A Dynamic Model of Conflict and Cooperation



we substitute those expressions into (A.3) yielding

0 = nan21 [-r - r (n - 1)] Za0(Z) + 1+

(32 ) [r (n + 2) + n (r 1) 2r] Z [n (1 a(Z)) δZ] a0(Z)

+ r (n - 1) [n (i a(z )) δz ] (δ + ρ) r n —1 z.                (A.5)

n2a                                   n2a

Further rearranging (A.5) gives rise to (10) in the text.

Appendix B: Linear Strategy

We will show below that (15) represents a linear strategy. Under symmetry, rewrite the HJB
equation
(6) as follows:

ρV (Z)= max [p (a1, a2,..., an) Z + V0(Z) {n (1 a) δZ}] .            (B.1)

ai[0,1]

Suppose that the value function is linear, that is, V (Z)=A+BZ,whereA and B are unknown
constants. Substitute this hypothetical value function into the above HJB equation to get

ρ [A + BZ] = max


— Z + B {n (1 a) δZ}


(B.2)


Substituting further the (interior) first-order condition (8) , that is, a = r (n 1) Z/Bn2 into

a in (B.2), we obtain

pA + pBZ = 1Z + B
n


n1


(n1) z\ δZ .

Bn2 J ʃ


(B.3)


Comparing the coefficient of Z and the constant in both sides of (B3) yields

pA Bn = 0 and pB---1-------+ + Bδ = 0.

nn

Solving the above simultaneous system of equations in terms of A and B to yield

23



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