(x\)'s.
gen xl = wl + cB
gen xh = wh + cB
We're going to need to know which demand \(A\) makes. That requires us to compare the payoffs. Recall, \(B\) always accepts the smaller demand. So when \(A\) sets \(x=\color{red}{\underline{x}}\), \(A\) simply gets \(\color{red}{\underline{x}}\). But \(\color{blue}{\overline{x}}\) is only accepted if \(B\) is relatively low in martial effectiveness (i.e., with probability \(\color{blue}{\phi}\)). When it is, \(A\) gets \(\color{blue}{\overline{x}}\), but when it is not, \(A\) fights a war (i.e., with probability \(\color{red}{1-\phi}\)) against the more martially effective type, and receives \(\color{red}{\underline{w}}-c_A\).

gen uxl = xl
gen uxh = phi*xh + (1-phi)*(wl-cA)
Of course, \(A\) risks war if and only if \(A\) expects to be better off when doing so.
gen risk = uxh>uxl
Finally, war occurs if and only if \(A\) risked war and just so happens to be facing the wrong type.
gen war = risk if low==0 
Still with me? I know that was a lot, and all we've done so far is set things up, so I apologize. You deserve a break. Go get yourself a drink, or something to eat. Then come back. I'll be here.

Ready?

Estimate the following model, which you'll note contains nothing but the one variable that we know we can measure pretty reliably and have been using for decades.
logit war balance
Did you get some nice results? Lots of pretty stars? I bet you did.

That's not a coincidence. I could prove to you algebraically that the model above logically implies that war is more likely to occur when \(A\) and \(B\) are near parity, but I won't. I've done so elsewhere, as have others, and this post is already pretty long. But if you're the type of person who believes stars in Stata over proofs in mathematical appendices, I hope that made an impression on you. Fearon's claim that war occurs when states get unlucky with their gambles does not imply that war is entirely determined by chance. It means that there is an element of chance. It means we will never have models that predict with perfect accuracy.

But so what? Did you really think that was the goal? If so, you're in the wrong business. War is not the only political phenomenon that we're extremely unlikely to ever predict with perfect accuracy.

In fact, the empirical models most of us use require there to be an element of chance. I'll spare you the statistical explanation. If you don't believe me, try estimating the following model and see what happens.
logit war risk low
In short, not only is it intellectually indefensible to dismiss Fearon's argument on the grounds that you'd have no reason to do the only kind of research you know how to do if he was right, it is actually not true that his argument suggests this anyway. I'm not saying his argument has no implications for the way the more inductive, less theoretically-minded among us ought to think about war. It absolutely does. But the notion that it's pointless to continue studying interstate conflict empirically because that would be the equivalent of trying to explain why some coin flips come up heads is not one of them.


1. If you're unfamiliar with this term, or too lazy to consult Wikipedia, this is the fancy name for arguments of the form: "well, if we believed \(p\), we'd have to believe \(q\), and \(q\) in turn implies \(c\), which is crazy, so obviously \(p\) is wrong."

2. He also understated them in a different way, necessitating this important followup.

3. I don't doubt that Davenport and Gates, as well as others who were working on civil conflict before it got sexy, frequently encountered the four claims in these two posts, but I've never heard anyone say anything like this myself. Next conference you attend, be it ISA, MPSA, APSA, or Peace Science, pay close attention to how many conflict papers study inter- versus intra-state conflict. Pay attention too to whether they are empirical or theoretical. Finally, pay attention to the presenter's age. I bet you'll find that the overwhelming majority of empirical papers by people under the age of 40 focus in intrastate conflict. If I'm wrong about that, I'll gladly buy you a beverage of your choice. 

4. This also presupposes that interstate war is inefficient and thus requires us to explain why agreements were not reached, while the same need not be said of civil conflict. I can't think of a good reason why that might be true, even if Fearon himself is famous for an account of civil war that is essentially decision-theoretic in nature.

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