Wednesday, December 12, 2007

Does "war in mixed strategies" make sense?

At yesterday's political economy workshop, our own Massimo Morelli presented a working paper that he co-authored with Matthew Jackson entitled "Strategic Militarization, Deterrence, and Wars" (paper is here). The centerpiece of the paper is an analysis of a mixed strategy equilibrium involving states playing "hawkish," "dovish", and "deterrence" strategies, with wars occurring when "hawkish" and "dovish" strategies interact.

This is the second paper that we've seen at the political economy workshop this Fall where fighting has occurred only in mixed strategies. The other paper was a working paper by Ernesto Dal Bo and Robert Powell on civil conflict resulting from a situation where governments hold private information on the size of a centrally controlled endowment from which contenders seek a share (paper is here).

This raises all the questions about whether such mixed strategy "fighting" equilibria are plausible characterizations of behavior. In the Dal Bo and Powell, paper, contenders mix over fighting and not fighting to "keep the government honest". However, the contender's decision to fight comes after she has already received her share of the endowment from the government. For the equilibrium to be believable, we have to accept that even after the contender's share has been paid out, she would still with positive probability initiate an insurgency that offers no additional gain. This strikes me as implausible.

In the Morelli and Jackson paper, things are a little better insofar as the timing of the game does not conflict so much with the logic behind the mixed strategy. But it's still a case where wars are initiated as the result of joint randomization of strategies.

However, Massimo argued for another interpretation. Rather than thinking in terms of randomizing strategies in the context of a bilateral interaction, think of one central actor facing many adversaries. Then, we can imagine a mixed strategy as being one where the central actor is playing a different strategy with each adversary such that the distribution of strategies conforms with the equilibrium "mix". My thought was that this interpretation was stretching things a bit. This interpretation would seem to require that the adversaries all act as if they are playing a bilateral game with the central actor; but clearly in any real world situation, the adversaries would condition their behavior on what the central actor is doing vis-a-vis the other adversaries. So, to be convinced, I would have see that the same equilibrium "mix" holds in a compelling, respecified game with one central actor and multiple adversaries and with the central actor choosing war with a subset of the adversaries in pure strategies. I just don't think that something like war makes any sense as an element of a mixed strategy.

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