Endogenous Heterogeneity in Strategic Models: Symmetry-breaking via Strategic Substitutes and Nonconcavities

4.1 The results

In this section we analyze the case in which the components of the payoff func-
tion, U and L, are not submodular. We first consider the case in which U and L
are supermodular and then the case where U and L do not have this property,
which is replaced by quasiconcavity.

Consider the following assumptions:

B1 U and L are supermodular.

B2 U and L are differentiable and U₁ (x, x) >L₁ (x, x), ∀x ∈ (0, c)

Define r₁ (y) = argmax{U (x,y) : (x,y) ∈ ∆U} and r₁ (y) = argmax{L (x,y) :
(x, y) ∈ ∆L}

B³ U2(r1 (У) ,У) < L2(rι (У) ,У),∀y ∈ Y

B4 U1(0,0) > 0,L1(c,c) < 0

B5 U1 (x, 0) > 0, ∀x<dand L1 (x, c) < 0, ∀x>d,for d : r1 (d - ε) >d>
r₁ (d + ε)

B 1 says that on either side of the diagonal, but not necessarily globally,
each player’s marginal returns to increasing his action increase with the rival’s
action. B2 holds that each player’s payoff, though globally continuous in the
two actions, has a valley-like shape along the diagonal. B3 is responsible for
the uniqueness of the jump in the reaction curves. B4 excludes the existence
of equilibria in (0, 0) and (c, c). Finally, B5 restricts the reaction curves to a
compact subset of the action space which enables us to prove the existence of
an asymmetric PSNE.

These assumptions form a sufficiently general framework to encompass many
of the studies mentioned in Section 1 as illustrated below. Furthermore, all
assumptions are possible to check in a particular model.

Lemma 4.1 If B1 - B4 hold, then there exists exactly one d ∈ [0, c] such that
r1 (d - ε) >d>r1 (d + ε), ∀ε>0.

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