A Dynamic Model of Conflict and Cooperation

trades the marginal increase in expected payoff from an increase in appropriation against the
marginal loss in the discounted value of the future stream of payoffs which results from a
reduction of productive effort. If the payoff gain from an increase in ai is larger than the
payoff loss implied by the decrease in li for all levels of ai , then agent i will rationally devote
all resources to appropriation. In contrast, the agent chooses a_i =0in cases where the
discounted marginal gain from productive investment exceeds the instantaneous marginal gain
from aggressive behavior for all levels of ai .

2.2 Equilibrium

We can then make use of (8) to characterize subgame perfect equilibria of the differential game.
Since we have started our analysis assuming identical agents, a natural focus is on symmetric
equilibria. The symmetry assumption allows us to drop the subscript i in the subsequent
discussion, and we will suppress this index unless strictly necessary for expositional clarity.

Let us first analyze interior solutions of ai . Differentiation of the interior first-order condi-
tion in (8) gives

At an interior solution of a (Z) we may apply the envelope theorem to characterize a⁰ (Z).

Using the symmetry assumption, we obtain

+ ⁺ n²a ZZ) [(1 ^- a ^(Z)) n ^{- (р +} 2^δ) Z]
n n a (Z )

r (n ~ 1) Z (a - δZ)

a²a (Z)^{2   V ,}

We will employ phase-plane methods to characterize the qualitative solution of the nonlin-
ear differential equation (10) and the associated MP strategies. For this purpose we have to
identify the steady state locus where Zi = 0, called C₁ in the following. Let us denote by C₂
the loci where a⁰ (Z) goes to plus infinity, and by C3 the loci where a⁰ (Z) equals zero in the

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