Tuesday, October 16, 2012

Deadlines and Wars of Attrition

Over at PV@Glance, Andrew Kydd argues against the Ryan's critique of the administration's Afghanistan policy.  Among other things, he raises concerns about the claim that announcing a deadline will encourage the enemy to keep fighting.  So far as I know, Kydd is right that we don't have any existing models that speak to this, but I think he's mistaken (as he acknowledges that he might be) when he speculates that this result could not emerge from the simple sort of model that the Ryan critique appears to be based on.

Before I go any further, though, let me be clear that my goal here is only to apply some game-theoretic thinking to the interesting strategic question Kydd raised in the middle of his post. Whether anyone should vote for Romney and Ryan over Obama and Biden because of the former's critique of the latter's policy in Afghanistan is another question -- one I'm not going to concern myself with here.

That is, I take no issue with Kydd saying that it's a bit strange for the Romney campaign to criticize Obama for pursuing what we have good reason to believe would be the same exact strategy that a Romney administration would pursue.  I'm not sure this applies quite as equally to McCain's critique of Obama's announced position in 2008, but I'm more than happy to set that aside.

Similarly, I think Kydd is right to stress that we can't ignore the strategic incentives facing Karzai's government.\(^1\)  Whether such concerns trump the impact of US policy on the incentives facing the Taliban is, to my mind, an open question, but a proper assessment of the situation would certainly need to take both into account.

So I'm not telling you who to vote for, and I'm not saying that administration's policy gets it wrong.\(^2\)  All I'm saying here is that we ought not be so quick to dismiss the claim that when it is known that one party to a conflict will quit fighting by a certain date, this may create an incentive for the other party to fight longer than they otherwise would have.

There are several variants of the basic war of attrition game.  But the following captures all the essential elements.

In each period \(t \in \{0, 1, 2, \ldots, T\}\), players A and B choose whether to fight or to quit.  The game ends once both players have quit.  At that time, each player receives their payoffs, which are equal to \(W_iv - qc_i\), where \(v>0\) is the value for the contested good\(^3\), \(c_i\in(0,v]\) is the per-period cost of fighting for \(i\), \(q\in\{0, T\}\) indexes the period in which the player quit, and \(W_i \in \{0, 0.5, 1\}\) is an indicator of whether player \(i\) won the war.  Specifically, \(W_i\) takes on a value of 1 if \(q_i>q_j\), where \(j\neq i \in \{A,B\}\).  If \(q_i < q_j\), then \(W_i =0\).  If \(q_i=q_j\), then \(W_i=0.5\).

Suppose we have complete information.  That's not terribly realistic, but it gives us a baseline against which to evaluate more realistic versions of the model.

No player can profit from fighting past the point that the cumulative costs of war would exceed the value of the disputed good.  More formally, no player can fight for longer than some maximum \(m_i\), where \(m_i \equiv v/c_i\).  Note that the players may not even fight for that long -- I stress that \(m_i\) is the maximum number of periods \(i\) might be willing to fight, not the number of periods for which \(i\) will actually fight in equilibrium.

If \(m_i > m_j\), then \(j\)'s optimal strategy is to quit in period \(0\), allowing \(i\) to quit in the first round and acquire the good at a much lower price than they would have been willing to pay.\(^4\)

Now suppose that, due to the vagaries of domestic politics, A expects to receive a slightly different payoffs.  Instead of receiving \(Wv - q_Ac_A\), A receives \(Wv - q_Ae_Ac_A\) if \(1 < q_A < x_1\) or if \(q_A > x_2\), where \(x_1 < x_2\) and \(e_A>0\).  That is, suppose that electoral considerations will impose an additional cost on A if they quit too soon or if they fight too long.  Suppose further that \(x_2 - x_1\) is precisely equal to 1.  That is, suppose that there is exactly one time period in which A can quit fighting and escape electoral sanction, unless of course they succeed in acquiring control of the disputed good straight away.  Let this period be denoted \(d_A\).

There are a number of possibilities here.  Imagine first that \(m_B < d_A < m_A\).  In this case, domestic political concerns are irrelevant.  B is going to quit in period \(0\) and A is going to acquire full control of the good without having to incur any significant costs.  It does not matter that the maximum number of periods for which A would have been willing to fight if unconstrained by domestic politics exceeds the maximum number of periods for which A will actually fight given the reality of those constraints.

Now suppose that \(d_A < m_A < m_B\).  Again, domestic political concerns are irrelevant.  A is going to quit in period \(0\), and would have done the same absent electoral considerations.

But what if \(d_A < m_B < m_A\)?

Provided \(e_A\) is sufficiently large, then A cannot credibly commit to outlasting B, and yet neither would A be willing to quit before \(d_A\).  So A will fight until period \(d_A\) and B will fight until period \(d_A+1\).  Absent electoral considerations, however, B would have quit straight off.  Thus, it is precisely because B knows that A will be forced by domestic political considerations to withdraw prematurely that B will hold out and thus secure victory.

Now, that doesn't quite speak to the claim that announcing your withdrawal date publicly emboldens the enemy.  In the preceding analysis, B is motivated to keep fighting by the reality of A's domestic political constraints.  I said nothing about A publicly pre-committing to withdrawal at a certain date.  And insofar as one might very well doubt that a Romney administration would be insensitive to public pressure to end the war, one might be tempted to shrug off the critique that Obama should not have announced that the US will do exactly what everyone already knew it would do.\(^5\)  But we shouldn't dismiss the idea that electorally-motivated deadlines can alter the outcome of a war of attrition.

