## Thursday, November 8, 2012

### Breaking Down Bueno de Mesquita 2005

This series has focused so far on interstate crisis bargaining.  There are some important pieces that I still want to cover in that area, but for now, let me turn my attention to terrorism.

In "Conciliation, Counterterrorism, and Patterns of Terrorist Violence," Ethan Bueno de Mesquita seeks to explain why governments offer concessions to groups that engage in terrorist violence despite the tendency for violence to increase afterwards.  If offering concessions only invites more terrorism, as appears to be the case, what reason could governments possibly have for doing so?

Brief Synopsis

The intuition here is that partial concessions can allow the government to buy-off one faction, thus gaining a greater ability to defeat the remaining faction.  The more extreme faction is likely to escalate its campaign of terrorist violence once they lose the support of the moderates, but the government is more likely to prevail in the end, and that may offset the short-term spike in violence.

As an example, Ethan discusses the Spanish government's experience with the ETA.  Between 1968 and 1977, attacks conducted by the ETA claimed the lives of 73 people.  Partial autonomy was granted to the Basque region in 1978, yet despite the fact that this represented a significant shift towards the desired outcome of the ETA, violence increased after that.  Over the next three years, the ETA would kill 235 people, and fatality levels remained elevated for decades.  Yet, the ETA has grown weaker over time, and more than 700 members are currently imprisoned.  Just last year, they announced a unilateral ceasefire, and unlike similar past declarations, this one seems to be sticking.  If the ETA's campaign of violence has indeed come to an end, it is largely because their capabilities have been badly degraded, not because they achieved their remaining goals.

This is, of course, a stylized account.  Many other factors, and many other actors, are at work.  But the basic dynamics of Ethan's argument can be see in this conflict.  The point is not to offer a comprehensive treatment of this complex conflict, but to offer an illustrative example of an argument that many find to be very counterintuitive.

Details of the Argument

As ever, the model I'm going to analyze relies on much stronger assumptions than the one Ethan analyzed.  Had he not published his piece, I would be bothered by some of the assumptions I'm about to make.  But the argument I'm about to make holds under more general conditions, so there's no sense in criticizing my treatment of it.  That's not to say that the argument is beyond criticism -- only that anyone who is skeptical of this argument ought to focus their attention on the more sophisticated treatment found in Ethan's paper.  The goal of this post is not to establish a new result, but to help those who might find Ethan's analysis a bit intimidating to understand his argument a bit better.

Suppose we have three actors: a government, $$G$$; and two factions of a dissident group, one of which, $$M$$, is relatively moderate, and one of which, $$E$$, is relatively more extreme.

At the outset of the game, $$G$$ offers some level of concessions, denoted $$x \in (0,1)$$, to the dissident group.  $$M$$ can then either accept or reject the offer.  I'll assume that if $$M$$ rejects it, there will be no agreement.  That is, for the sake of simplicity, I assume away the possibility of $$E$$ accepting terms that $$M$$ rejected.  However, if $$M$$ accepts, it is possible that $$E$$ will as well.  Thus, if $$M$$ rejects $$G$$'s terms, the game ends with the two factions continuing a moderate campaign of violence against the government, while if $$M$$ accepts, the game either ends with $$E$$ accepting as well, thus ending the campaign altogether, or with $$E$$ splitting off from $$M$$ to conduct a more intense campaign against the government on their own.

Let $$\nu_i>0$$ denote the value of the issue to actor $$i \in \{G, M, E\}$$.  If the game ends with both $$M$$ and $$E$$ accepting $$G$$'s terms, then the payoffs are straightforward: $$G$$ receives $$\nu_G(1-x)$$; $$M$$ receives $$\nu_Mx$$ and $$E$$ receives $$\nu_Ex$$.

In the event that $$M$$ rejects $$G$$'s terms, things are also relatively straightforward.  Let $$t$$ denote the expected outcome of a terror campaign involving both $$M$$ and $$E$$, and let $$c_i$$ denote $$i$$'s cost therefrom, with the assumption that $$c_G$$ takes on a smaller value here than it would if $$E$$ lacked $$M$$'s support, while $$t$$ takes on a larger value.  That is, I assume that there will be less violence if $$M$$ continues to exert a moderating influence on $$E$$, but I also assume that $$G$$ will have a harder time defeating the group than it would if it co-opted $$M$$.  Thus, $$G$$'s payoff here is $$\nu_G(1-\overline{t}) - \underline{c}_G$$, while $$M$$'s is $$\nu_M\overline{t} - c_M$$ and $$E$$'s is $$\nu_E\overline{t} - c_E$$.

Finally, if $$M$$ accepts $$G$$'s terms but $$E$$ does not, I assume that both $$M$$ and $$E$$ enjoy the benefits of whatever concessions G grants immediately and both factions will derive utility from the additional concessions $$E$$ will eventually force $$G$$ to make (which may be quite small).  In principle, $$G$$ could offer benefits exclusively to $$M$$, and the moderates might be assumed to have no further interest in the outcome of the conflict once they lay down arms, since any future concessions $$G$$ might make may be offered exclusively to $$E$$.  By structuring the payoffs the way I do, I'm assuming that any benefits gained from the government come in the form of (partial) changes in the policy to which the group objected.  Note that similar results would obtain so long as $$M$$ and $$E$$ retain partial interest in seeing $$G$$ grant concessions to the one another, though the model cannot really speak to situations where the factions have no interest in the outcomes achieved by one another.

