Colonel Blotto in the War on Terror: Implications for Event Frequency

Michael R. Powers Advanta Center for Financial Services Studies Temple University [email protected] 215-204-7293 215-204-4712 (fax)

Zhan Shen Department of Risk, Insurance, and Healthcare Management Temple University 1. Introduction

The research literature on terrorism risk is small but growing. Although probabilistic models are widely used to quantify natural catastrophe risk, such approaches are insufficient in the study of terrorism. Because of the human intelligence involved in a terrorist’s decision making, some scholars (see Harris, 2004; Kunreuther and Erwann, 2004; Major, 2002; Sandler and Arce, 2003; Woo, 2002) have argued that game theory, in conjunction with probability theory, should be used to develop successful models of terrorism risk.

1.1. Probability Models

Paté-Cornell (2002) presents a classic Bayesian model for better detection and interpretation in the fusion of terrorist-attack information. The model allows computation of the posterior probability of an event given its prior probability (before the signal is observed), and the quality of the signal is characterized by the probabilities of false positives and false negatives. Another method is to divide the terrorist activity into several steps and evaluate the corresponding probability associated with each node on a scenario tree. Harris (2004) notes that if the conditional probabilities of detecting the terrorist at each node can be evaluated, then the probability that this security method will fail can be determined. Haimes and Horowitz (2004) apply hierarchical holographic modeling (HHM) to track terrorists. HHM first sets up the multi- view image of a system hierarchy of terrorist threats to generate a large number of scenarios, and then ranks them by likelihoods and consequences. Florentine et al. (2003) use Bayes’ Theorem as the key tool for analyzing observables identified through the HHM diagram to assess the likelihood of a terrorist attack. Major (2002) proposes that the probability of a successful terrorism attack be represented as the product of the probability of remaining undetected and the conditional probability of destruction. Kunreuther and Erwann (2004) argue that the probability of success is determined not only by the amount of the resources the terrorist group allocates to

-2- the attack but also by the resources its opponent allocates to detecting terrorist activity and defending the target.

1.2. Game Theory

Armstrong (2002) argues that gaming terrorism through “role-playing” is more appropriate for conflict situations. Oster (2002) discusses the potential of applying game theory to account for insurance protection against terrorist attacks. Kunreuther and Erwann (2004) point out that the terrorist group gains value or utility from the damage inflicted on its adversaries. Both parties are constrained by their own resources and the game becomes one of strategic decisions as to how to deploy those resources. Major (2002) develops a zero-sum game between terrorists and defenders with a numerical example requiring sophisticated numerical analysis. Yusef (2004) claims that the game conflict between U.S. and radical Islamic groups possesses a Nash Equilibrium of continuous warfare. Sandler and Arce (2003) construct a game- theoretic model that includes different choices of targets (e.g., businesspeople, officials, and tourists).

1.3. Other Issues

Woo (2002) argues the terrorists are adaptive and will “follow the path of least resistance” by attacking the most vulnerable targets. He also proposes that the partial derivative of the probability that the defense is unable to prevent or stop the attack with respect to defensive resources is a power of the utility function. Harris (2004) observes that terrorists can attack multiple targets simultaneously, which has not been taken into account by prior models.

2. The Frequency Model

Extending Major’s (2002) analysis, we write the probability of a successful terrorist attack on a particular target, i , as

-3- f = PrSuccessful Attacki at= p p(i) p(i) p(i) , i { } 1 2 3 4

where:

p = Pr 1 Attack 1 {і }

i p( ) = Pri Attacked 1 Attack, 2 { і }

(i) p3 = Pr{Attack ati Undetected 1 Attack i Attacked}, і З

(i) p4 = Pr{Attack ati Successful і 1 AttackЗ i Attacked Attack ati Undetected . З }

The first of the above probabilities, p1 , is essentially the underlying probability of

terrorist action during a given time period. In the commercial risk analyst’s model, this

probability is generally captured by an overall “outlook” analysis for a particular future time

period.

