## Sunday, January 6, 2013

### Breaking Down Slantchev 2012

I began this series of posts by discussing Fearon's influential "Rationalist Explanations for War", wherein he argues that we cannot understand war if we cannot answer the question "Why war and not negotiation?"  Fearon identified three classes of answers, and provided three specific answers within one of those classes.

That is, Fearon first acknowledges both that leaders who are prone to various errors or pathologies might fight wars no matter how inefficient it is to do so and also that war need not be seen as inefficient from the perspective of individual leaders.  In other words, he is quite transparent about the fact that the three "rationalist" explanations for war he provides are necessary to explain war if and only if we view war as inefficient.$$^1$$

Why might we assume that war is inefficient?  There are two reasons.  First, it has been conventional at least since Clausewitz, if not earlier, to view war as a means to an end rather than an end unto itself.  Second, war is costly.  Some of those costs are born by the leader's subjects rather than the leader herself, but not all of them.  If nothing else, wars cost money.  Often quite a lot of it.

This brings us to Slantchev's "Borrowed Power: Debt Finance and the Resort to Arms."  One thing Fearon overlooked is the fact that wars are often financed through loans -- and that these loans are repaid less often by the vanquished than the victorious.  That simple insight gives us an explanation for war that differs from those identified by Fearon and also has the virtue of allowing us to distinguish ex ante between observations where the argument is likely to apply and those where it is not.

Brief Synopsis

The intuition behind Slantchev's argument is that war is not always a simple means to an end -- it sometimes creates indirect benefits that are not available through peaceful agreements.  Note that the same is true of wars fought in the shadow of a rapid shift in power (see also here and here) and diversionary wars (see also here).  But the mechanism here is a bit different.  Slantchev's argument is not about war forestalling adverse shifts in power or enhancing a leader's ability to retain office, but reducing the expected cost of servicing debts taken on in peacetime.  In essence, he argues that sometimes it is peace that is inefficient.

The model Slantchev analyzes assumes that states who lose wars are not responsible for their debts all.  But as I'll show below, this assumption is not critical to his argument.  All we need assume is that the expected cost of servicing debts after fighting a war is lower than it is in times of peace.

As is the norm for these posts, I won't walk you through the actual model in Slantchev's article.  Rather, I will analyze a simpler model so that the proofs will be easy to follow.

Details of the Argument

Traditionally, we assume that war is a costly lottery.  Side 1 wins full control of the dispute benefits with probability $$p$$, side 2 wins with the complementary probability, and both incur some costs, denote $$c_1>0$$ for 1 and $$c_2>0$$ for 2.  We typically write the war payoffs as $$p - c_1$$ for 1 and $$1 - p - c_2$$ for 2, reduced from $$p(1-c_1) + (1-p)(-c_1)$$ and $$p(-c_2) + (1-p)(1-c_2)$$.

Now suppose that those who lose repay a fraction of their debts, the full cost of which would be $$d_1$$ for 1 and $$d_2$$ for 2.  (Alternatively, we could assume that some fraction of those who lose repay none of their debts, but it makes little difference either way).  The war payoffs would then be $$p(1-c_1-d_1) + (1-p)(-\delta_1d_1 - c_1)$$ and $$p(-c_2 -\delta_2d_2) + (1-p)(1-c_2-d_2)$$, where $$\delta_1 \in (0,1)$$ is the portion of 1's debt that 1 will repay if 1 loses the war and $$\delta_2 \in (0,1)$$ is the portion of 2's debt that 2 will repay if 2 loses the war.

Without loss of generality, we can simply say that 1 expects to control $$w$$ of the good following a war.  Maybe that war will be fought to the bitter end, in which case $$w$$ is simply 1's probability of victory, but maybe it's not.  It doesn't matter.  We can also say that 1 expects to repay $$\gamma_1 \in (0, \delta_1)$$ of 1's debt, while 2 expects to repay $$\gamma_2 \in (0,\delta_2)$$ of 2's debt.  We can forget about victory and defeat and work with slightly cleaner, reduced form payoffs of $$w - c_1 - \gamma_1d_1$$ for 1 and $$1 - w - c_2 - \gamma_2d_2$$ for 2.

If 1 receives $$x - d_1$$ from a peaceful agreement, and 2 receives $$1 - x - d_2$$, then war is inefficient if and only if we can identify some range of $$x$$s that ensure $$x - d_1 \geq w - c_1 - \gamma_1d_1$$ and $$1 - x - d_2 \geq 1 - w - c_2 - \gamma_2d_2$$ simultaneously.

