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

A3 holds, we know that W increases in y quicker then V does. It means that
when the overall reaction curve jumps down along the diagonal, it never jumps
up again. ■

Proof. (Theorem 6.1) Consider restricted reaction curves r ₁1 ∆U (y) and r₂∣ ∆_u (x),
both decreasing as implied by A1⁰ and Topkis’s Monotonicity Theorem. From
Lemma 8.5 and the monotonicity it follows that c ≥ rι∣∆U (0) ≥ rɪ ∣∆U (d) > d
and 0 < r2∣∆_u (0) ≤ r2∣∆U (d) ≤ d, therefore rι∣∆U(y) : [0, d] → [d, c] and
r₂ ∣_∆U (x): [d, c] → [0, d] are well defined. Define the mapping B :[d, c] → [d, c],
B(x) = rι∣∆U ◦ r2∣∆U (x), which is increasing given that each of rι∣∆U and r2∣∆U
is decreasing. From Tarski’s Fixed Point Theorem, we know that there exists x
such that B(x) = rι∣∆U ◦ r2∣∆_u(x), therefore (x,r2 (x) ∣δ_u) is a PSNE.

From Lemma 8.5, there is no symmetric PSNE in [0, c]. Hence, there must
exist at least one pair of PSNEs. ■

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