The Foster-Hart Measure of Riskiness for General Gambles

TL;DR

This paper extends the Foster-Hart riskiness measure from discrete to continuous distributions, showing it often equals the worst-case risk and remains effective in dynamic, repeated gamble scenarios.

Contribution

It provides a consistent extension of the Foster-Hart risk measure to continuous variables and dynamic settings, avoiding bankruptcy in repeated gambles.

Findings

The original Foster-Hart measure has no solution for many continuous distributions.

The extended measure often equals the worst-case risk measure.

The dynamic extension prevents bankruptcy in repeated gambles.

Abstract

Foster and Hart proposed an operational measure of riskiness for discrete random variables. We show that their defining equation has no solution for many common continuous distributions including many uniform distributions, e.g. We show how to extend consistently the definition of riskiness to continuous random variables. For many continuous random variables, the risk measure is equal to the worst--case risk measure, i.e. the maximal possible loss incurred by that gamble. We also extend the Foster--Hart risk measure to dynamic environments for general distributions and probability spaces, and we show that the extended measure avoids bankruptcy in infinitely repeated gambles.