A Dynamic Model of Conflict and Cooperation

(ii)If aE > 1, thus r (ρ + δ)(n - 1) > [1 - r (n - 1)] δ, then there exist uncountably many non-
linear Markov perfect strategies, coupled with strategy a(Z) = 1, that are globally defined over

,7   ,∙         Г  , ,      11,,1,,    'Tl ■    ■       _П ⅛

the entire domain of a state space, leading to steady state equilibria ranging over 0, Z

Г7 , Г7 , Г7        1

ZS < Z<ZE , where

r (т - 1) (δ + ρ)
( - 1) rρ + ρ

Proposition 1 implies that there exist multiple MP strategies in the space of nonlinear
strategies including the linear strategy, even in the case where the domain is globally defined.
It can be best understood that multiplicity of MP strategies and of the associated steady state
equilibria arise from the incomplete transversality condition, as pointed out by Tsutsui and
Mino (1990). Since there are unaccountably many asymptotically stable nonlinear Markov
strategies that converge to different steady states, it is impossible to uniquely pin down the
constant of integration c₁ in (14) in addition to the constant c₃ in (C6) of Appendix C,
so that we are unable to identify the unique path supported by a particular nonlinear MP
strategy which converges the unique steady state. The emergence of a continuum of steady
states supported by multiple non-linear MP equilibrium strategies significantly distinguishes
our results not only from the results of Hirshleifer (1991, 1995) and Skaperdas (1992) using
a static model but also from those of Maxwell and Reuveny’s (2005) using a dynamic model
with myopic agents in which the unique one-shot Nash equilibrium prevails or is repeated
every period.

Another important aspect of Proposition 1 is that given any initial stock of Z, the economy
approaches the range of steady state equilibria where the common-pool stock takes a posi-
tive value and individual aggressiveness takes an intermediate value between zero and one.
In this sense, (implicit) ‘partial cooperation’ can be seen as a best response to the risk of
appropriation. In affluent economies where the level of the stock variable is sufficiently large,
investment in aggression reaches the maximum possible level (i.e., a = 1) in finite time. It then
is decreasing until the steady state S is reached. Put differently, in affluent societies where
there is a large amount of the common pool stock, a full fighting strategy will be rationally
and inevitably chosen during the transition to the steady state. On the other hand, if the
initial stock level is relatively low at the start of the game investment in aggressive behavior

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