On rho in a Decision-Theoretic Apparatus of Dempster-Shafer Theory

TL;DR

This paper clarifies the role of the parameter rho in decision-making within Dempster-Shafer theory, showing that assuming a uniform distribution for rho simplifies the process of choosing the most preferable option without unwarranted assumptions.

Contribution

It demonstrates that assuming a uniform probability distribution for rho suffices for decision-making, eliminating the need for unwarranted assumptions about its value.

Findings

Uniform distribution assumption for rho is sufficient for decision preference.

No additional assumptions about rho are necessary for optimal decision-making.

The approach is applicable when rational choices of future decision makers are considered.

Abstract

Thomas M. Strat has developed a decision-theoretic apparatus for Dempster-Shafer theory (Decision analysis using belief functions, Intern. J. Approx. Reason. 4(5/6), 391-417, 1990). In this apparatus, expected utility intervals are constructed for different choices. The choice with the highest expected utility is preferable to others. However, to find the preferred choice when the expected utility interval of one choice is included in that of another, it is necessary to interpolate a discerning point in the intervals. This is done by the parameter rho, defined as the probability that the ambiguity about the utility of every nonsingleton focal element will turn out as favorable as possible. If there are several different decision makers, we might sometimes be more interested in having the highest expected utility among the decision makers rather than only trying to maximize our own expected utility regardless of choices made by other decision makers. The preference of each choice is then determined by the probability of yielding the highest expected utility. This probability is equal to the maximal interval length of rho under which an alternative is preferred. We must here take into account not only the choices already made by other decision makers but also the rational choices we can assume to be made by later decision makers. In Strats apparatus, an assumption, unwarranted by the evidence at hand, has to be made about the value of rho. We demonstrate that no such assumption is necessary. It is sufficient to assume a uniform probability distribution for rho to be able to discern the most preferable choice. We discuss when this approach is justifiable.