We can rewrite the first inequality as $$x \geq \underline{x}$$, where $$\underline{x} \equiv w - c_1 + d_1(1-\gamma)$$.  That is, any peaceful agreement must allocate at least $$\underline{x}$$ of the good to 1.

We can rewrite the second inequality as $$x \leq \overline{x}$$, where $$\overline{x} \equiv w + c_2 - d_2(1-\gamma_2)$$.  That is, no peaceful agreement cannot allocate more than $$\overline{x}$$ to 1.

Provided that $$\underline{x} < \overline{x}$$, divisions of the good exist that satisfy both conditions.  The bargaining range will be non-empty.  If, on the other hand, $$\overline{x} < \underline{x}$$, the only divisions of the good large enough to be acceptable to 1 will be too large for 2 and war will be unavoidable.

Through simple algebraic manipulation, it is straightforward to show that $$\overline{x} < \underline{x}$$ is equivalent to $$c_1 + c_2 \geq d_1(1-\gamma_1) + d_2(1-\gamma_2)$$.  Intuitively, if the costs of war are sufficiently large, then war remains inefficient and there exists a range of proposals that are mutually preferred to war.  But if the cost of war is sufficiently low, peace is inefficient and no such range exists.  How large the costs of war must be to guarantee the existence of a bargaining range depends both on how costly it is to service the debts each side has taken on and the degree to which fighting a war is expected to reduce these costs.  The costlier the debts, and the smaller the fraction that each side expects to repay in the event of war, the more likely it is that no agreement can be reached.

But if states know this, wouldn't they have an incentive to avoid taking on debt?  Well, not necessarily.

Slantchev allows states to choose precisely how much to borrow, and also allows them to choose precisely how much of the financial resources they have available (which includes their initial endowments as well as whatever they borrowed) to spend in war preparations.  That's far more appropriate than what I'm about to do.  But since we already know that I'd get the same result if I modeled things more appropriately, I'm going to make some incredibly strong assumptions in order to illustrate his result more simply.

I'm going to assume that states face one simple choice -- to borrow or not -- rather than two continuous choices.  Further, I'm going to assume that if both borrow, neither gains an advantage.  That needn't be the case, as one state might have greater access to credit.  But it keeps things simple and we'd get the same result if I didn't make such a strong assumption.  However, if one side borrows and the other does not, the one that borrows gains an advantage.  Specifically, we replace $$w$$ with $$\overline{w}$$ if 1 borrows while 2 does not and with $$\underline{w}$$ if 2 borrows and 1 does not, where $$\underline{w} < w < \overline{w}$$.

Let's assume that neither side faces a sufficiently large cost of borrowing, or sufficiently large reduction in said cost when war occurs, to unilaterally eliminate the bargaining range.  However, let's assume that the bargaining range will disappear if both sides borrow.  Finally, let's assume that that 1 issues ultimata to 2 in the bargaining phase.  That's not important to the result, but it's a pretty standard assumption, and this last step of the analysis is cleaner if the entire surplus is absorbed by one state.

If borrowing decisions are made simultaneously, we then have

\begin{array}{ccc}
& \mbox{no borrow} & \mbox{borrow}\\
\mbox{no borrow} & w+c_2, & \underline{w}+c_2,\\
& 1-w-c_2 & 1 - \underline{w} - c_2 - d_2\\
\mbox{borrow} & \overline{w}+c_2-d_1, & w-c_1-\gamma_1d_1, \\
& 1-\overline{w}-c_2 & 1-w-c_2-\gamma_2d_2
\end{array}

Mutual borrowing constitutes a Nash equilibrium, despite the fact that it guarantees war where peace would have been certain if either state had chosen not to borrow, provided $$w - \underline{w} \geq c_1 + c_2 + \gamma_1d_1$$ and $$\overline{w}- w \geq \gamma_2d_2$$.  Intuitively, if the distributive implications of facing a debt-financed opponent are sufficiently large relative to the expected costs of repaying a loan after fighting a war (and the costs of war if we're talking about player 1, who consumes all of the surplus associated with avoiding a war under the ultimatum bargaining protocol), then both sides will borrow even though it leads to tragedy.  In effect, the explanation for war becomes the Prisoner's Dilemma.  Both sides are strictly better off under mutual cooperation, but if loans can be repaid cheaply and additional armaments make a big difference to the outcome of a military contest, then neither side can credibly commit to forego the benefit of borrowing and arming.

