Lecture
One of the main components of a decision-making problem is identifying the DM's system of preferences. These preferences can be expressed explicitly or implicitly. The formalization of selecting the most preferable options from the available alternatives is the construction of a choice function. There are several approaches to this.
The most common way of identifying preferences is to analyze the possible alternatives using binary relations. The use of binary relations makes it possible to formalize such concepts as «worse», «better», «no worse», «identical», and others. However, this toolkit is not without its drawbacks, since it is highly labor-intensive, and in some problems a pairwise comparison of the available solution alternatives does not lead to a global selection of the best among all the available alternatives.
In cases where it is possible to numerically assess the utility of each alternative, a binary preference relation can also be constructed on the basis of this utility.
The choice function of a DMP is specified either directly by the DM, or by means of some decision rule, which can be constructed using binary preference relations or by optimizing some objective function. The complexity and uncertainty of a DMP lies in the fact that the resulting decisions will depend on which decision rule is used. For one and the same problem, different rules can lead to different decisions. Thus, the resulting decisions are always subjective.
An even more difficult task is decision-making under conditions of uncertainty, when the resulting outcome is also influenced by external conditions that the DM cannot control. However, even in this case there are approaches to identifying preferences on the set of alternatives. If a probabilistic structure can be defined on the set that determines the influence of external conditions (the environment), then preference relations and choice functions can be constructed using stochastic dominance, which is an analogue of Pareto dominance, discussed in detail in Chapter 9, or by means of utility and risk functions.
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