1. The Mathematics of Terrorism Risk
Michael R. Powers
Department of Finance
School of Economics and Management
Tsinghua University

2. Equilibrium Force Allocations and Attack Probabilities
Part II
Coauthor: Bruce Gudmundsson, Marine Corps University

3. Objective
Given probability density function (histogram) of available target values, f(W) ~ W-t, find probability density function (histogram) of destroyed target values, g(W) ~ W-y

4. Outline
Theoretical Development Colonel Blotto Games of: Powers and Shen (2006) Powers and Gudmundsson (2008) Empirical Results Comparison of theory to observations of: Kaizoji and Kaizoji (2008), w.r.t. f(W) Johnson et al. (2005), w.r.t. g(W)

5. Colonel Blotto Game
Powers and Shen (2006): n targets; “volumes” Vi , values Wi  Vi Attackers allocate constrained Ai Defenders allocate constrained Di Targets cannot be partially damaged Zero sum; gain/loss function V,  > Case (i) All i attacked simultaneously; Case (ii) Exactly one i attacked w.p. i

6. Cournot-Nash Equilibrium
Theorem 1: If all targets are attacked simultaneously, and  = 1/2, then a Cournot-Nash equilibrium is formed by Ai  (Vi)1/2 and Di  (Vi)1/2 Theorem 2: If exactly one target is attacked w.p. i  (Vi)1/2-, and   (0, 1/2], then a Cournot-Nash equilibrium is formed by Ai  (Vi)1/2 and Di  (Vi)1/2

7. Cournot-Nash Equilibrium (Cont.)
Theorem 3: If the target-selection probability i is endogenized as a strategy of the attackers, then no pure-strategy Cournot-Nash equilibrium exists

8. Revised Colonel Blotto Game
Assumptions: n targets; “volumes” Vi , values Wi  Vi Attackers allocate constrained Ai Defenders allocate unconstrained Di Attacked targets at least partially damaged Zero sum; gain/loss function V,  > Exactly one i selected w.p. i

9. Cournot-Nash Equilibrium
Theorem 1: If exactly one target is attacked w.p. i  (Wi)r, and  > 0, then a Cournot-Nash equilibrium is formed by Ai  (Wi)a and Di  (Wi)d, where r, a, and d are chosen so that pi = 0 Theorem 2: Given that the p.d.f. of target values is f(W) ~ W-t, the attackers can maximize their expected gain in Theorem 1 by selecting r = t’  for any t’ ≥ t

10. Cournot-Nash Equilibrium (Cont.)
From the first-order conditions of Theorem 1, we know d + r +  = q + 2a d = (q + a)/2 In conjunction with Theorem 2, we then obtain Corollary 1: In Cournot-Nash equilibrium, Ai  (Wi)t’-1 and Di  (Wi)(q+t’-1)/2, where i  (Wi)t’-1-

11. Hypothetical Parameter Values
Assume: q = t’ = t = 2.35 (see Kaizoji and Kaizoji, 2008)  = 0.5 for peacetime govts. (risk prone)  = 1.5 for wartime govts. (risk averse) Then: Ai  (Wi)t’-1 = (Wi) Di  (Wi)(q+t’-1)/2 = (Wi) i  (Wi)t’-1- = (Wi)0.85 in peacetime i  (Wi)t’-1- = (Wi)-0.15 in wartime

12. Empirical Literature
Johnson et al. (2005) found that the p.d.f. of destroyed (rather than available) targets is g(W) ~ W for less-developed wartime countries g(W) ~ W for more-developed wartime countries To compare our estimates with these observations, consider that g(W) =  f(W) ~ (Wi)-0.15(Wi)-2.35 = (Wi) for more-developed wartime countries

13. Conclusions
In both peacetime and wartime, government defenders tend to allocate forces in slightly lower proportion to high-value targets than do terrorist attackers. In peacetime, terrorist attackers tend to give substantial weight to high-value (“trophy”) targets; however, such targets actually are avoided in wartime.

14. References
Johnson, N. F., Spagat, M., Restrepo, J. A., Becerra, O., Bohórquez, J. C., Suárez, N., Restrepo, E. M., and Zarama, R. (2005), “Universal patterns underlying ongoing wars and terrorism,” arXiv:physics/ v1, available at Kaizoji, T. and Kaizoji, M. (2008), “A mechanism leading from bubbles to crashes: the case of Japan’s land market,” arXiv:cond-mat/ v2, available at Major, J. A. (2002), “Advanced techniques for modeling terrorism risk,” Journal of Risk Finance, Vol. 4, No. 1, pp Powers, M. R. (2008), “Lanchester resurgent? the mathematics of terrorism risk,” Journal of Risk Finance, Vol. 9, No. 3. Powers, M. R. and Shen, Z. (2006), “Colonel Blotto in the war on terror: implications for event frequency,” paper presented at the American Risk and Insurance Association Annual Meeting, Washington, DC.