On Concave Functions Over Lotteries

This note discusses functions over lotteries that are concave and continuous, but not necessarily superdifferentiable. Machina [1984] and Maccheroni [2002]’s analyses of utility functions over lotteries and the statement of Theorem 2 in Frankel and Kamenica [2019]’s proposed measure of information and uncertainty all claim or imply that if a function over lotteries is concave and continuous, then it can be written as the minimum of affine functions. However, Section 3 gives an example of a concave and continuous function that cannot be written as the minimum of affine functions, because there is no tangent hyperplanes that dominates the functions at the boundary. Section 4 shows that concavity and upper semi-continuity are equivalent to a representation as the infimum of affine functions, and 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 “mimimum” representation is equivalent of the existence of local utilities (i.e., supporting affine functions) at every lottery, a property that is equivalent to superdifferentiability