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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? The former seems like a logarithmic sort of object, and the latter a linear sort of object, so they don't really type check in my mind.

Is there an economic application that makes obvious why valuations with diminishing marginal utility relate to valuations with constant marginal utility?

Or is this a phenomenon 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 possible?

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  • $\begingroup$ Did you read the example and proof that follow lemma 3.3? $\endgroup$
    – Golden_Ratio
    Commented Mar 5, 2023 at 3:09
  • $\begingroup$ @Golden_Ratio Yeah and I watched that part of the lecture the notes are from twice, but I think a geometric or applied example to anchor the algebra to would be helpful. $\endgroup$
    – user10478
    Commented Mar 5, 2023 at 7:18

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