2.1. The Colonel Blotto Game

Let V i denote the intrinsic value of the target, and let A i and D i denote the intrinsic

amounts of resources allocated to the target by the attacker and defender, respectively. We posit

that A i < Di < V i.

For an attacker that reaches its target undetected, we model the ensuing combat by

assuming: (1) that the two sides engage in a sequence of micro contests in which each side

stakes one unit, and has an equal probability of winning, (2) that the attacker initially possesses

c c Ai units (for some c і 0), and (3) the defender initially possesses D i units. From the classical

gambler’s ruin model, it follows that the probability that the attacker will win the macro combat

-4- c i ( ) Ai 1 p4 = c c . (1) Ai + Di

2.2. Detecting Terrorists

We model the defender’s search for the attacker by assuming: (1) that the searchable size

s of the target’s domain consists of V i (for some s і 0) independent units, (2) that the attacker

s s possesses Ai units to distribute across the target’s V i units at random, and (3) that the defender

s possesses D i units, each of which it uses in succession to try to detect one of the attacker’s units.

s Ai The probability that the first unit deployed by the defender will detect the attacker is thus s . V i

s Ai Given that the first unit fails, the probability that the second unit will succeed is s ; given V i -1

s Ai that the first two units both fail, the probability that the third unit will succeed is s ; and so V i - 2

s forth. Consequently, the probability that all D i units will fail to detect is given by

s s s s (i) ж Ai цж Ai цж Ai ц ж Ai ц p3 = з1 - sчз 1 - s чз 1 - s чL з1 - s s ч. V V -1 V - 2 V - D +1 и i ши i ши i ш и i i ш Since the value of the target is invariably much greater than the values of the resources allocated

s s s s by the attacker and defender, respectively, it is reasonable to assume V i >> Ai and V i >> Di , from which it follows that

1 Note that other stochastic combat models could be used to derive the same expression for the attacker’s probability of success. For example, if we consider the macro-contest to consist of a “race” in time to a certain random event, c where that event is given by a Poisson process with parameter (rate) Ai t for the attacker and by a Poisson process c with parameter (rate) D i t for the defender, then the probability of victory for the attacker is exactly the same as in (1).

-5- s s (i) ж Ai Di ц p3 » expз- s ч. (2) V и i ш

3. Equilibrium Solutions

Treating the approximation in (2) as exact, and combining it with (1) yields

s s c ж Ai Di ц Ai fi = p i expз- s ч c c , V A + D и i ш i i

where (i) . p i = p1 p2

Now consider the defender’s post-attack effective loss,

ж n ц L = l V I D D зе i iч, иi =1 ш

where l D is an increasing and concave-downward function, and the I i are Bernoulli( fi) random

variables. If the I i are i.i.d. (as in the case of simultaneous attacks, addressed in Theorem 1

2 below), then, for small values of fi,

n E L » fl V , (3) [ D ] е i D ( i) i=1

n and we will take this approximation to be exact. If the I are such that I =1 (as in the case of i е i i=1 one random attack, addressed in Theorem 2 below), then the approximation in (3) is in fact

exact.

We model the problem as a zero-sum game in which the attacker must select the vector

n A = A, A,K , A E L A = A [ 1 2 n] to maximize [ D ] subject to е j , and the defender must select the j=1

2 This is generally a reasonable assumption in the case of terrorism risk.

-6- n D = D ,D ,K ,D E L D = D vector [ 1 2 n] to minimize [ D ] subject to е j . We consider two variations, j=1

one with simultaneous attacks at all targets and one with a random attack at only one target. In

3 the former case, p i = p1 for all i . In the latter case, the vector of probabilities,

n p = p ,p ,K ,p p = p 4 [ 1 2 n] (subject to е j 1 ), is determined exogenously. We show that if j=1

p = [p1,p 2,K ,p n] is endogenized as a strategy of the attacker, then strategic equilibrium cannot

exist.

