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

8 Appendix

8.1 Summary of supermodular/submodular games

We give an overview of the main definitions and results in the theory of su-
permodular games that are used in the paper, in a simplified setting that is
sufficient for our purposes. Details may be found in Topkis (1978).¹⁴

Let I₁ and I₂ be compact real intervals and F : I₁ × I₂ → R. F is
(strictly) supermodular if ∀x1,x2 ∈ I1,x2 >x1 and ∀y1 ,y2 ∈ I2,y2 >y1 we
have F(x2,y2) - F(x2,y1)(>) ≥ F(x1,y2) - F (x1,y1) .Fis (strictly) submod-
ular if -F is (strictly) supermodular.

Theorem 8.1 (Topkis’s Characterization Theorem) Let F be twice con-
tinuously differentiable. Then

(i) F12 = ∂x∂_y ≥ 0 [ ≤ 0] for all х,У ⇔ F is supermodular [submodular].

(ii) F12 = ∂x∂_y > 0 [< 0 ] for all х,У ⇒ F is strictly supermodular [sub-
modular].

The supermodularity property is not preserved by monotonic transforma-
tions of the function F. An alternative notion (ordinal) is the single crossing
property defined as follows: F has single crossing property [dual single crossing
property] in (x, y) if ∀x1,x2 ∈ I1,x2 >x1 and ∀y1,y2 ∈ I2,y2 >y1 we have

F(x1,y2) - F (x1, y1) ≥ 0[≤ 0] ⇒ F(x2,y2) - F(x2,y1) ≥ 0[≤ 0] .

The single crossing property does not have a correspondent differential char-
acterization and thus it is often more difficult to check. Now we present the
main monotonicity theorems.

Theorem 8.2 (Topkis’s Monotonicity Theorem) If F is continuous in y
and (strictly) supermodular [submodular] in (x, y), then arg maxy∈I₂ F(x, y) has
(all of its) maximal and minimal selections increasing [decreasing] in x ∈ I1 .

¹⁴ Other aspects of the theory may be found in Topkis (1979), Milgrom and Roberts (1990)
and Vives (1990).

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