Cournot Oligopoly and the Theory of Supermodular Games

STOCHASTIC CAPITAL ACCUMULATION GAMES 121hence, since v ∈ CM 1 , by a well-known characterization of weak convergenceof measures,∫∫v(x ′ )F(dx ′ | x n −c n − h(x n )) −→ v(x ′ ) dF(x ′ | x −c−h(x)). (3.5)Continuity of the supremand of (3.4) follows from (3.5). Furthermore, the feasibleset [0, K 1 (x)] clearly represents a continuous correspondence, in view ofA.3. Hence, by the Maximum theorem, T (v) is continuous.Next, we show that T (v) is nondecreasing. Let x 1 ≥ x 2 . Then, by AssumptionA.2 (i), Lemma 0.1, and the fact that v ∈ CM 1 , we have, since x 1 − h(x 1 ) ≥x 2 − h(x 2 ), for every c,∫u 1 (c) + δ 1 v(x ′ ) dF(x ′ | x 1 −c−h(x 1 ))∫≥ u 1 (c) + δ 1 v(x ′ ) dF(x ′ | x 2 −c−h(x 2 )). (3.6)Since T (v)(x 1 ) is the sup of the LHS of (3.6) over c ∈ [0, K 1 (x 1 )], and T (v)(x 2 )is the sup of the RHS of (3.6) over c ∈ [0, K 1 (x 2 )], and since [0, K 1 (x 2 )] ⊂[0, K 1 (x 1 )] by A.3, we have T (v)(x 1 ) ≥ T (v)(x 2 ). This shows that T mapsCM 1 into itself.Next, observe that CM 1 , endowed with the uniform distance, defined byd(γ 1 ,γ 2 ) = sup |γ 1 (x) − γ 2 (x)|, is a closed subset of the Banach space C,consisting of bounded continuous functions on [0, +∞), endowed with the supnorm. Hence CM 1 is a complete metric space.A standard argument in discounted dynamic programming theory shows thatT is a contraction with unique fixed-point V h ∈ CM 1 which thus satisfies(3.3).A best response of Player I to h is defined as any argmax of (3.3), for x ∈[0, +∞). For the next results, it is convenient to define two alternative choicevariables for Player I. Instead of choosing c ∈ [0, K 1 (x)], Player I may bethought of as choosing z = x − c, z ∈ [x − K 1 (x), x]. Then (3.3) may berewritten in the equivalent form∫V h (x) = max{u 1 (x − z) + δ 1 V h (x ′ ) dF(x ′ | z−h(x)): z ∈ [x − K 1 (x), x]}.(3.7)Likewise, Player I may be viewed as choosing joint investmenty = x − c − h(x), with y ∈ [x − K 1 (x) − h(x), x − h(x)].Then (3.3) becomes∫V h (x) = max{u 1 [x − y − h(x)] + δ 1 V h (x ′ ) dF(x ′ | y):y ∈ [x − K 1 (x)−h(x), x − h(x)]} (3.8)