Cournot Oligopoly and the Theory of Supermodular Games

122 RABAH AMIRThe next result shows that, if h ∈ LCM 2 , there is a unique best response g.LEMMA 1.2.Player I.If h ∈ LCM 2 , there is a unique stationary best response g byProof. We show that the maximand in (3.8) is a strictly concave function ofy, for each x ∈ [0, +∞). Since u 1 is strictly concave, it suffices to show that theintegral term is concave in y. To this end, first note that F(x |·)being conveximplies that for any y 1 , y 2 ∈ [x − K 1 (x) − h(x), x − h(x)], any λ ∈ [0, 1], andany x ′ ∈ [0, +∞)F[x ′ | λy 1 + (1 − λ)y 2 ] ≤ λF(x ′ | y 1 ) + (1 − λ)F(x ′ | y 2 ). (3.9)Then, consider∫λ∫V h (x ′ ) dF(x ′ | y 1 )+(1−λ)∫=∫≤V h (x ′ ) dF(x ′ | y 2 )V h (x ′ )d[λF(x ′ | y 1 )+(1−λ)F(x ′ | y 2 )]V h (x ′ ) dF[x ′ | λy 1 +(1−λ)y 2 ],as a consequence of (3.9), Lemma 0.1, and the fact that V h is nondecreasing (infact V h ∈ CM 1 , by Lemma 1.1).This shows that the integral term is concave in y. Since the feasible set [x −K 1 (x) − h(x), x − h(x)] is a convex interval for fixed x ∈ [0, +∞), there isa unique argmax in (3.8) and hence also in (3.3). This completes the proof ofLemma 1.2.LEMMA 1.3.LCM 1 .If h ∈ LCM 2 , the unique best response g by Player I is also inProof. From Lemma 1.2, we know that g is single-valued. We now show thatg is nondecreasing. To this end, observe that the concavity of the integral termof (3.8) implies the supermodularity of the integral term of (3.3) in (c, x), asaconsequence of Lemma 0.2. Furthermore, the correspondence x → [0, K 1 (x)]is clearly ascending since K 1 is nondecreasing. Hence, by Topkis’ theorem, g isnondecreasing.Next, we show that no slope of g can exceed one. Both terms of the maximandof (3.7) are supermodular in (x, z), again as a consequence of Lemma 0.2 andthe fact that h is nondecreasing (since h ∈ LCM 2 ). Also, the correspondence[x − K 1 (x), x] is ascending in x, since the function x − K 1 (x) is increasing,as a consequence of Assumption A.3. Hence, by Topkis’ theorem, the argmaxin (3.7) is nondecreasing, which is equivalent to saying that the slopes of g (theargmax in (3.3)) are all less than one, since z = x − c.