A Dynamic Model of Conflict and Cooperation

the value function associated with strategy baL is bounded. Hence we can apply the sufficiency
conditions in Theorem 3 of Rowat’s (2007) to demonstrate its subgame perfectness.

Applying the above sufficiency conditions and using the cornered strategy a =1,wecan
construct more subgame perfect equilibrium strategies. As shown in Lemma 2 of Appendix
C, the range of the state variable Z where the cornered strategy a =1 remains as an optimal
one (i.e., (∂p_i∕∂ai) Z > V0 (Z)) depends on the chosen value of c₃ in (C6). In particular, the
cornered strategy a =1 remains subgame perfect over the domain (0, ∞) if the constant c₃
is larger than the infimum of the set of c3 satisfying inequality (C7) in Appendix C (see Fig.
3). When c3 is smaller than that infimum, more of the a4-families of strategies patched by
strategy a =1 will be subgame perfect as c₃ increases to that infimum, as shown in Fig.3.

In Fig.1 (i.e., when aE < 1) the strategy a4 patched by the cornered strategy a =1is
qualified as a subgame perfect equilibrium strategy over the domain (0, ∞) if c₃ is greater
than that infimum. If this is a case, the linear MP strategy ends up with the highest level of
steady state durable goods stock. On the other hand, in Fig.2 the patched strategy ^₄ is still
qualified if c₃ is greater than that critical value. In addition, the strategies left off the strategy
^₁ which is tangent to the constraint a = 1 and then connected with the strategy a = 1 at
that tangent point in Fig.2 are also subgame perfect. As a result, there are non-linear MP
strategies which entail higher levels of durable goods stock in the steady state such as Z as
compared to that supported by the linear strategy a_L.

We summarize with the following theorem:

Proposition 1 Consider the dynamic game defined by (5) and assume that the constant of
integration c₃ is chosen so as to satisfy inequality (C7) in Appendix C. Symmetric Nash
equilibria in continuos and asymptotically stable Markov perfect strategies must satisfy the
following:

(i)If a_E 5 1, thus r (ρ + δ)(n - 1) 5 [1 - r (n - 1)] δ, then there exist uncountably many
non-linear Markov perfect strategies and the unique linear Markov perfect equilibrium strategy,
coupled with strategy a(Z) = 1, that are globally defined over the entire domain of a state space,
leading to steady state equilibria ranging over (0,Z_S];and

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