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

set and hence we may use Tarski’s Fixed Point Theorem to conclude that there
exists a pair (x,y) such that B(x,y) = (rι(y), r2(x)). (x, y) is a pure strategy
Nash equilibrium.

Finally, by Lemma 4.1 we know that no equilibrium can be symmetric. ■
Proof. (Theorem 4.2) Define R, rι(y)∣∆U and r2(x)∣∆_u as before. From
assumption B5, rι(y)∣∆U ∈ R and r2(x)∣∆_u ∈ R and from assumption B1⁰
they are continuous. Consider the mapping B : R → R such that B(x, y)=
(r₁(y ), r₂(x)). B is a continuous correspondence, given that both its compo-
nents are continuous, R is a compact set and hence we may use Brouwer’s
Fixed Point Theorem to conclude that there exists a pair (x∕, y∕) such that
B(x∕, y∕) = (r1 (y∕), r2 (x∕)). (x∕, y∕) is a pure strategy Nash equilibrium.

By Lemma 4.2 we know that no equilibrium can be symmetric. ■

8.4 Proofs of Section 6

To prove the theorem we first formulate a useful lemma.

Lemma 8.5 If A10, A2 and A4 hold, then there exist exactly one point d ∈ [0, c]
such that r1(d - ε) >d>r1(d + ε),ε > 0.

Proof. (Lemma 8.5) Consider ∆U . From A1⁰ and Topkis’s Monotonicity The-
orem we know that reaction curve is decreasing in this area. The generalized
first order condition for a maximum to occur in (x, x) is (in the absence of dif-
ferentiability in this point) U₁ (x, x) ≤ L₁(x, x). Assumption A2 rules out this
possibility, thus no x ∈ (0, c) can be a best response to itself. Moreover, neither
(0, 0) nor (c, c) can be an equilibrium, since from A3 follows, that for player 1
it is always profitable to deviate from any of these points. Hence, the reaction
curve does not cross the 45^o line, and there must exist a point, call it d ∈ [0, c]
such that r1(d - ε) >d>r1(d + ε).

Now, we prove uniqueness of this point. Consider r₁(y) and r₁(y) defined
as in Section 4. Denote W(y) = L(rι(y),y) and V(y) = U(rι(y),y). From the
Envelope Theorem, ^dwy(^y) = L2(r₁(y), y), and ^dV_y(^y) = U₂(r₁(y),y). Hence, if

41

<<   38   39   40   41   42   43   44   >>

More intriguing information