Just a minor point to add to Daniel Larison’s typically sensible post about the folly of issuing empty threats over Ukraine.

I hope that everyone agrees that bluffing is dangerous, because a bluff can be called and, if it is, the bluffer must either make good on the bluff – which, presumably, is very strongly counter to his interest, else he wouldn’t be bluffing but making threats in earnest – or suffer exposure as someone whose threats are not to be taken seriously. If we say, “don’t cross this line in the sand or else” and the person we are threatening crosses it, and we do nothing, then he’ll be that much less inclined to pay any attention when we draw such lines in future sands. (Note: I’m not arguing that our credibility is some unitary factor independent of the characteristics of individual conflicts; other actors in the system can presumably make rational estimates of where our “real” interests lie. Nonetheless, it isn’t a good thing to get a reputation for making empty threats.)

On the other hand, bluffing is a useful tool because, as in poker, it enables you to “play” a somewhat stronger hand than the one you actually have. If there is some uncertainty about whether a threat is a bluff or not, the threat may be accepted as real, and you gain the benefit of the threat at a lower cost than it would take to accumulate the cards necessary to make it good. Moreover, the acceptance of the bluff as true itself sends a signal to other potential opponents: our last opponent backed down in the face of our threats. He thought we were serious. Maybe you should, too?

Looked at this way, there’s a case for judicious bluffing – that is to say, as a matter of calculated risk. If I bluff, my opponent may call – but he may treat the bluff as serious, and back off, and if he does so then the “power” of my declared threats has been enhanced. In other words, there’s a risk of loss, but also a risk of gain. If the action we’re trying to deter is sufficiently damaging, it becomes relatively easy to make the case for bluffing – because a successful bluff deters the action and also enhances one’s credibility, while a failed bluff only results in a loss of credibility; the negative action would presumably happen anyway if the bluff hadn’t been made in the first place.

A bit of simple math might be helpful to explain this point of view. Assume, for simplicity’s sake, that the loss or gain to credibility (“c”) is symmetric – we gain just as much from a successful bluff as we lose from a bluff being called – and that the opponent’s action (“a”) is certain if either the bluff is called or no threat is made. We’ll use P(s) to represent the probability of the bluff’s success. In that case, you get the following:

No bluff: cost = a (opponent takes the action)
Bluff called: cost = a+c (opponent takes the action *plus* we lose credibility)
Bluff successful: cost = -c (i.e. we gain credibility because our opponent backed down)
Total cost of bluffing = P(s)*(-c) + (1-P(s))*(a+c) = a+c-P(s)*a-2*P(s)*c

Since the cost of not bluffing is “a,” to compare bluffing to not bluffer we subtract “a” from both sides. Result: bluffing makes sense if c is less than the sum P(s)*a+2P(s)*c.

That looks like a pretty big number relative to c. To illustrate, take the following example: c is twice as large as a – i.e., the cost to credibility of a failed bluff is twice as large as the cost of the action we’re trying to deter in the first place – and the probability of success is only 50%. Should you bluff?

No bluff: cost = 1
Bluff called: cost = 3
Bluff successful: gain = 2
Total cost of bluffing = 50% * 3 – 50% * 2 = 1.5-1.0 = 0.5

Your indifference point in this ridiculously simplified analysis would be a 40% chance of success. In other words, this analysis leads to the conclusion that you should bluff in circumstances where the bluff is 50% more likely to fail than to succeed, and where the total cost of a failed bluff is three times as large as the cost of never making a threat.

You can see how someone might rationally conclude that bluffing is a pretty good strategy, in a lot more cases than you might initially suspect. Indeed, notwithstanding the many excellent points in Paul Pillar’s refresher course in Cold War deterrence, he gives the inaccurate impression that America was did not do a lot of bluffing in that multi-decade standoff. Whereas, in fact, there is considerable question whether America’s core deterrent ever was truly credible, in the sense that it was never clearly rational to actually escalate to a nuclear exchange for the sake of Western Europe or Japan, and yet America threatened first use of nuclear weapons in response to a conventional Soviet assault.