3.1. Simultaneous Attacks

First, consider the case in which the terrorists attack all targets simultaneously; i.e., such

that p i = p1 and

s s c ж Ai D i ц Ai fi = p1 expз- s ч c c . (4) V A + D и i ш i i

n n Theorem 1: Define a = A V 1 2 and d = D V 1 2 , and let p = p and l V = kV 1 2 (for е j е j i 1 D ( i) i j=1 j=1

some positive constant k ) for all i . If the conditions

й sщ (i) c>1- s1 - (ad) , лк ыъ

c 1+ (a d) (ii) c< c , and 1- a d ( )

3 This is because ( i) for all i . p2 = 1 n i 4 p( ) = 1 This is because е 2 . i=1

1 s< (iii) s 1- ad ( ) hold, then the vectors

1 2 1 2 1 2 1 2 1 2 1 2 A = aV ,aV ,K ,aV , D = dV ,dV ,K ,dV [ 1 2 n ] [ 1 2 n ] form a strategic equilibrium.

Proof: See the appendix. 

Note that for small values of ad and a d , conditions (i), (ii), and (iii) can be replaced by the simpler (but less sharp) alternatives

s 0, 1 О ( ] and

c 1- s+ ad, 1 О [ ].

Note further that in equilibrium,

c (a d) fi = p1 exp(-ad) c , (5) 1+ a d ( ) which is both constant across i and independent of the parameter s . A closer examination of (5) reveals that for small values of ad ,

c (a d) Ac fi » p1 c = p1 c c . (6) 1+ a d A + D ( )

3.2. Random Attack on One Target

Now consider the case in which the terrorists decide to attack only one target, chosen

n randomly with probability p (such that p = p ), so that i е j 1 j=1

-8- s s c ж Ai Di ц Ai fi = p i expз- s ч c c . V A + D и i ш i i

n n n 1 2 1 2 1 2-l й 1щ Theorem 2: Define a = A V j , d = D V j , and p = p1 V j , where l О 0 , , and е е е к 2ъ j=1 j=1 j=1 л ы

let p = pV 1 2-l and l V = kV l (for some positive constant k) for all i. If conditions (i) i i D ( i) i

through (iii) of Theorem 1 hold, then the vectors

1 2 1 2 1 2 1 2 1 2 1 2 A = aV ,aV ,K ,aV , D = dV ,dV ,K ,dV [ 1 2 n ] [ 1 2 n ]

form a strategic equilibrium.

n Proof: Replace the factor p l V = p kV 1 2 in each term of the objective function ( fl V ) 1 D ( i) 1 i е i D ( i) i= 1

1 2-l l by plD (V i) = pV i kVi . The proof is then completely analogous to that of Theorem 1. 

Note that in equilibrium,

c 1 2-l (a d) fi = pV i exp(-ad) c , (7) 1+ a d ( )

1 2-l s which is proportional to V i and independent of the parameter . For small values of ad ,

c (a d) Ac f » pV 1 2-l = pV 1 2-l . (8) i i c i c c 1+ a d A + D ( )

p = p ,p ,K ,p Finally, consider what happens if we attempt to endogenize [ 1 2 n] (subject to

n p = p е j 1 ) as a strategy for the attacker. j=1

-9- n p = p ,p ,K ,p p = p Theorem 3: If the attacker is free to select [ 1 2 n] (subject to е j 1 ) as a j=1

strategy, then an equilibrium solution cannot exist.

2 ¶ E[LD ] Proof: Assume that an equilibrium exists. Since for all i, it follows that the 2 = 0 ¶p i

p = p ,p ,K ,p optimal [ 1 2 n] must occur at a boundary. In other words, the attacker does best by

s s c ж Ai Di ц Ai (1-b ) 2 selecting p i = p1 for the target i for which expз- s ч c c V i is maximized, and by V A + D и i ш i i

selecting p i = 0 otherwise. However, as soon as the attacker selects p i = p1 for some target i ,

the defender’s best response is to allocate all of its resources to i . As a result,

s s c ж Ai Di ц Ai (1-b ) 2 expз- s ч c c V i is no longer maximized at i , and we have a contradiction.  V A + D и i ш i i

There really is no need to endogenize p = [p1,p 2,K ,p n] because the probabilities

1 2-l p i = pV i identified by Theorem 2 are imposed automatically by the identification of

l V = kV l . In other words, once we know l V = kV l , the choice of p = pV 1 2-l is forced, D ( i) i D ( i) i i i

p = p ,p ,K ,p and endogenizing [ 1 2 n] actually over-specifies the model.

