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

Lemma 8.4 Given A1 - A3, there exists exactly one point d ∈ (0,c),suchthat
ri(d - ε) >d>ri(d + ε),i =1, 2.

Proof. (Lemma 8.4) From Topkis’s Monotonicity Theorem and Lemma 8.3 , all
the selections from the best reply correspondences are downward sloping. Hence,
both ri(d - ε) and ri (d + ε) exist for any selection of ri and are independent of
the selection.

From assumption A3, we know that (0, 0) ∈/ Graph ri and (c, c) ∈/ Graph ri
(i.e. ri does not go through (0, 0) or (c, c)). These two properties imply that ri
cannot be identically 0 or c.

We next show that the reaction correspondence r₁ (say) cannot ever cross the
45^o line at an interior point, i.e. in (0, c). The generalized first order condition
for a maximum of F (say) to occur at a point (x, x) with x ∈ (0, c), which applies
even in the absence of differentiability, is that U1 (x, x) ≤ L1 (x, x) . Assumption
A2 rules out this case . Hence no x ∈ (0, c) can ever be a best reply to itself,
meaning that the reaction curves do not cross the 45^o line at any interior point.

Since r1 starts strictly above 0 (for y =0)and ends strictly below c (for
y = c), the above properties of r₁ imply that there exists exactly one d ∈ (0, c)
such that r1(d - ε) >d>r1(d + ε). In words, there must exist a jump in the
best reply function past the diagonal as in figure 1. ■

Using the Lemmas 8.3 and 8.4 we can now prove Theorem 3.1.

Proof. (Theorem 3.1) From Lemma 8.3 we have overall submodularity of the
payoff function. This guarantees that a PSNE exists.

Consider now the behavior of the reaction curves in the area ∆U .The
same conclusion follows for ∆_L by symmetry. Define the following restricted
reaction curves: rι∣∆U (y) : [0, d] → [d, c] and r2∣∆_u (x) : [d, c] → [0, d] both
decreasing as implied by Lemma 8.3. Define the mapping B :[d, c] → [d, c],
B(x) = r₁∣∆U ◦ r₂∣∆U (x), which is increasing given that each of r₁∣∆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) ∣∆U) is a PSNE. From
Lemma 8.4, there is no symmetric PSNE in [0, c]. Hence, there must exist at

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