A Dynamic Model of Conflict and Cooperation

we substitute those expressions into (A.3) yielding

0 = n_an2¹ [-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,..., a_n) 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

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

a in (B.2), we obtain

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

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