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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