A Dynamic Model of Conflict and Cooperation

obtain a general solution to (10):

(n — 1) rZ (n — δZ) ^δ (ρ + δ)

ρ+ δ

n [1 — r (n — 1)] (n — δZ) ~ + (n — 1) r (ρ + δ) c₁

where c₁ represents an arbitrary constant of integration and may take a positive, zero or
negative value. When c1 =0, (14) simplifies to

r(n— 1)(ρ+δ)
Z Z Tvi Z.
[1 — r(n — 1)]n

It is easy to confirm by the well-established guessing method for a value function that strat-
egy aL stands for the linear strategy (see Appendix B). Moreover, the left branch of the
linear strategy a_L to the left of the steady state line C₁ starts from the origin, and then
reaches point S on the steady state line C1 , while its right branch starts from any initial
value Z0 >ZS [if we do not here take into account the resource constraint (1)], then reaching
point S also. Moreover, it can be verified by substitution that the linear strategy a_L also goes
through the singular point E .

The a₄- (a₃-) family of strategies represents the solution curve of (14) coupled with c₁ > 0
when n<δZ (n>δZ), while the a₁- (a₂-) family of strategies represents the solution curve
of (14) coupled with c1 < 0 when n<δZ (n > δZ). Moreover, the left-branch of the a4-
family of strategies also starts from the origin, while its right-branch starts from any initial
value Z0 < ZE, both of which reach the same point on line C1 .Theleftbranchofthea1-family
of strategies starts from the origin and reaches a point on the steady state line C1 , while its
right branch starts from point (n∕δ, 0) and reaches the same point on line C₁; therefore, those
strategies never hit the horizontal axis except for the origin and point (n∕δ, 0). On the other
hand, when the a₂- and a₃-families of strategies start from any initial value Z₀ >Z_E , the a₂-
family of strategies approaches point (n∕δ, 0), while the a₃-family of strategies goes to plus
infinity, as illustrated in Figs.1 and 2.

Nevertheless, not all integral curves in Figs.1 and 2 are qualified as MP equilibrium strate-
gies. There are three additional requirements which have to be met. The first prerequisite is
that strategies should not violate the resource constraint (1). This implies that a (Z) should

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