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

least one pair of asymmetric PSNEs. ■

Theorem 3.2 rules out the existence of multiple pairs of asymmetric equilib-
ria.

Proof. (Theorem 3.2) Once again we concentrate on the area ∆U . Conclusions
follow for the area ∆L by symmetry. Whenever r1 [r2] is interior, first order con-
dition U1 (r1 (y) ,y)=0[L1 (r2 (x) ,x)=0], together with the implicit function
theorem and the assumptions (3) and (4), implies that r1 [r2] is differentiable
in ∆ττ and that r' (∙>Λ — Ui2(^rι(^y)^,y) ≥   1  _al₄∩ _r∣ (ʃ) _ L12(^r2(^x)^,x) ≥   1

in да и anu. niai r 1 y — — 77—7—≥—ʌ—ʌ ≥ — ι, aιso r о ιx ι — — 7—7—7—ʌ—ʌ ≥ — ι.

^U                   1 y          U11(r1 (y),y)           ,           2               L11(r2(x),x)           .

Hence, rι(y)∣∆U and ^r2 (x) ∣∆U are contractions. Using Banach’s fixed point
theorem we can conclude that there exists exactly one pure strategy Nash equi-
librium in ∆U.¹⁶ In the same way there exists exactly one pure strategy Nash
equilibrium in ∆L . Concluding, we have exactly one pair of pure strategy Nash
equilibrium. ■

Finally we provide a proof of Theorem 3.3.

Proof. (Theorem 3.3) Since x* > d > y*, we have F(x*,y*) — U(x*,y*) and
F(y*,x*) — L(y*,x*). Also U(ri(d),d) — L(rι(d),d) if d denotes the unique
point of jump of reaction curve between ∆_U and ∆_L, as defined in Lemma 8.4.

Then

F (x*,y*) — U (x*,y*) —

— U(ri(y*),y*) ≤ U(ri(d),d) —

— L(ri(d),d) ≤ L(rι(x*),x*) — L(y*,x*) — F(y*,x*)

where both inequalities follow from the monotonicity of U (ri(y), y) and L(ri(y), y).
■

8.3 Proofs of Section 4

First we prove Lemma 4.1 and Lemma 4.2 since these proofs are similar. We
then move to proving Theorems 4.1 and 4.2.

¹⁶ (Banach’s Fixed Point Theorem): Let S ⊂ Rⁿ be closed and f : S → S be a contraction
mapping, then there exists x ∈ S : f(x)=x.

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