For this reason, I'm not sure it's entirely correct to say that this isn't a commitment problem.  But I won't quibble about terminology.  Whether we think of this is a commitment problem (though a fundamentally different one than that induced by the anticipation of a rapid shift in power) or not, this is an important contribution.  It also yields observable implications that someone out there might want to go and test.

Specifically, war should be more common when the state with worse access to credit still has very good access to credit (which can be measured using bond yields), particularly when the two states are near parity (which can be measured using CINC or, ahem, my M scores).  Focusing on such conditions would isolate cases where borrowing costs are low (since both sides have access to credit) and where the implication of refusing to borrow is large (since the marginal impact of military expenditures on the likely outcome of war is greatest when states are near parity).

1.  Why then do people continue to criticize Fearon for overlooking the possibility that leaders might behave pathologically, as if the idea that the inefficiency puzzle need not always apply is something he was completely unaware of?  Yeah, I don't know either.

1. I glanced at the Slantchev article when my copy of APSR arrived and now I've looked at this post, albeit rather quickly.

I'm a bit stupid about these things and I'm not quite sure I get the argument. As I understand it, the argument, put intuitively and simplistically, is that a state that loses a war is more likely to default on its debts than a state that wins. Thus a state that appears to be losing may, under certain conditions, have an incentive to keep fighting so as to avoid defaulting after defeat. And the reason it wants to avoid a default is that a default would raise the cost of any borrowing it wants to do subsequently. (When you say at the beginning "the expected cost of servicing debts after fighting a war is lower than it is in times of peace" this strikes me as true, if at all, only for the victor; the loser's cost of debt service will go up after a war, esp if it defaults. [Or maybe I'm confused.])

Anyway, the whole argument and model, whether I've understood it properly or not, would seem to apply only to wars with traditional state belligerents on both sides, and those kinds of wars aren't fought much anymore. In an Afghanistan-style conflict, say, the argument would seem to have little application since the 'insurgents', ie the Taliban, are probably relying mostly for finance on opium sales, extortion in the form of charging people protection money, illicit smuggling of various kinds etc. The Taliban isn't floating bond issues, I wouldn't think. Slantchev here has devoted a lot of scholarly energy to producing a model that applies to interstate wars of a sort that are quite rare now. That still cd be a useful exercise, I suppose, but it does seem to limit its current applicability.

1. Hi LFC,

The argument is a little different than that. It's not about how expectations of default influence behavior during war, but whether war occurs in the first place. You're certainly right that, in the long run, a state that defaults on its loans will pay a cost for having done so. Slantchev has effectively assumed that away (as he assumes away many complicating factors).

The point is that *sometimes* avoiding war is costlier *in the short run* than going to war because avoiding war requires you to repay your loans in full.

Allowing for the long term implications of default would further restrict the conditions under which war would occur (it would apply more to states who care relatively little about the future than to those who are more concerned about the long-run), but the fundamental logic of the argument wouldn't change.

As to the scope of the argument, you are right that it's unclear how this applies to conflicts involving non-state actors. If it does at all. I don't think this model tells us much about the war in Afghanistan.

But, yes, I do still think it's a useful exercise. First of all, the fact that traditional interstate conflict is rarer today than in the past does not make it irrelevant. Interstate conflicts still occur, and there are many crises active today that could become interstate conflicts. We have no way of knowing for sure that the decline is permanent either. It may be. But interstate war might be more common again in the future.

Even if interstate war is on its way to extinction though (which would be great!), it's still worthwhile to improve our understanding of it. The better we understand the simplest possible case, the better positioned we'll be for studying more complicated, and more relevant, models.

2. Phil,
Thanks for clarifying the argument. I do understand your statement here, which is very clear.

In looking, admittedly quickly, at the article in hard copy, I had found the abstract (which is usually a help, even in technical articles) somewhat hard to follow, and I obviously didn't succeed in following the parts of the text that I looked at either. Maybe I was thrown off by the way he deployed the historical examples at the beginning (e.g. WW1) but I'm going to assume, and I'm sure it's the case, that this was my fault for not taking the time to figure it out. I'm usually pretty good with prose (as opposed to numbers and notations), but not this time.

Also I agree that it's still worthwhile to improve our understanding of interstate war.

P.s. Writing a post on structural realism. Anyone still interested in that, um, stuff? (tempted to use worse word here :)).

3. Perfectly understandable. Glad my clarification helped.

There are still some people who are interested in structural realism, yes. Fewer now than in the past, of course, but they exist. I look forward to seeing what you have to say.

2. P.s. Though I suppose (am not sure) the Pakistan govt, which of course is involved in the war, may be borrowing through more or less conventional means. But its ambiguous position might complicate the analysis.