## Wednesday, December 14, 2011

### Resolve and Victory

How does an actor's resolve influence the likelihood that they achieve victory in war?

You'd think this is a simple question.  How could the answer not be that more resolved actors do better in war?

Well, there aren't many studies that have tried to offer a systematic evaluation of this relationship, and there's at least one study that finds the opposite of what you'd expect.

What should we make of that?

Not only does Slantchev find a statistically significant effect that runs in the wrong direction, but the effect isn't especially small either.  To quote Slantchev:

The substantive impact of salience is quite large.  When the initiator starts a war over a nonsalient issue, the probability of a favorable outcome exceeds the probability of concessions or defeat for the first three years of the war. In contrast, when the initiator starts a war over an issue more salient to itself, the corresponding period is less than two years.When a war over a nonsalient issue lasts a year, the probability of victory drops from 45% to 25%, compared to a drop from 25% to less than 10% when the war is over a salient issue. Similarly, the probability of defeat rises above 25% after four years of fighting over a nonsalient issue compared to three years of fighting over a salient issue.
It may surprise you to know that this is the relationship that Slantchev hypothesized.  Why is that, you ask?  Well, his argument is essentially one of selection effects.
Thus, players would initiate nonsalient conflicts only when the probability ofwinning is rather high. If initiators calculate their chances correctly on the average, this would imply that such conflicts would tend to end in their favor. If, on the other hand, they are compelled to fight over a more salient issue, then (because the payoff is greater), the probability of winning does not have to be that high.
What I find so interesting about this is that Slantchev argues that this hypothesis is "also counterintuitive to the extent that few would accept [it] without the logic provided by the theory."

But what theory is he referring to?  In the preceding pages, he describes work applying bargaining models to the study of interstate conflict.  One can (and I indeed have) derive(d) results very similar to those of each of his other five hypotheses.*  But his explanation for this particular finding is pretty much decision-theoretic.  Can we derive an equivalent expectation from a bargaining model?

Before I go further, let's take a step back and clarify some terminology.

The concept of resolve is difficult to define precisely.  The norm that has developed in formal models is to assume that actors differ with respect to the subjective value they attach to the good in dispute, and to capture this either by introducing valuation terms (often denoted v or with the Greek letter nu, since nu looks like a v), or by assuming that the subjective loss of utility associated with incurring the costs of war (often denoted simply c) is inversely related to resolve.

The determinants of war outcomes are understood to be more complex than this, but there is also a norm to assume that the expected share of the disputed good following a war (often, but not always, interpreted as the likelihood of winning an all-or-nothing contest) as being strictly a function of each side's military capabilities (typically denoted m).

Take some challenger, C, and some defender, D.  C issues an ultimatum, denoted x.  D either accepts, which gives us peace, or rejects, leading to war.

If D accepts, C's payoff is v_Cx, while D's is v_D(1-x).

If D rejects, C's payoff is v_Cw - c_C, while D's is v_D(1-w) - c_D, where w = (e_Cm_C)/(e_Cm_C + e_Dm_D).

Different authors use slightly different notation, but this is pretty standard stuff so far.  (If you want more detail, or find my particular notation confusing b/c you're used to seeing it done somewhat differently, see either the lecture slides from my War & International Security Class, particularly Lectures 7 - 10, or the General Logic of Deterrence section of this paper).

Suppose C is uncertain about e_D, believing it to take on a relatively low value (denoted e_D-small) with probability phi, and a relatively large value (denoted e_D-large) with probability 1 - phi.  This implies that w = w-small with probability 1 - phi and w = w-large with probability phi.  That is, C expects better outcomes when facing types of D with relatively low levels of martial effectiveness.  (Again, if this discussion is too perfunctory for you, I refer you to my lecture slides or my paper on deterrence).

I tend to express the results of these types of models by saying that C selects a value of x that she knows D will accept if and only if e_D is relatively small provided that phi is larger than some cutpoint, which I'd call phi-hat, and would select a value of x that D is sure to accept regardless of type when phi is less than or equal to that cutpoint.  But in the interests of assessing the relationship between resolve (v_C) and the expected war outcome (w-small -- which is the only value of w that's relevant since C never fights wars against the type of D that is relatively low in martial effectiveness in equilibrium), let's establish a cutpoint over w-small.

It turns out that C risks war with her optimal proposal if and only if:

w-small < w-large + ((c_C + c_D(v_C/v_D))(phi - 1))/(v_Cphi).

Well, that's an eyesore.  What does it tell us though?

It turns out that the right hand side is increasing in v_C.  In other words, when C ascribes a relatively high value to the good in dispute, she's willing to risk war for a wider range of values of w-small.  When C ascribes less value to the good in dispute, she's willing to risk war under a smaller range of values.  Intuitive enough.

But note that the inequality is more likely to hold when w-small is relatively small in comparison to all that junk on the right hand side.  In other words, there is more likely to be a positive probability of war in equilibrium when C expects to do relatively poorly in that war.  And this is more true the less resolved C is.  So this relatively simple bargaining model tells us that the variance in war outcomes should be greater the more highly resolved C is (which Slantchev did not test), whereas when C is relatively low in resolve, we should only expect war to occur in those cases where the expected outcome of war will be particularly unfavorable for C.

In other words, this is not the result Slantchev found.

But his argument made so much sense.  How can that be?

Let's try a simple decision-theoretic take.  Suppose war occurs if and only if C's war payoff is greater than 0. Taking the same payoff as above (and ignoring incomplete information), this is true iff v_Cw - c_C>0, or > c_C/v_C.  Here, it is clear that an increase in v_C allows the inequality to be satisfied for ever lower values of w, consistent with Slantchev's argument.

