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

Figure 3: Reaction curves are partially increasing, but posses an unique jump
down, which precludes any symmetric equilibrium.

[c, d), ordered from left to right. The other player’s action space order remains
without change. It can be verified, that the game satisfying B 1 - B5 then
becomes a game of strategic substitutes.

Now consider another property of U and L.

B1’ U and L are quasi-concave.

The assumption B1 can be replaced by the assumption B1⁰ which guarantees
that player’s best replies are continuous (not overall) and still the result holds.
In other words, the supermodularity of U and L is not necessary (even though
often observed in applications) for the existence of only asymmetric PSNE in
this framework. In particular, assuming that U and L are quasiconcave implies
that the reaction curves are partially continuous (even though not monotone)
and thus, as long as the unique jump across the diagonal exists and the reaction
curves are completely contained in compact subsets of the domain, we can show
the existence of asymmetric PSNE. Since monotonicity cannot be guaranteed
anymore Tarski’s Fixed Point Theorem must be replaced by Brouwer’s Fixed
Point Theorem in showing the existence of PSNE.

Lemma 4.2 If B1⁰ and B2 - B4 hold, then there exists exactly one d ∈ [0, c]
such that ∀ε>0: r1 (d - ε) >d>r1 (d + ε),

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