be bounded to the nonnegative region below a horizontal line with intercept 1 in Figs.1 and 2.
The second requirement is that strategies should cover the entire (or global) domain [0, ∞)
in a continuous way.7 At first glance this requirement seems to eliminate all strategies aj ,
j =1,...,4,andaL. Nevertheless, strategies can potentially be continuously extended either
by the cornered strategy a = 1 along the resource constraint (1), and/or by the non-aggressive
strategy a =0on the horizontal axis (see Rowat, 2007). Both potential extensions are triggered
by the corner solutions where the equality in (8) does not apply. Despite this, the strategies
of the a3-family that does not reach the resource constraint (1) are immediately eliminated
because they can neither cover the global domain [0, ∞) by themselves nor be extended by
any strategy in a continuous way.
Furthermore, the non-aggressive strategy a(Z)=0on the horizontal axis is eliminated,
since this strategy does not satisfy (8) for Z ∈ (0, ∞), as shown in Lemma 1 in Appendix C.
As a result, extending the a1- and a2-families of strategies by the patching strategy a =0to
the global domain is not possible.
Turn to the linear strategy baL , where the hat indicates those strategies extended by the
patching cornered strategy a =1. Since strategy baL can continuously pass through point E
in Fig. 1, the coordinates of which are given by (13), strategy baL is continuos over the entire
domain [0, ∞). This property is also obtained in the case illustrated in Fig. 2 where strategy baL
does not go through point E . Here, the patching strategy a = 1 instead of the interior
strategy aL will cross locus C2 and thus strategy baL is continuos over the entire domain of Z
in Fig.2 as well. Taken together, the extended linear strategy baL survives as a candidate for a
subgame perfect strategy which will be discussed below.
The third and final requirement is subgame perfection. We have to show that there do not
exist profitable deviations from strategy baL . Strategy baL is stable in the sense that from an
arbitrary initial value of Z strategy baL can reach the steady state point S in the long run. As a
result, the convergence towards the finite steady state point S, together with (B4), ensures that
7 Tsutsui and Mino (1990), Itaya and Shimomura (2001) and Rubio and Casino (2002) restrict the state
space in order to get continuos, stable and subgame perfect equilibrium strategies. In particular, Tsutsui
and Mino treat the domain of a state variable as endogenous to get different stable Markov perfect strategies
associated with different steady states. Unfortunately, this approach prevents comparison of payoffs between
strategies.
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