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 A
10 and Topkis’s Monotonicity Theorem. From
Lemma 8.5 and the monotonicity it follows that c
rιU (0) rɪ U (d) > d
and
0 r2u (0) r2U (d) d, therefore rιU(y) : [0, d] [d, c] and
r
2 U (x): [d, c] [0, d] are well defined. Define the mapping B :[d, c] [d, c],
B
(x) = rιU r2U (x), which is increasing given that each of rιU and r2U
is decreasing. From Tarski’s Fixed Point Theorem, we know that there exists x
such that B
(x) = rιU r2u(x), therefore (x,r2 (x) δu) 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. ■

42



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