4. Discussion

4.1. Intrinsic Value

Although the intrinsic value unit proposed in Section 2 is somewhat vague, it facilitates

the comparison of qualitatively different types of assets (money, land, buildings, life, health,

-10- etc.). For simplicity, and especially in an insurance context in which losses are compensated with money, it is reasonable to think of the intrinsic value actually as some transformation of monetary value.

4.2. Results

Despite their differences regarding the attack mechanism, Theorems 1 and 2 provide identical equilibrium allocations of resources for the defender and the attacker. In both cases, the attacker and defender simply allocate their resources in direct proportion to the square root of the intrinsic values of the various targets.

Under the conditions of Theorem 1, the equilibrium frequency – i.e., the probability of a successful attack, given by equations (5) and (6) – is exactly the same for every target, depending only on the ratio of the total resources available to the attacker to the total resources available to the defender, and increases as the attacker’s relative resources increase. Consequently, the frequency of an event at target i is independent of the target’s value, which means that in setting insurance premiums, the frequency component for every insured target should be the same.

Naturally, the model underlying Theorem 1 is somewhat unrealistic because terrorists do not have sufficient resources to attack all possible targets simultaneously.

The conditions underlying Theorem 2 address this shortcoming by allowing the terrorist to attack only one target at a time. In this case, the equilibrium frequency – given by equations

(7) and (8) – increases with the value of the target. This makes intuitive sense because one would expect a terrorist organization that can choose only one target to seek a high return on its invested resources. The result also indicates that in setting insurance premiums, the frequency

1 2-l component should increase in proportion to V i . Since Theorem 3 ensures that there is no equilibrium solution in which the attacker endogenizes the attack probability, the most important remaining issue is to estimate the parameter l of the loss function.

-11- 4.3. Estimating the Loss Function

Although many different types of loss functions exist, it is generally practicable to approximate any increasing and concave downward loss function with the power function of

Theorem 2. We give the following example.

l V = lnV Consider the loss function proposed by Woo (2002); i.e., D ( i) ( i). We can

l 6 12 approximate this function by lD (V i) = kVi over the catastrophe loss interval 10 , 10 by [ ] solving the following two equations simultaneously:

l l ln106 = k 106 and ln1012 = k 1012 . ( ) ( ) ( ) ( )

This yields

l 12 12 ln(10 ) (10 ) l = 2 = 106 6 l Ю ( ) , ln(10 ) 106 ( ) which implies

1 l = log10 (2) » 0.05 6 and

6 6 ln(10 ) ln(10 ) k = = » 6.91 1 . log10 (2) 2 106 6 ( )

5. Conclusions

In extending Major’s (2002) work, we allow the terrorists to attack multiple targets at the same time. In addition, since both players want to maximize or minimize their payoffs (expected gains or losses), we employ the method of Lagrange multipliers to solve the problem. Also, we update the conditional probability of destruction given remaining undetected based upon the

-12- classical gambler’s ruin model. Our analytical results show that the defenders should allocate their resources to all targets proportionally, which contradicts Woo’s (2002) rule of following the path of least resistance.

Nonetheless, there are some limitations in our model. First, Yusef (2004) argues that the terrorism game is a non-zero-sum nature, as are most games in the real world because of substantial differences between the utility functions of the opponents. Second, although we recognize that targets have different values, we still assume the targets to be homogeneous apart from their values. Sandler and Arce (2003) suggest that there might be different types of target choices, such as business and tourism, which cannot be treated in the same way. These issues should be addressed through further research.

