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.
Highlights
In many practical situations, decision making is a matter of urgent and important choices being based on vague, fuzzy and mostly empirical information
Our construction of a total ordering on loss distributions will crucially hinge on an embedding of random variables into ÃIR, where a natural ordering and full fledged arithmetic are already available without any further efforts
Our proposed preference relation is designed for IT risk management
Summary
Decision making is a matter of urgent and important choices being based on vague, fuzzy and mostly empirical information. Our ordering will be total, so that the preference between two actions with random consequences R1, R2 is always well-defined and a decision can be made As it has been shown in [8, 9], there exist several applications where such a framework of decision making on abstract spaces of random variables is needed. Risk management is concerned with extreme events, since small distortions may be covered by the natural resilience of the analyzed system (e.g., by an organization’s infrastructure or the enterprise itself, etc.) For this reason, decisions normally depend on the distribution’s tails. The main contribution of this work is twofold: while any stochastic order could be used for decision making on actions with random variables describing their outcome, not all of them are suitable in a risk management context. This work is a condensed version of [14, 15] (provided as supporting information S1 File), whereas it extends this preliminary research by practical examples and concrete algorithms to efficiently choose best actions despite random consequences and with a sound practical meaning
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