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3, we introduce a two-stage stochastic dynamic game, as a special case of the general discrete time LQG game of Sect. 2, which is solved completely for its SPE in both pure and mixed strategies, some of the CE type and others non-CE (see Theorem 1 for the complete solution). Analysis of the two-stage game allows us to develop insight into the intricate role information structures play in the characterization and existence of SPE for the more general ZSDGs of Sect. 2, and what CE means in a game context.
Proof of Theorem 2 Observe first that for any c ∈ FL the function y → inf ξ ∈P S J (c) s q ds |y, ξ is concave. (22) Indeed, by (A3) J (c) s q ds |y, ξ S = Θ S J (c) s μθ1 ds l + S k=2 J (c) s μθk ds − S J (c) s μθ1 ds gk (y) ξ(dθ ) for any ξ ∈ P. By Assumption (C) and Lemma 11, we obtain that S J (c) s μθk ds − S J (c) s μθ1 ds ≥ 0 (23) for all θ ∈ Θ and k = 2, . . , l. Since g2 , . . , gk are concave, we infer that l y→ k=2 S J (c) s μθk ds − S J (c) s μθ1 ds gk (y) is concave for all θ ∈ Θ, and therefore (22) is follows.
S. (2003). On a new class of nonzero-sum discounted stochastic games having stationary Nash equilibrium points. International Journal of Game Theory, 32, 121–132. Nowak, A. S. (2006a). On perfect equilibria in stochastic models of growth with intergenerational altruism. Economic Theory, 28, 73–83. Nowak, A. S. (2006b). A multigenerational dynamic game of resource extraction. Mathematical Social Sciences, 51(3), 327–336. 22 Ł. Balbus et al. Nowak, A. S. (2010). On a noncooperative stochastic game played by internally cooperating generations.