This Lemma guarantees that the reaction curves possess a jump and that this
jump is unique. The flavor of the proof is the following: given supermodularity
of U and L, we know that reaction curves are increasing in ∆U and in ∆L .
Assumption B2 implies that there is no interior symmetric equilibrium due to
the presence of a canyon along the diagonal. Furthermore, assumption B4 rules
out symmetric equilibrium on the boundary. This is equivalent to saying that
the reaction curves do not intersect the diagonal on the whole strategy space.
Hence, there must be a jump across the diagonal. Finally assumption B3, that
entails an idea of monotonicity of maxima along y, guarantees that in case a
jump occurs, it is unique. From this Lemma we can conclude that the reaction
curves possess a jump across the diagonal in a point d and that once it occurs,
the reaction curves never jump back again. The point d is useful to define
subsets of the strategy space, which are necessary in the next theorem.
The following theorem implies that no symmetric PSNE exists for a game
where assumptions B 1 - B5 hold and that a PSNE exists. From these premises
we can conclude that there are only asymmetric PSNEs.
Theorem 4.1 Assume that B1 - B5 hold, then there exist at least one pair of
asymmetric PSNEs and no symmetric one.
The intuition behind this result is the following: from Lemma 4.1 we have
that the reaction curves are partially increasing (as they increase in each side of
the diagonal) and possess a unique downward jump at point d. Since reaction
curves are not overall increasing we cannot guarantee without further assump-
tions the existence of a PSNE. Introducing assumption B5 guarantees that the
reaction functions are well defined in R = {(x, y):d ≤ x ≤ c, 0 ≤ y ≤ d},R ⊂
∆u and R0 = {(x,y) :0 ≤ x ≤ d, d ≤ y ≤ c},R0 ⊂ Δl, in the sense that
they are completely contained in these compact subsets of the strategy space.
We obtain, hence, increasing maps in compact sets and Tarski’s Fixed Point
Theorem can be applied to show that within these sets a PSNE exists.
Note that we can reorder one player ’s action set in a nonstandard way to
obtain a submodular game. Consider action space of (say) player 1 as (d, 0] ∪
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