-13- References

Armstrong, J. Scott, “How to Avoid Surprises in the War on Terrorism: What to Do when You Are between Iraq and a Hard Place,” 2002, the Wharton School, University of Pennsylvania, Philadelphia, PA, online at: http://www.chforum.org/library/gaming.html.

Florentine, Christopher, Isenstein, Mindy, Libet, Jared, Neece, Steve , Zeng, Jim, Haimes, Yacov Y., and Horowitz, Barry M. (2003), “A Risk-Based Methodology for Combating Terrorism,” Proceedings of the 2003 Systems and Information Engineering Design Symposium.

Haimes, Yacov Y. and Horowitz, Barry M. (2004), “Adaptive Two-Player Hierarchical Holographic Modeling Game for Counterterrorism Intelligence Analysis,” Journal of Homeland Security and Emergency Management, 1, 3.

Harris, Bernard (2004), “Mathematical Methods in Combating Terrorism,” Risk Analysis, 24, 4, 985-988.

Kunreuther, Howard and Erwann, Michel-Kerjan (2004), “Insurability of (Mega)-Terrorism Risk: Challenges and Perspectives,” OECD Task Force on Terrorism Insurance, p. 56.

Major, John A. (2002), “Advanced Techniques for Modeling Terrorism Risk,” Journal of Risk Finance, 4, 1.

Paté-Cornell, Elisabeth (2002), “Fusion of Intelligence Information: A Bayesian Approach,” Risk Analysis, 22, 3, 445-454.

Sandler, Todd and Arce, Daniel G. (2003), “Terrorism and Game Theory,” Simulation and Gaming, 34, 3.

Oster, Christopher (2002), “Can the Risk of Terrorism Be Calculated by Insurers? Game Theory Might Do It,” the Wall Street Journal, online at http://www.gametheory.net/news/Items/011.html.

Woo, Gordon (2002), “Quantitative Terrorism Risk Assessment,” Journal of Risk Finance, 4, 1, 7-14.

Yusef, Moeed (2004), “The United States and Islamic Radicals: Conflict Unending?” International Social Science Review, 79, 1-2, 27-43.

-14- Appendix

Proof of Theorem 1

(1) Define the probability function for success of destruction

Based on the analysis above, we set up the probability function based on two theories: Search theory and the Colonel Blotto game: s s c s c Ai ЧDi Ai pi = pi Чpi =exp(- s )Ч c c , Vi Ai + Di s c where pi represents the probability of undetected and pi represents the probability of successful destruction when undetected.

(2) Terrorism model using Lagrange multipliers (generalized)

L = f ℓ (V ) = p k p V 1/ 2 p k е i D i 1 Ч Че i Ч i (Note: 1 and are constants.) s s c Ai ЧDi Ai pi = F(Vi , Ai , Di ) =exp(- s )Ч c c Vi Ai + Di

Max L s.t. g(A) = Ai = A A   е

Min L s.t. g(D) = Di = D D   е grad(L) - lA Чgrad(g(A)) - lD Чgrad(g(D)) = 0   й¶L ¶L ¶L ¶L щ к ,⋯, , ,⋯, ъ- l A [1,⋯,1,0,⋯,0]- l D [0,⋯,0,1,⋯,1]= 0 л¶A1 ¶An ¶D1 ¶Dn ы

We look for the solutions for Ai and Di that satisfy the conditions below. First-Order Conditions:

¶L ¶pi 1/ 2 = p1k ЧVi = lA , "i , ¶Ai ¶Ai

¶L ¶pi 1/ 2 = p1k ЧVi = lD , "i . ¶Di ¶Di Second-Order Conditions: 2 2 2 ¶ L ¶ pi 1/ 2 ¶ pi = p1k ЧVi ө < 0 , "i , ¶A2 ¶A2 ¶A2 i i i 2 2 2 ¶ L ¶ pi 1/ 2 ¶ pi = p1k ЧVi ө > 0 , "i . ¶D2 ¶D2 ¶D2 i i i

