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

B 1⁰⁰ U and L have the single crossing property

Theorem 6.2 If the assumptions B1⁰⁰ and B2 - B5 hold, then there exist at
least one pair of asymmetric PSNEs and no symmetric one.

From Milgrom and Shannon’s Theorem Assumption B1⁰⁰ implies that re-
action curves are partially increasing in ∆_U and ∆_L . From this fact and as
long as they are well defined in a compact subset of the strategy space, we can
obtain existence of PSNEs. The valley-shape of the profit precludes symmetric
equilibria in the same spirit as Theorem 4.1.

7 Conclusion

Our theorems assert that, under specific conditions, heterogeneity in agent’s
behavior might arise even when they are a priori identical. This paper con-
stitutes, hence, a contribution to the discussion about the sources of diversity
across economic agents and disparities in economic performances. While pre-
vious literature stands on arguments related to multiplicity of equilibria and
strategic complementarities (Cooper, 1999) or on strategic substitutability and
stability of equilibria (Matsuyama, 2002), our approach stands on the existence
of a fundamental nonconcavity of the payoff function and on some form of strate-
gic substitutability. It is, thus, similar in spirit to Matsuyama’s work. However,
we show that endogenous heterogeneity does not rely on the idea that only
stable equilibria are observable as in Matsuyama. With respect to Cooper’s ap-
proach, where agents can still choose symmetrically, our results guarantee that
symmetric equilibria can never arise in a two player setting. Even though we
have, in our model some form of strategic substitutability (notice that when
talking about two-player games strategic substitutability can be converted into
complementarity through a simple inversion of one agent’s strategy space), the
critical assumption for the inexistence of symmetric equilibria is the noncon-
cavity of the payoffs. In fact, we show that strategic substitutability can be
replaced by partial quasiconcavity and still the results follow.

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