Cournot Oligopoly and the Theory of Supermodular Games

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126 RABAH AMIRLEMMA 2.1. For every v = (v 1 ,v 2 )∈CM 1 ×CM 2 ,the game G v has a Nashequilibrium in LC M 1 × LCM 2 .Proof. Repeat the argument for proving Proposition 1 step for step uponreplacing V h by v 1 (and similarly for Player II, V g by v 2 ). Observe that the proofhere is a special case of that of Proposition 1 since (v 1 ,v 2 )is exogenously fixed.In particular, for the continuity step (Lemma 1.4) v 1 does not depend on n whileV 1 does.The actual details of this proof are left to the reader.Define the best response map Bv x for the game Gx v (which is best thought ofas the game G v with x fixed)asBv x :[0,K 1(x)]×[0, K 2 (x)] −→ [0, K 1 (x)] × [0, K 2 (x)](c 1 , c 2 ) −→ (c1 ∗ , c 2 ∗ ),wherec ∗ 1 = arg max {u 1 (c 1 ) + δ 1∫c ∗ 2 = arg max {u 2 (c 2 ) + δ 2∫}v 1 (x ′ ) dF(x ′ | x −c 1 −c 2 ): c 1 ∈ [0, K 1 (x)](3.13a)}v 2 (x ′ ) dF(x ′ | x −c 1 −c 2 ): c 2 ∈ [0, K 2 (x)] .(3.13b)Here, the single-valuedness of Bv x above is due to the fact that the maximandsin (3.13) are strictly concave functions of c 1 and c 2 , respectively; see the proofof Lemma 1.2.LEMMA 2.2. For each fixed x ∈ [0, +∞) and (v 1 ,v 2 ) ∈ CM 1 ×CM 2 , themap Bv x is nonexpansive and monotone nonincreasing.Proof. We show the desired conclusion for the map c 2 → c1 ∗ . From Lemma1.2, we know that ∫ v 1 (x ′ ) dF(x ′ |·)is a concave function. Hence, by Lemma 0.2,the integral term in (3.13a) is supermodular in (c 1 , −c 2 ). The feasible set [0, K 1 (x)]is independent of c 2 , thus ascending in c 2 . Therefore, by Topkis’ theorem, c1 ∗ isnondecreasing in (−c 2 ), or nonincreasing in c 2 .As in the proof of Lemma 1.3, it is convenient here to rewrite (3.13a) with thedecision variable being joint investment for Player I, i.e., y = x − c 1 − c 2 ,as∫y ∗ = arg max{u 1 (x − y − c 2 ) + δ 1 v 1 (x ′ ) dF(x ′ | y):}y ∈ [x − K 1 (x)−c 2 ,x −c 2 ] . (3.14)
STOCHASTIC CAPITAL ACCUMULATION GAMES 127The maximand in (3.14) is supermodular in (y, −c 2 ) by Lemma 0.2. Moreover,the feasible set [x − K 1 (x) − c 2 , x − c 2 ] is ascending in (−c 2 ), for fixed x ∈[0, +∞). Hence, by Topkis’ theorem once more, y ∗ is nondecreasing in (−c 2 ),or nonincreasing in c 2 . Note here that y ∗ (c 2 ) is single-valued since the maximandin (3.14) is strictly concave in y. Now, since y ∗ (c 2 ) = x − c1 ∗(c 2) − c 2 , the factthat y ∗ is nonincreasing in c 2 means thatc1 ∗(c 2) − c1 ∗(c′ 2 )c 2 − c2′ ≥−1, ∀c 2 ,c 2 ′ ∈[0, K 2(x)].Together with the fact that c1 ∗ is nonincreasing in c 2 (which is the first step in thepresent proof), this yields− 1 ≤ c∗ 1 (c 2) − c1 ∗(c′ 2 )c 2 − c2′ ≤ 0. (3.15)Obviously, a similar conclusion holds for c2 ∗(c 1). The desired conclusion is easilyseen to follow from (3.15).LEMMA 2.3. For each fixed x ∈ [0, +∞) and (v 1 ,v 2 ) ∈ CM 1 ×CM 2 , thegame Gv x has a unique Nash equilibrium in pure strategies.Proof. Existence of a Nash equilibrium for the game Gv x follows fromTarski’s fixed-point theorem applied to the composite map Bv x ◦ Bx v , which isnondecreasing. A fixed-point of the composition is easily seen to be a Nashequilibrium. This argument is due to Vives (1990).Alternatively, one may also use Brouwer’s fixed-point theorem since Bv x is obviouslycontinuous, as a consequence of (3.15), and the action spaces [0, K i (x)]are compact. To establish uniqueness of Nash equilibrium for Gv x , assume, foreventual contradiction, that Gv x has two distinct Nash equilibria (a 1, b 1 ) and(a 2 , b 2 ) with, say a 1 > a 2 and hence, b 1 the best response functionis nonincreasing). By (3.15), we have− 1 ≤ c∗ 1 (b 1) − c1 ∗(b 2)b 1 − b 2= a 1 − a 2≤ 0b 1 − b 2(3.16a)and similarly for Player II− 1 ≤ c∗ 2 (a 1) − c2 ∗(a 2)a 1 − a 2= b 1 − b 2≤ 0.a 1 − a 2(3.16b)It follows from the first inequalities in (3.16a) and (3.16b) that a 1 − a 2 =−(b 1 − b 2 ), i.e., that the first inequalities in (3.16a) and (3.16b) are actuallyequalities. This in turn, implies that, say for Player I,c1 ∗(c 2) − c1 ∗(c′ 2 )c 2 − c2′ =−1, for all distinct c 2 , c 2 ′ in [b 1, b 2 ]. (3.17)
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Cournot Oligopoly and the Theory of Supermodular Games

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