of Γ.
Theorem 5.1 Assume that the following assumptions hold:
1. F is strictly quasi-convex in own strategy
2. F(c,0) >F(0,0) andF(0,c) >F(c,c)
Then, the game has no symmetric equilibrium and it has exactly one pair of
asymmetric equilibria given by (0, c) and (c, 0).
Proof. From the definition of strict quasi-convexity, we know that:
F (λz1 +(1- λ) z2) < max{F(z1),F(z2)}, ∀z1, z2 ∈ S × S
Let z1 =(0,y) and z2 =(c, y), then we know that any x ∈ (0, c) yields a
lower payoff than x =0or x = c, ∀y ∈ [0, c]. This means that ∀y ∈ [0, c],
r1 (y)=0or r1 (y)=c. Analogously, for player 2 we have the same result, due
to symmetry.
Finally, from Assumption 2 we have that (c, 0) and (0, c) are the only PSNEs
of the game. ■
Figure 4 illustrates the results of this section. Notice that we depicted reac-
tion curves which are not continuous, specifically, they possess an odd number
of jumps. This is a direct consequence of the Assumptions 1 and 2 that imply
either that the reaction curves are continuous, or that, if they jump, they must
jump an odd number of times.
5.1 Applications
This theorem generalizes the results of Amir (2000) and Mills and Smith (1996),
whose models are usually presented within the literature about endogenous het-
erogeneity of firms.13 In these papers it is considered a two-stage duopoly game.
In the first stage, firms make long term investment decisions that affect the pro-
duction costs. In the second stage, firms compete à la Cournot. Both firms face
13 Another example where the game can be reduced to a two person normal form game
is presented in Boyer and Moreaux (1997). Conditions for the non-existence of symmetric
equilibria are the same, even if the payoff function is not convex.
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