Decisions with Uncertain Consequences -- A Total Ordering on Loss-Distributions

TL;DR

This paper introduces a method to totally order certain loss-distributions, enabling decision-making under uncertainty by comparing effects modeled as random variables, with demonstrated practical applicability through simulations.

Contribution

It develops a theoretical framework for restricting the space of distributions to establish a total order, facilitating decisions under uncertain consequences.

Findings

Practical total ordering of loss-distributions is achievable.

Simulation results demonstrate the method's applicability.

The approach aids decision-making with uncertain outcomes.

Abstract

Decisions are often based on imprecise, uncertain or vague information. Likewise, the consequences of an action are often equally unpredictable, thus putting the decision maker into a twofold jeopardy. Assuming that the effects of an action can be modeled by a random variable, then the decision problem boils down to comparing different effects (random variables) by comparing their distribution functions. Although the full space of probability distributions cannot be ordered, a properly restricted subset of distributions can be totally ordered in a practically meaningful way. We call these loss-distributions, since they provide a substitute for the concept of loss-functions in decision theory. This article introduces the theory behind the necessary restrictions and the hereby constructible total ordering on random loss variables, which enables decisions under uncertainty of consequences. Using data obtained from simulations, we demonstrate the practical applicability of our approach.