A Dynamic Model of Conflict and Cooperation



Appendix A: Derivation on the HJB equation

In this appendix we show how to derive (10) in the text. Assuming an interior solution, we
solve
(8) for each agent to get the optimal strategy ai = ai (Z). By substituting this optimal
strategy into
(6), the HJB equation (6) associated with agent i is transformed into

ρVi(Z)=pi(a1(Z),...,an(Z))Z+Vi0(Z)


n

(1-aj(Z))-δZ
j=1


(A.1)


By differentiating (A.1) with respect to Z and applying the envelope theorem to the resulting
expression, we obtain

nn
ρVi (Z) = ∑dpiaj (Z) Z + p,(.) + V (Z) £(1
a

- aj (Z)) - δZ


j=1    j                                  j=1

n

+Vi0(Z) -    a0j(Z)-δ .

(A.2)


j=1

Substituting (8) and (9) into Vi0 (Z) and Vi00 (Z) in (A.2), respectively, and exploiting symmetry
yields

2pi


∂ai2


"=(■ - <Z -  Z


a0i(Z)+(n - 1)


2pi


∂ak∂ai


a0 (Z) + p(.)+


a0k(Z) Z[n (1 - a(Z)) -δZ]


+dpi [n (1 - a(Z)) - δZ] - (δ + ρ)∂piZ, k = i.
a
i                             ai

(A.3)


Since the assumption of symmetry further allows us to make use of the following simple
expressions:

pi
2pi
∂ aO2


1 ∂pi   r (n - 1) ∂pi r

n , ∂ai n2a , ∂ak     an2 ,

n (r - 1) - 2r   2pi    r2 (-n +2)

n3a2     , ∂ak ∂ai       n3a2


r(n- 1)


(A.4)


22




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