Notes on supermodularity and increasing differences in expected utility

•I provide axiomatic foundations for supermodularity of Bernoulli utility functions.•Axioms for quasi-supermodularity and supermodularity are compared.•For parametric utilities, I provide axioms for increasing differences.•Axioms for increasing differences and single crossing are compared.

Many choice-theoretic and game-theoretic applications in Economics invoke some form of supermodularity or increasing differences for objective functions defined on lattices. These notes provide axiomatic foundations for these properties on expected-utility representations of preferences over lotteries.