Cournot Oligopoly and the Theory of Supermodular Games

120 RABAH AMIRHere, the upper bound on the possible values that Player i can get follows fromthe definition of payoffs and Assumption A.3. It is also useful to define theclosure of CM i under pointwise convergence:We are now ready forM i : {v: [0,+∞) → [0, +∞) such that 0 ≤ v ≤ u i (C i )1−δ iand v is nondecreasing}.PROPOSITION 1. The infinite-horizon game has a Nash equilibrium in stationarystrategies which are elements of LC M 1 × LCM 2 .It is convenient to break down the proof of Proposition 1 into three distinctlemmas, labeled 1.1–1.3. We first define the associated best-response optimizationproblem.Suppose that Player II, say, uses a stationary strategy h ∈ LCM 2 . Then PlayerI’s value function for optimally responding to h (in the infinite-horizon game),to be denoted V h , is defined byV h (x) = sup E ∑ ∞t=0 δ 1u 1 (c 1 t )subject to x t+1 ∼ q(· |x t −c 1 t −h(x t )), t = 0, 1,...(3.2)with x 0 = x,where the expectation is over the unique probability measure induced by x, hand a stationary strategy by Player I (see Section 1), and the supremum may betaken over the space of all stationary policies (see, e.g., Bertsekas and Shreve,1978, Stokey et al., 1989).LEMMA 1.1. In the optimization problem (3.2), assume that h ∈ LCM 2 .Then V h ∈ CM 1 and V h is the unique solution to the functional equation∫V h (x) = max{u 1 (c) + δ 1 V h (x ′ ) dF(x ′ | x −c−h(x)): c ∈ [0, K 1 (x)]}.(3.3)Proof. Define the map T : CM 1 →CM 1 by∫T (v)(x) = sup{u 1 (c) + δ 1 v(x ′ ) dF(x ′ | x −c−h(x)): c ∈ [0, K 1 (x)]}.(3.4)First, we show that T indeed maps CM 1 into itself. To this end, we start by provingthat the supremand in (3.4) is continuous in (c, x). Let c n → c and x n → x.Then, since h ∈ LCM 2 , x n − c n − h(x n ) → x − c − h(x). By Assumption A.2(see also (2.1)), we have F(· |x n −c n −h(x n )) → F(· |x−c−h(x)), and