A Dynamic Model of Conflict and Cooperation

A = ПВ and B = 1 ^- r (^{n - 1}).

ρ               (ρ+δ)n

Further substitution of these expressions into the first-order condition, a = r (n - 1) Z/Bn²,
yields (15). The corresponding value function is given by

Appendix C: Subgame Perfect Solutions

To prove the existence of MP Nash equilibria we apply a sufficiency theorem stated in Theorem
3 of Rowat’s (2007). To do this, we first have to show that the strategy is feasible, which is
defined in Definition 1 of Rowat’s (2007). Although it may be appropriate to distinguish
between the value function associated candidates strategies and that associated with the well-
defined strategies like those of Rowat’s, it is omitted for the sake of notational simplicity
but with understanding that the solutions have to pass the further tests to qualify for an
equilibrium strategy satisfying the properties stated in the text.

Lemma 1 The strategy a(Z)=0for Z ∈ (0, ∞) is not an equilibrium strategy.

Proof. Suppose that (∂p_i∕∂a_i) Z — Vi⁰ (Z) < 0. Then it follows from (8) that α_i = 0. In

this case, the HJB equation (6) becomes

ρVi(Z) = nZ + Vi⁰ (Z)(n — δZ).                        (C.1)

By integration and imposing symmetry, we have

_v∕_z∖ _ (^{n - δ}Z)2 [^{n +} (^{ρ +} 2^δ) ^Z] , δ _δZ∖-ρ                 ∕C₂∖

^v (^z )= n (ρ + 3δ)(ρ + 2δ) ^{+ c}2^{(n - δ}Z ) ^{δ ,}              (C.2)

where c₂ represents a constant of integration. When c₂ 6=0, lim V (Z)=±∞.Thisimplies
Z→n∕δ

that the strategy a(Z)=0for Z ∈ [0, ∞) is not an equilibrium strategy whenever c2 6=0,

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