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

Let F be defined as in (1) and consider the following assumptions:

A10 U and L have dual single crossing property

A4 U2(r1(y)^,y) <L2 (rι(y)^,y), ∀y ∈ [^0,c]

The Assumption A1⁰ is alternative to Assumption A1 defined in Section 3
and provides the ordinal condition for the reaction curves to be decreasing. As-
sumption A4 expresses monotonicity of U and L with respect to the opponent’s
action along the best replies.

We now show the main result of this section in the following steps: first we
conclude about decreasing best-replies; then we observe that there is a downward
jump in the reaction function at point d and that this jump is unique. Finally
we use Topkis fixed point theorem to conclude of the existence of equilibria in
the subsets of the strategy space defined using point d.

Finally we treat the setting of Section 4, extending its results to the ordinal
definitions of complementarity.

Theorem 6.1 If A1⁰,A2,A3 and A4 hold, then the game Γ has at least one
pair of asymmetric PSNEs and no symmetric one.

The idea of the proof is that within ∆_U and ∆_L the dual single crossing
property of the payoff function F holds from assumption A1⁰. As Milgrom
and Shannon (1994) showed, the dual single crossing property allows us to
draw the same conclusions as the submodularity of the payoff function in terms
of monotonicity of the reaction curves. Furthermore the dual single crossing
property has the advantage of being more general and preserved by monotonic
transformations. Then assumption A2 implies that there exists a jump down in
the reaction curves at a certain point d and A4 implies that this jump is unique.
With point d we define compact subsets of the strategy space where reaction
curves are decreasing and thus we can guarantee the existence of a PSNE by
Tarski’s Fixed Point Theorem.

Some results from Section 4 can also be extended into an ordinal version.

Let F be defined as in (1) and consider the following assumption:

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