Figure 7: If F satisfies submodularity on the square on the diagonal, this implies
it satisfies submodularity on the rectangle.
situation depicted in figure 5 as we now show, say for F . Consider the case of
figure 7 with the four points (x, z), (z, z), (x, y) , (z, y) as shown. With z<x<y,
we know from A1 that, since F = U on ∆U ,
F(x, y) - F(x, x) ≥ F(z, y) - F(z, x).
From Lemma 8.2, submodularity holds for the configuration of the square (x, x),
(z, x), (z, z), (z, x), hence we have
F(x, x) - F (x, z) ≥ F(z, x) - F(z, z).
Adding the two inequalities yields
F(x, y) - F (x, z) ≥ F(z, y) - F(z, z),
which is just the definition of submodularity for the original points (x, z), (z, z),
(x, y) and (z, y).
It can be shown via analogous steps that the submodularity of F for any
other configuration of points can be reduced to showing submodularity for
squares with two vertices on the diagonal. The details are left out. ■
The next result allows us to conclude that the two reaction curves always
admit a discontinuity that skips over the diagonal, a key step for our endogenous
heterogeneity result.
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