A Dynamic Model of Conflict and Cooperation

This finding is quite intuitive, and is also consistent with the static conflict models of Hirshleifer
(1991, 1995).

An increase in the number of agents augments the aggregate endowment in proportion to n,
which causes an outward shift of the aggregate resource constraint C1 (i.e., scale effect), since
each entrant provides one additional unit of the endowment. The larger aggregate endowment
will increase the payoff each agent can expect to obtain from a given investment in aggression,
thereby intensifying each agent’s aggressive behavior and thus making the linear strategy aL
steeper. Since the former effect causes a counter-clockwise turn of line C1 around point (0, 1),
these two effects together intensify individual appropriation, but the long run effect on the
common-pool stock Z is ambiguous:⁸

r (δ + ρ) δ∆^-2 > 0,

£{1 — r (n — 1) — nr} (—rρ + δ) — pn²r²] ∆^-2 R 0.

A higher depreciation rate causes a reduction in the level of the common-pool stock Z
available to contenders, thereby discouraging appropriation. This negative prize effect causes
a clockwise turn of line C1 around point (0, 1) (i.e., the aggregate resource constraint C1 moves
inward toward the origin). At the same time, a higher δ implies that the cost of reproducing
the common-pool stock increases relative to the cost of aggressive behavior, which in turn
strengthens an incentive for aggressive behavior, thus making the linear strategy aL steeper.
Although these two effects on appropriation operate in opposite directions, the following result
indicates that the former effect will outweigh the latter effect in the long run:

—pr (n — 1) [1 — r (n — 1)] ∆ ² < 0,
—n [1 — r (n — 1)] ∆^-2 < 0.

A decrease of the subjective rate of time preference makes the linear strategy aL steeper,
⁸This effect has been also found in Result 4B of Hirshleifer (1995).

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