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) ) < L2(rι (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>
r
1 (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. B
2 holds that each player’s payoff, though globally continuous in the
two actions, has a valley-like shape along the diagonal. B
3 is responsible for
the uniqueness of the jump in the reaction curves. B
4 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.

20



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