Appendix (available upon request)
This appendix contains the proof of theorem 2. We start with the following auxiliary result.
Lemma 7 Assume that the parameter values satisfy κi < κi, I ∈ {1, 2, }, as well as
condition (22). Then it holds that βi > γiqi and R — n 1 ∕31 — n2β2 < min{ρ 1 ,ρ2}.
Proof: Using the definitions of β 1 and βi it is straightforward to verify that the two
inequalities in (22) are equivalent to β1 > γ 1 q1 and β2 > γ2q2, respectively. Furthermore,
because of the first inequality in (22), γ1 ≥ 0, and A1 ≥ 0 we have
n 2 ρ 2 — ( n 2 — 1) ρ 1 — R < n 2 A 2 - ( n 2 - 1) A1 - ( n - 1) Y1 11
≤ n2A2 — (n2 — 1)A1
= n2A2 + n1A1 — (n — 1)A1
≤ n2A2 + n1A1.
Using the definitions of β1 and /3i it is easy to verify that this condition is equivalent to
R — n 1 ∕3ι — n2β2 < ρ 1. The inequality R — n 1 β1 — n2β2 < ρ2 can be proved in analogous
way by starting from the second inequality in (22) instead of the first one. This completes
the proof of the lemma. <ι
Proof of Theorem 2: Without loss of generality we may assume that player i is of type
1. If all players j = i use their equilibrium strategies φj , then it follows that player i faces
the following optimal control problem P1:
maximize
subject to
+∞
— κ1xi(t)] dt
e-ρ1t[U1 (ci(t),yi(t)+γ1z(t))
0
z( t ) = R z ( t ) — xi ( t )
yi ( t ) = Г1 yi ( t ) + Xi ( t ) — Ci ( t )
yj(t) = ri(j)yj(t) + βi(j)z(t) — ⅛(j)[Uj(t) + Yi(j)z(t)] for j = i
xi(t) ≥ 0 , Ci(t) ≥ 0 , y(t) ≥ 0 , z(t) ≥ 0,
where R1 = R — (n1 — 1)β1 — n2β2. The Hamiltonian function of this problem is
H = U1(Ci, Ui + γ 1 z) — κ 1 Xi + λ(R1 z — xi) + μi(r 1 yi + xi — ci)
+ ∑ μj [ ri ( j ) yj + βi ( j )z — qi ( j )( yj + γi ( j )z )],
j=i
where λ and μk, k = 1, 2 ,...,n, are the costate variables corresponding to z and yk,
respectively. Note that the Hamiltonian is jointly concave in (z, y,Ci,Xi). The theorem
is therefore proved if there exist costate trajectories λ : [0, + ∞) → [0, + ∞) and μk :
[0, +∞) → [0, +∞), k =1, 2,...,n, such that the feasibility conditions of problem P1 as
19
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