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.
39