On concave functions over lotteries

Abstract

This note discusses functions over lotteries that are concave and continuous, but are not necessarily superdifferentiable. Earlier work claims that concave continuous utility for lotteries that satisfy best-outcome independence can be written as the minimum of affine functions. We give a counter-example that cannot be written as the minimum of affine functions, because there is no tangent hyperplane that dominates the functions at the boundary. We then review the fact that concavity and upper semi-continuity are equivalent to a representation as the infimum of affine functions, and show that these assumptions imply continuity for functions on finite-dimensional lotteries. Therefore, in finite-dimensional simplices, concavity and continuity are equivalent to the “infimum” representation. The “minimum” representation is equivalent to the existence of local utilities (supporting affine functions) at every lottery, a property that is equivalent to superdifferentiability.