(3) Relationship among Vi , Ai , and Di (analytical solution for Ai and Di )

The analytical solution for Ai and Di might take the following form: 1/ 2 Ai = aVi 1/ 2 Di = dVi where a and d are positive small numbers. Thus, we can easily get

-15- n n 1 2 1 2 a = A V j d = D V j . еj=1 еj=1

(4) First-order condition validation

(a) First-order conditions w.r.t Ai

s s s-1 s c s s c-1 c ¶pi Ai ЧDi -sAi ЧDi Ai Ai ЧDi cAi ЧDi = exp(- s )Ч s Ч c c + exp(- s )Ч c c 2 ¶Ai Vi Vi Ai + Di Vi (Ai + Di ) s s s+c-1 s c-1 c Ai ЧDi й-sAi ЧDi cAi ЧDi щ = exp(- )Ч + V s кV s (A c + D c ) (A c + D c )2 ъ i л i i i i i ы s s s+c-1 s c-1 c ¶L ¶pi 1/ 2 Ai ЧDi й-sAi ЧDi cAi ЧDi щ 1/ 2 = p1k ЧVi = p1k exp(- )Ч + ЧV i ¶A ¶A V s кV s(Ac + D c) (Ac + D c)2 ъ i i i л i i i i i ы 1/ 2 s+c-1 1/ 2 s 1/ 2 c-1 1/ 2 c s s й-s(aV i ) Ч(dV i ) c(aV i ) Ч(dV i ) щ 1/ 2 = p1kexp(-a d )Ч + ЧV i кV s((aV 1/ 2 )c + (dV 1/ 2 )c) ((aV 1/ 2 )c + (dV 1/ 2 )c)2 ъ л i i i i i ы -sa s+c-1 d s ca c-1 d c = p kexp(-a sd s) й Ч + Ч щ 1 Чк c c c c 2 ъ л (a +d ) (a +d ) ы (b) First-order conditions w.r.t Di s s s s-1 c s s c c-1 ¶pi Ai ЧDi -sAi ЧDi Ai Ai ЧDi й cAi ЧDi щ = exp(- )Ч Ч + exp(- )Ч - ¶D V s V s A c + Dc V s к (A c + D c )2 ъ i i i i i i л i i ы s s s+c s-1 c c-1 Ai ЧDi й-sAi ЧDi cAi ЧDi щ = exp(- )Ч - V s кV s (A c + D c ) (A c + D c )2 ъ i л i i i i i ы s s s+c s-1 c c-1 ¶L ¶pi g Ai ЧDi й-sAi ЧDi cAi ЧDi щ 1/ 2 = p1k ЧVi = p1k exp(- )Ч - ЧV i ¶D ¶D V s кV s(Ac + D c) (Ac + D c)2 ъ i i i л i i i i i ы 1/ 2 s+c 1/ 2 s-1 1/ 2 c 1/ 2 c-1 s s й-s(aV i ) Ч(dV i ) c(aV i ) Ч(dV i ) щ 1/ 2 = p1kexp(-a d )Ч - ЧV i кV s((aV 1/ 2 )c + (dV 1/ 2 )c) ((aV 1/ 2 )c + (dV 1/ 2 )c)2 ъ л i i i i i ы -sa s+c d s-1 ca c d c-1 = p kexp(-a sd s) й Ч - Ч щ 1 Чк c c c c 2 ъ л (a +d ) (a +d ) ы

-16- (5) Constraints on parameters (derivation from second-order conditions)

2 2 2 ¶ L ¶ pi 1/ 2 ¶ pi (a) Second-order conditions w.r.t Ai = p1k ЧVi ө < 0 , "i ¶A2 ¶A2 ¶A2 i i i

