2.1 Solution Concept
We solve the differential game using the notion of a (stationary) MP Nash equilibrium, because
we think that this equilibrium concept captures the essential strategic interactions over time.
MP strategies are decision rules such that each agent’s decision is the best response to those of
the other players, conditional on the current payoff-relevant state variable Z (see, e.g., Chapter
4 in Docker et al., 2000). Markovian strategies rule out path dependence in the sense that
they depend only on the current values of the state variables rather than strategy choices in
history. As a result, it does not matter how one gets to a particular point, only that one gets
there.
MP equilibrium strategies must satisfy the Hamiltonian-Jacobi-Bellman equation given by:
ρVi (Z)= max
ai∈[0,1]
pi (a1 ,...,an) Z + Vi0 (Z)
X(1 - aj) - δZ
(6)
where Vi denotes the maximum value agent i attributes to the game that starts at Z . Notice
that
d2Pi 7 7
Z =r (n
1) n (r -1)2- 2r Z < 0 for n
n3ai2
=2∧r>0,
(7)
>2 ∧ 0 <r<n/(n - 2),
implying that the r.h.s. of (6) is concave in ai ∈ [0, 1].
We assume that r<1/(n - 1) in
what follows,6 guaranteeing not only that the second-order condition (7) holds but also that
the linear strategy of each agent, which plays an important role in the later analysis, is a
nonnegative value. The first-order necessary condition for agent’s choice of appropriation is
given by
the l.h.s. of which is evaluated for all ai ∈ [0, 1]. According to (8), each agent, when choosing ai,
_
∂pi Z
∂ ai
|
=0 = |
⇒ ai ∈ [0, 1] , | |
|
V0 (Z) < |
>0= |
⇒ ai =1, |
|
< 0 = |
⇒ ai = 0, |
(8)
6 Tullock (1980) assumes the same condition in his two-agent, rent-seeking game. Hirshleifer (1991, 1995)
and Gonzalez (2007) also assume that r<1 in their two-agent games. Condition r<n/(n - 1) reduces to
r< 1 when n =2.