Obviously, there are a dozen ways to poke holes my analysis above (and I should be clear, that analysis is not something I’m defending, just something I cooked up to illustrate a point that I then wanted to debate). The effect on credibility could be asymmetric, for example – the gain from a successful bluff could be of much lower magnitude than the loss from a called bluff. Or you can question the whole framework by emphasizing the inherent uncertainty of all the numbers involved (which, after all, will most likely be pulled out of the analyst’s posterior). But one hole that should get poked more often is the unwarranted assumption that threats can only decrease, and not increase, the likelihood of the action you’re attempting to deter.

Suppose your opponent is contemplating action “a” that would accrue some gain to him at some cost on you – but not a large enough cost to be worth fighting him over. Nonetheless, you threaten to fight if he takes that action. If he allows himself to be deterred, in our analysis above your credibility is enhanced – you experience a gain in power. But at whose expense?

First and foremost: your opponent’s. After all, every other actor in the system can rationally conclude that you might well be bluffing just as easily as your opponent can. They can’t be certain – but they know there’s a good chance. If your opponent backs down in a situation where a bluff is fairly probable, that results in a substantial blow to his credibility. Even if the cost of fighting with you is sufficiently high that it would mean a substantial cost to the opponent to take an action that leads to war, he can’t afford simply to absorb the cost of backing down in the face of a possible bluff. He has to play the odds.

Well, let’s run the odds from his perspective, using the same kind of over-simplified analysis. Assume that calling our bluff or backing down generates symmetrical gain and loss, and that the value of the action itself is still 1/2 the cost of backing down to a bluff. Assume, further, than the cost of war is 10x the value of the action. We’ll use P(b) to indicate the opponent’s estimate of the probability that we are bluffing. (We already know that the true probability is 1.) Well?

Back down: loss = 2 (loss in credibility)
Call bluff, no war: gain = 3 (1 for action itself, 2 for gain in credibility)
War: loss = 10
Total value of calling bluff: P(b)*3-(1-P(b)*10) = P(b)*13-10

Since the loss due to backing down is 2 (a value of -2), it’s worth calling the bluff if P(b)*13 is greater than 8, or, in other words, if there’s a greater than 8/13 chance we are bluffing (in which case the expected loss from calling the bluff is also 2).

Think about that. In this ridiculously over-simplified, zero-sum analysis, we should bluff if we think there’s at least a 40% chance of the bluff succeeding, even if we rule out in advance the option of making good on the threat and even though, if our bluff is called, we’ll lose 3 times what we would have lost if we had never bluffed at all. And our opponent should call our bluff if they think there’s at least a 60% chance of it being a bluff, notwithstanding that if they’re wrong and we go to war they’ll lose 10 times what they would have gained from taking the action if we’d never threatened them. Moreover, if our opponent estimates at least a 70% chance that we are bluffing, the relative value of taking the action becomes *higher* to them than it would have been had we never bluffed in the first place, because of the incremental value to their prestige and credibility in having defied our threats.┬áNow, recall that we’re likely to have positively-biased estimates of our own ability to bluff. Does it still seem reasonable to assume that threats will at least reduce, and not increase, the likelihood of our opponent taking a given action?

Again, there are a dozen holes that can be poked in such an admittedly over-simplified analysis. But the important point is that there are entirely rational reasons to suspect that issuing a threat can increase the likelihood that the opponent takes the action you are trying to deter. For any such action, “a,” there are a variety of potential costs and benefits to the actor – a vast penumbra of uncertainty about outcomes that might in itself be sufficient to deter many actors from many potentially beneficial actions. By issuing a threat, you’ve made one of those potentialities much more concrete: inaction will definitely result in some loss. Depending on how large that loss looms, and what the opponent figures are the odds that you’re bluffing, the threat itself could be sufficient to motivate the action you intended to deter – or some other action of equal or greater cost to you.

With America defining its interests in such a global fashion, it’s very likely that this dynamic plays an important part in our opponents’ responses. It certainly seems to have been relevant in Georgia, where part of Russia’s motivation in provoking Georgia into launching a war was precisely the desire to call America’s bluff. (How much, after all, is South Ossetia itself really worth to anybody, even the South Ossetians?) The same might prove true in Eastern Ukraine if we handle the situation in the way that some hawks prefer.