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 a_i = a_i (Z). By substituting this optimal
strategy into (6), the HJB equation (6) associated with agent i is transformed into

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

j=1    ^j                                  j=1

n

+Vi⁰(Z) -    a⁰j(Z)-δ .

j=1

Substituting (8) and (9) into V_i⁰ (Z) and V_i⁰⁰ (Z) in (A.2), respectively, and exploiting symmetry
yields

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

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

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