Invariant utility functions and certain equivalent transformations

This paper defines invariant utility functions to continuous monotonic transformations. We also define transformation invariance as the condition in which the certain equivalent of a lottery follows a continuous monotonic transformation that is applied to its outcomes. We show that invariant utility functions uniquely satisfy transformation invariance, and we illustrate how knowledge of an invariance criterion determines the functional form of the utility function. This formulation extends the widely used notions of invariance to shift and scale transformations on the outcomes of a lottery to more general monotonic transformations. Moreover, we interpret any continuous and strictly monotonic utility function as an invariant utility function to a composite monotonic transformation. Furthermore, we show how this composite transformation uniquely characterizes the utility function up to a linear transformation. We derive the invariance formulations that lead to the assignment of hyperbolic absolute risk-averse (HAM) utility functions, linear plus exponential utility functions, and a two-parameter power-logarithmic utility function that generalizes the logarithmic utility function. We work through several examples to illustrate the approach.