According to this paper, "every submodular function can be represented as a maximum of additive valuations." It gives an algebraic description as well, but I am having trouble internalizing the idea even at a high level.

Is there a simple geometric intuition for this fact? Is there an economic application that makes obvious why valuations with diminishing marginal utility relate to valuations with constant marginal utility, or is this in the spirit of Fourier Series, in that it turns out we can build one type of function out of the other, but it is not at all a priori clear that this should be the case?



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