So this raises two questions.

First: what the heck is driving the result in the case of the bargaining model?  Why is C willing to risk war more often when she expects to do poorly in war?

Second: should we believe the bargaining model fil just showed us does a better job of explaining war onset than a simple decision-theoretic model?

Let's take these in turn.

The fundamental insight of bargaining models is that we cannot explain war without explaining why states fail to negotiate.  The fact that one of the actors finds war more attractive does not, in and of itself, tell us anything about whether the two are likely to negotiate.  It might tell us something about what we expect the terms of any negotiated agreement to be, sure.  But not whether they will actually reach one in equilibrium.

The easiest way to think about what the informational story tells us is that war is more likely when the challenger has greater incentives to gamble.  And she has greater incentives to gamble, all else equal, when she expects the gamble to payoff more often (i.e., believes that there's a relatively high probability that the defender is low in resolve or martial effectiveness or capabilities or whatever quality it is that C is uncertain about), when the upside to gambling is large (i.e., the terms D accepts when D is of one type are markedly different from the terms he accepts when he is of another type), and when the downside to gambling is small (i.e., when the costs of war are relatively palatable).

Intuitively, the better C does against those types of D that actually might reject her terms in equilibrium, the less difference there is between what C can get the different types to agree to, and the smaller is the upside to gambling.  Thus, war becomes less likely.  As C expects to do worse against the types of D that might plausibly reject her terms, the upside to gambling increases, and war becomes more likely.  And, as I said above, this is especially true the less resolved C is.  If C was highly resolved, she wouldn't need such a large difference between the terms one type of D would accept versus those another type would accept in order to be willing to gamble.

So, assuming you got this far into the post, this is the point where you say, "I guess that makes sense.  But I'm not sure that's how states actually behave."

Fair enough.  I'm not sure it is either.

But is it a better approximation than purely decision-theoretic accounts?

There's an awful lot of literature to suggest that it is.  Not least of which, the very Slantchev article we're discussing.  Several of the findings he presents in that paper provide support for hypotheses that are consistent with non-obvious implications of bargaining models.  These are generally not hypotheses that you'd be able to derive from the simple decision-theoretic model that accounts for the strange result w/r/t to resolve and victory.

Going past that one article, what about the fact that parity is strongly associated with war onset, as so many authors have found?  That can readily be explained by bargaining models.  But a purely decision-theoretic view says that initiators are more likely to go to war when they expect to win (all else equal -- this being less true for the more highly resolved).  Why then don't we see that wars are most likely to occur when the initiator is much stronger than the target?

I won't claim my simple little model provides a perfect account of war onset.  It doesn't.  But I'm fairly confident that it does better than any even simpler model, one that assumes away all strategic interaction between states.  For all I know, it's entirely possible that some richer model, one that is more realistic than both the ultra-simple bargaining model I discussed and the even simpler decision-theoretic model implicit in Slantchev's story, would lead us to expect that the wars that occur in equilibrium are more likely to end poorly for the initiator if the initiator is relatively high in resolve.  I haven't yet seen such a model, but that doesn't mean much.

But consider this.  One of the many many ways in which the bargaining model I discussed abuses reality is that  it assumes that and m (and thus w) are uncorrelated.  That's almost not true empirically, as a number of scholars (foremost among them being Slantchev) are beginning to argue.  That is, if we allow states to endogenously select the amount of material capabilities they bring to bear in a conflict (m) we will generally find that those who ascribe a greater value to the issue in dispute (larger v) will be the ones who are most willing to commit resources to the war effort, and will therefore tend to receive better outcomes (higher w) as a result.  In other words, while I don't doubt for a moment that the simple model I analyzed above has abstracted away from a great many important elements of the real data-generating process, one of the most obvious ways in which it did so in fact biases it against concluding that more resolved states will be more likely to win the wars they fight in equilibrium.

Now, those models can get messy.  The willingness to devote resources to the war can itself be a function of how well the war is already going.  And once we allow for domestic politics, they can get really ugly.  Do states (at least those that are accountable to the public) ramp up their effort when they see that things aren't going as well as they'd initially expected?  There's a lot of interesting work yet to be done on this topic.  I'm not sure I see any reason to believe that extending our existing models in this way is going to lead us to believe that relatively resolved states expect worse outcomes from the wars they fight in equilibrium, but who knows.

Pending future work that might persuade me otherwise, then, I'm inclined to say that we should indeed expect states that ascribe greater value to the issue in dispute to receive better outcomes in war, contrary to the results reported by Slantchev.  But then, I'm the type of person who is very reluctant to read much into results obtained from the analysis of observational data when I don't understand what would lead me to expect those results.

I suppose that those of you who are more inclined to believe whatever empirical analysis tells us, theory be damned, might have a different reaction.  Given that Slantchev's is just about the only study to even attempt to evaluate the relationship between resolve and victory in a systematic fashion, you probably ought to conclude that resolved states fare poorly in the wars they fight.

*For the most part.  There's one other exception, though it is more of a semantic issue since the observable implication is the same either way.  That is, Hypothesis 4 tells us that states with greater reserves will be slower to update their priors, and as a result will demand better settlements at any given point in time.  It's not hard to show that states with greater stocks of available resources would receive better deals at any given point in time, but I'm not sure how one would arrive at the conclusion that this has anything to do with their willingness to update their beliefs.