Given that assumption, $$G$$'s value for having the game end with $$E$$ escalating violence while $$M$$ lays down arms is $$\nu_G(1-x-\underline{t}) - \overline{c}_G$$, while $$M$$'s is $$\nu_M(x+\underline{t})$$ and $$E$$'s is $$\nu_E(x + \underline{t}) - c_E$$.  Note that $$M$$ no longer incurs any costs, though both $$G$$ and $$E$$ naturally do, and, as discussed above, $$G$$'s cost here is strictly greater than when $$M$$ exerted a moderating influence on $$E$$.  However, with the support of $$M$$, $$G$$ is in a better position to defeat $$E$$, and so expects a better outcome from the more violent terror campaign (thus $$t=\underline{t}$$, which is less than $$\overline{t}$$ by assumption).

Got all that?

Let's start with $$E$$'s decision of whether to keep fighting in the event that $$M$$ accepts $$G$$'s terms.  Provided $$\nu_E(x+\underline{t}) - c_E \geq \nu_Ex$$, it is in $$E$$'s interest to lay down arms as well.  Unsurprisingly, given our assumption that $$E$$ gets to enjoy the partial concessions granted by $$G$$ even if they keep fighting, the size of $$x$$ has no bearing on this decision.  Through some simple algebraic manipulation, we see that the previous inequality cannot hold if $$\nu_E > \displaystyle \frac{c_E}{\underline{t}}$$.  Intuitively, if $$E$$ is extreme enough -- if they value the issue in dispute sufficiently -- then they will continue their campaign of terror even after they lose the support of $$M$$.  Since the puzzle we've set out to address is why $$G$$ would offer concessions when doing so leads to increased violence, let's assume that this is the case from this point forward.

Given our assumption that $$E$$ will keep fighting, what would it take to peel $$M$$ away?  Well, it follows what we've said so far that $$M$$ will accept $$G$$'s terms if and only if $$\nu_M(x+\underline{t}) \geq \nu_M\overline{t} - c_M$$.  This can be rewritten as $$x\geq \overline{t} - \underline{t} + \displaystyle \frac{c_M}{\nu_M}$$.

For notational convenience, let $$\hat{x}$$ denote $$\overline{t} - \underline{t} + \displaystyle \frac{c_M}{\nu_M}$$.  We can therefore say that $$M$$ accepts if and only if $$x \geq \hat{x}$$.  The only remaining question is whether $$G$$ prefers to set $$x\geq\hat{x}$$.

First of all, if $$G$$ wants to buy-off $$M$$, and any $$x\geq\hat{x}$$ will achieve that goal, it's clear that $$G$$ would never set $$x>\hat{x}$$.  That is, if $$G$$ is going to co-opt the moderates at all, they will do so by offering the smallest amount of concessions necessary.  So what we really want to know is whether $$G$$ prefers to have the game end with $$M$$ accepting $$\hat{x}$$, which implies a bloodier campaign against an unfettered $$E$$, or to forego concessions altogether and face-off against the combined forces of $$M$$ and $$E$$.  They will, if and only if $$\nu_G(1-\hat{x}-\underline{t}) - \overline{c}_G \geq \nu_G(1-\overline{t}) - \underline{c}_G$$.

The previous inequality simplifies to $$\nu_G \geq \displaystyle \frac{\overline{c}_G - \underline{c}_G}{\displaystyle\frac{c_M}{\nu_M}}$$.

Notice something surprising about that?  This inequality, the one that determines when $$G$$ will negotiate with terrorists and be rewarded for doing so with an increase in violence (at least in the short term) is more likely to hold the more resolved the government is.  At least, if we take $$\nu_G$$, which stands for the value $$G$$ assigns to the issue at stake, as an indicator of how "resolved" the government is, as is pretty standard in these types of models.

The intuition behind that result is that $$G$$ finds the benefit of seeing $$t$$ drop from $$\overline{t}$$ to $$\underline{t}$$ to be sufficient to offset the increase in $$c_G$$ from $$\underline{c}_G$$ to $$\overline{c}_G$$ only if $$G$$ cares a lot about the issue at stake.  Put differently, if the reason that governments offer concessions is that they are doing so in order to co-opt the moderates, thereby accepting a short-term increase in violence in exchange for an increased likelihood of defeating the remaining faction, we should expect that governments who ascribe a greater value to issue at stake to make that tradeoff.  The less the government cares about the issue at stake, the less willing they will be to take away the extremists' only reason for moderating their behavior in hopes of achieving a better outcome further on down the road.  Put this way, hopefully it makes sense to you.  But it's still true that the model is telling us that it's relatively resolved governments who negotiate with terrorists, and who are more likely to preside over sudden and dramatic increases in violence (as a result of their successful attempt to drive a wedge between the moderates and the extremists).

As I said, this is a much simpler model than the one Ethan analyzed, and it relies upon stronger assumptions.  If you're not persuaded by this discussion, I encourage you to go read his paper, which not only offers a more sophisticated theoretical treatment but also an application to the Israeli-Palestinian conflict.