2 s s s-1 s s+c-1 s c-1 c ¶ pi Ai ЧDi -sAi ЧDi й-sAi ЧDi cAi ЧDi щ = exp(- )Ч( ) + ¶A2 V s V s кV s (A c + D c ) (A c + D c )2 ъ i i i л i i i i i ы s s s+c-2 s c c s+2c-2 s c-2 c c c 2 2 2c-2 c c c Ai ЧDi м s(s + c -1)Ai ЧDi (Ai + Di ) - csAi ЧDi c(c -1)Ai ЧDi (Ai + Di ) - 2c Ai ЧDi (Ai + Di )ь + exp(- )Ч - + V s н V s (A c + D c )2 (Ac + D c)4 э i о i i i i i ю s2 A2s+c-2 D 2s ckAs+c-2 D s+c s(s+ c-1)As+c-2 D s м i Ч i - i Ч i - i Ч i ь s s пV 2s(Ac + D c) V s(Ac + D c)2 V s(Ac + D c) п Ai ЧDi п i i i i i i i i i п = exp(- )Ч V s н csAs+2c-2 D s c(c-1)Ac-2 D c 2c2 A2c-2 D c э i п+ i Ч i + i Ч i - i Ч i п п V s(Ac + D c)2 (Ac + D c)2 (Ac + D c)3 п о i i i i i i i ю

( plug in the solutionsof Ai and Di ) s2 (aV 1/ 2 )2s+c-2 (dV 1/ 2 )2s cs(aV 1/ 2 )s+c-2 (dV 1/ 2 )s+c s(s+ c-1)(aV 1/ 2 )s+c-2 (dV 1/ 2 )s м i Ч i - i Ч i - i Ч i ь пV 2s((aV 1/ 2 )c + (dV 1/ 2 )c) V s((aV 1/ 2 )c + (dV 1/ 2 )c)2 V s((aV 1/ 2 )c + (dV 1/ 2 )c) п s s п i i i i i i i i i п = exp(-a d )Ч н cs(aV 1/ 2 )s+2c-2 (dV 1/ 2 )s c(c-1)(aV 1/ 2 )c-2 (dV 1/ 2 )c 2c2 (aV 1/ 2 )2c-2 (dV 1/ 2 )c э п+ i Ч i + i Ч i - i Ч i п п V s((aV 1/ 2 )c + (dV 1/ 2 )c)2 ((aV 1/ 2 )c + (dV 1/ 2 )c)2 ((aV 1/ 2 )c + (dV 1/ 2 )c)3 п о i i i i i i i ю exp(-a sd s)a c-2d c csa sd s csa s+cd s-c c(c-1) 2c2a c = мs2a 2sd 2s-c - - s(s+ c-1)a sd s-c + + - ь V (a c +d c) н a c +d c a c + d c a c +d c (a c +d c)2 э i о ю csa sd s csa s+cd s-c c(c-1) 2c2a c ө s2a 2sd 2s-c - s(s+ c-1)a sd s-c + й- + щ+ й - щ< 0 (1) лй ыщ к c c c c ъ к c c c c 2 ъ л a +d a +d ы лa +d (a +d ) ы

-17- 2 2 2 ¶ L ¶ pi 1/ 2 ¶ pi (b) Second-order conditions w.r.t Di = p1k ЧVi ө > 0 , "i ¶D2 ¶D2 ¶D2 i i i

2 s s s s-1 s+c s-1 c c-1 ¶ pi Ai ЧDi -sAi ЧDi й-sAi ЧDi cAi ЧDi щ = exp(- )Ч( ) - ¶D2 V s V s кV s (A c + D c ) (A c + D c )2 ъ i i i л i i i i i ы s s s+c s-2 c c s+c s+c-2 c c-2 c c 2 2 c 2c-2 c c Ai ЧDi м s(s -1)Ai ЧDi (Ai + Di ) - csAi ЧDi c(c -1)Ai ЧDi (Ai + Di ) - 2c Ai ЧDi (Ai + Di )ь + exp(- )Ч - - V s н V s (A c + D c )2 (Ac + D c)4 э i о i i i i i ю s2 A2s+c D 2s-2 csAs+c D s+c-2 s(s-1)As+c D s-2 м i Ч i + i Ч i - i Ч i ь s s пV 2s(Ac + D c) V s(Ac + D c)2 V s(Ac + D c) п Ai ЧDi п i i i i i i i i i п = exp(- )Ч V s н csAs+c D s+c-2 c(c-1)Ac D c-2 2c2 Ac D 2c-2 э i п+ i Ч i - i Ч i + i Ч i п п V s(Ac + D c)2 (Ac + D c)2 (Ac + D c)3 п о i i i i i i i ю

