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

Theorem 8.3 (Milgrom and Shannon) The conclusion of Topkis’s Theo-
rem continues to hold when supermodularity [submodularity] is replaced by the
[dual] single crossing property.

We can introduce now the notion of supermodular game and of its properties.
A two player game is supermodular (submodular) if both payoff functions are
continuous, supermodular (submodular) and both action spaces are compact
real intervals. ¹⁵ The fixed point theorems associated with this framework are
due to Tarski (1955).

Theorem 8.4 (Tarski’s Fixed Point Theorem) Let f : I1 × I2 → I1 × I2
be an increasing function, then f has a fixed point.

Theorem 8.5 A two player supermodular (submodular) game has a pure strat-
egy Nash equilibrium.

In general, this theory dispenses with assumptions of concavity or differen-
tiability of payoff functions, making it an extremely general framework to study
the properties of equilibria.

8.2 Proofs of Section 3

The proof of Theorem 3.1 is organized as follows: we begin with proving four
preliminary lemmas, and then present the main proof in two steps: first we show
existence of PSNE and afterwards that all PSNEs must be asymmetric.

The first lemma states that for a small enough square of points on the
diagonal we have submodularity.

Lemma 8.1 Consider the following points as depicted in figure 5. If A1 - A2
hold, then for small enough α>0: U (x, x - α) - U (x - α, x - α) ≥ L(x, x) -
L(x - α, x), ∀x ∈ [0, c].

¹⁵ Compactness is not necessary, it is required in order to use a simplified version of Tarski’s
Fixed Point Theorem, without referring to lattices.

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