Of course, the above not only assumes away third parties and incomplete information and many other factors that would complicate the analysis even if they would not necessarily alter the basic substantive conclusion, but it also assumes that we're talking about standard war of attrition models.  Kydd explicitly makes reference to such models, but, at times, he seems to be envisioning a different sort of model altogether.That is, Kydd says that the reason why deadlines don't matter is
because [B] is comparing the utility for quitting now with the utility for holding out one more instant, and quits when they are equal, and [A] has announced that it will quit at some time later than the next instant.
In standard war of attrition games, that's not how decisions are made.\(^6\) Actually, to the best of my knowledge, that's not how decisions are made in any class of model typically applied to the study of war.  In certain iterated games, we assume that actors continue fighting if and only if the current expected value of choosing "fight" as this period's strategy exceeds the current expected value of choosing "quit" as this period's strategy.  That is, there are models in which we as analysts do not try to work our way back from the end but instead try to characterize marginal decisions in any given period.  But that's not quite the same as saying that the players compare the utility for quitting now with the utility of holding out for one more instant (or period), because actors who choose to fight in this period may also choose to fight in the next period.  And the one after that.  And they take this into account when calculating the current expected values for the two options they face this period.\(^7\)

In ordinary English, if the Taliban expect that quitting today means they will never get any more of what they want, and quitting tomorrow also means that they will never get any more of what they want, but holding out until the day after tomorrow will bring them all of what they want, then I see no basis in extant game-theoretic models of war for dismissing the possibility that the Taliban would fight until the day after tomorrow.  To be sure, if the marginal costs of fighting are large enough, they might be dissuaded from doing so.  One might even argue that the entire purpose of the surge is to increase the costs the Taliban must incur if they are to wait the US out. But we can't just dismiss the possibility that the Taliban will wait the US out by saying that the decisions made in these types of situations are based only on the difference between quitting right now and quitting one instant from now.

While we can have a reasoned debate about what the available evidence suggests we should assume about the area of the parameter space this particular case occupies, the fact that there exist some parameter values for which mainstream models would predict that B would be motivated to fight longer if B knew that A would quit by some fixed date isn't really open to dispute.

1. On a related note, see Bapat's game-theoretic analysis of the Afghan surge.

2. Or, rather, that's not the goal of this post.  I do actually have serious concerns about the administration's policy in Afghanistan, for reasons I've discussed previously (see here, here, here, and here, among others), but at the moment, all I'm trying to do is clarify our thinking about the impact of one party announcing ahead of time the date at which they will cease fighting on the behavior of the other party.  Whether you buy the argument I've been pushing for some time now -- that the primary reason the US hasn't already withdrawn from Afghanistan is that Obama would be unlikely to be re-elected if he had done so -- need not have any bearing on whether you accept my claims about deadlines in bilateral wars of attrition (which may or may not be an appropriate model for the war in Afghanistan).

3. We could, of course, allow each actor to assign a different value to the good.  However, since the cost terms are measured as a share of the disputed good, we have effectively already accounted for variation in the players' subjective value for the good.

4. For the sake of simplicity, I'm ignoring the special case where \(d_i\) is equal to \(d_j\).

5. If one thinks it is clear that the US cannot credibly commit to remain in Afghanistan long enough for \(d_A\) to exceed \(m_B\).  Speaking not as a formal theorist but simply as an observer of current events, my sense is that we can in fact assume that the Taliban is well aware that no US president can commit to keeping US troops in theater long enough to exceed their breaking point.  But I will freely admit that I could be mistaken about that.

6. See here for an application of a standard war of attrition game.  

7. See here for a recent example.

8. I'm not sure what the basis for an optimistic assessment would be, but perhaps I've been reading the wrong news sources.  Or reading them through biased eyes.


  1. This is a good point, Phil. I just want to add that I think it could be made even more simply with a continuous-time model, which ends up being isomorphic to an all-pay auction: whoever "pays more" (decides to wait longer before withdrawing) gets the prize.

    It's easy in this case to show that there isn't an equilibrium in which either side employs a non-zero pure strategy, which can be thought of as corresponding to a fixed withdrawal date. If I choose t > 0 and lose, I would be better off just choosing 0. If I choose t > 0 and tie (assuming that the prize is distributed with some probability 0 < p < 1 in case of a tie), I would be better off choosing t + epsilon and winning for sure. And finally, if I choose t > 0 and win, that means I could still win by choosing t - epsilon, which would be preferable.

    That said, I'm skeptical this is a good model of the effects of Afghanistan policy. And I'm skeptical that the lesson of this toy model -- i.e., use a mixed strategy -- corresponds to Romney/Ryan's supposedly preferred policy. But I certainly agree that we can at least come up with basic models to support the critique of announcing a fixed withdrawal.

    1. Hey Brenton. Good to hear from you.

      Good point about the all-pay auction. I should have mentioned that.

      I too am skeptical that this is a good model of the effects of Afghanistan policy. As I said above, I think Kydd is right that we need to think about how US policy influences the behavior of the Afghan government. And my guess is that, if elected, Romney would adopt essentially the same policy as Obama anyway. My only point was that I thought Kydd was being too dismissive of the notion that it could make sense to try to wait out an opponent who you know will quit fighting at a certain point in time.