( plug in the solutionsof Ai and Di ) s2 (aV 1/ 2 )2s+c (dV 1/ 2 )2s-2 cs(aV 1/ 2 )s+c (dV 1/ 2 )s+c-2 s(s-1)(aV 1/ 2 )s+c (dV 1/ 2 )s-2 м i Ч i + i Ч i - i Ч i ь пV 2s((aV 1/ 2 )c + (dV 1/ 2 )c) V s((aV 1/ 2 )c + (dV 1/ 2 )c)2 V s((aV 1/ 2 )c + (dV 1/ 2 )c) п = exp(-a sd s) п i i i i i i i i i п Чн 1/ 2 s+c 1/ 2 s+c-2 1/ 2 c 1/ 2 c-2 2 1/ 2 c 1/ 2 2c-2 э cs(aV ) (dV ) c(c-1)(aV ) (dV ) 2c (aV ) (dV ) п+ i Ч i - i Ч i + i Ч i п п V s((aV 1/ 2 )c + (dV 1/ 2 )c)2 ((aV 1/ 2 )c + (dV 1/ 2 )c)2 ((aV 1/ 2 )c + (dV 1/ 2 )c)3 п о i i i i i i i ю exp(-a sd s)a cd c-2 csa sd s csa sd s c(c-1) 2c2d c = мs2a 2sd 2s-c + - s(s-1)a sd s-c + - + ь V (a c +d c) н a c + d c a c +d c a c +d c (a c +d c)2 э i о ю csa sd s csa sd s c(c-1) 2c2d c ө s2a 2sd 2s-c - s(s-1)a sd s-c + й + щ+ й- + щ> 0 (2) лй ыщ к c c c c ъ к c c c c 2 ъ лa +d a +d ы л a +d (a +d ) ы

-18- (c) Weak sufficient conditions for parameter constraints

Suppose that all parameters are positive. From inequality (1), we get:

(i) s2a 2sd 2s-c - s(s + c -1)a sd s-c < 0 s s s s s s Ю sa d < s + c -1 Ю c >1- s + sa d =1- s(1-a d ) s s Ю c >1- s(1-a d ) (3) csa sd s csa s+cd s-c (ii) - + < 0 a c +d c a c +d c a c ж ц a < d Ю з ч <1 Ю (This is true, since D > A.) (4) иd ш c(c -1) 2c2a c (iii) - < 0 a c +d c (a c +d c )2 c c c 2 c c -1 a Ю c(c - 1) (a +d )< 2c a Ю < 2c a c +d c Ю (c - 1)d c < (c +1)a c Ю c(d c - a c )< a c +d c a c a c 2ж ц 1+ ж ц c c c з ч з ч a +d 2a иd ш иd ш Ю c < c c =1+ c c =1+ c = c (5) d -a d -a жa ц жa ц 1- з ч 1- з ч иd ш иd ш From inequality (2), we get:

(iv) s2a 2sd 2s-c - s(s -1)a sd s-c > 0 s s s s Ю s -1< sa d Ю s(1-a d ) <1 1 Ю s < s (6) 1- (a d ) csa sd s csa sd s (v) + > 0 . (This is always true.) a c +d c a c +d c c(c -1) 2c2d c (vi) - + > 0 a c +d c (a c +d c )2 c c c c c c c -1 d (7) Ю (c -1)(a +d ) < 2cd Ю (c - 1)a < (c +1)d Ю < c +1 a c c -1 d c This follows because <1< (from (4)), which is always true whenever (i) is valid. c +1 a c