Lecture
Decision rule - is a method or model that makes it possible to choose the decision that is most preferable in some sense.
A decision rule is called subjective if, based on the same initial information, different researchers arrive at different conclusions.
The following variants are possible:
1. Model – objective, decision rule – objective.
2. Model – objective, decision rule – subjective.
3. Model – subjective, decision rule – objective.
4. Model – subjective, decision rule – subjective.
The decision-making problem (DMP) under conditions of uncertainty can be regarded as a game against nature. Therefore the choice of a decision rule can also be regarded as the strategy chosen by the Decision Maker (DM) in this game. At the same time, it is impossible to choose a strategy that would lead to minimal loss or maximal payoff regardless of what the state of nature turns out to be. One way of solving this problem is to choose a rule that makes it possible to determine how good a strategy is «on average». Earlier, such concepts as the risk measure and utility were defined, and it is these that provide such estimates. On their basis we shall now define the following decision rules, which allow the DM to find the best alternatives.
1. Extremal optimization by risk measure. According to this rule, the best alternatives are considered to be those x* € X, for which, given certain predetermined expected value M and loss function ?, the risk measure is minimal
It is also possible to set some threshold risk value c* e and consider as best those alternatives for which the condition is satisfied
2. Extremal optimization by utility. In this case, the best alternatives are considered to be those for which some predetermined utility function u attains its maximum
or exceeds some predetermined threshold value c* e ChL:
As an example of a decision-making problem under conditions of probabilistic uncertainty, let us consider the problem of constructing an optimal investment portfolio.
Suppose there are n securities, each characterized by its own return. The return of the 1st security when a unit of capital is invested in it is described by the random variable ξi, i = 1,2, ..., n. The returns of individual securities are, in general, not independent random variables. Thus, there is a random vector 
whose characteristics are determined by the state of the financial market. Each specific realization of it, i.e., the resulting set of returns, is precisely the realization of the state of the environment.
A unit of capital is invested in these securities such that y, — is its share invested in the i-th type of security. An investment portfolio y is called a set of numbers y1,y2,.... yn satisfying the condition y1+ y2 +... + yn = 1. Note that some yi may be zero, which means that capital is not invested in the given security, or negative, which means selling the security short (without cover).
The decision-making problem in this case consists in constructing an investment portfolio, i.e., determining how the unit of capital should be distributed among the securities. The alternatives here will be specific values of the portfolio, i.e., specific sets of capital invested in the available securities.
The result of the decision on forming the investment portfolio will be its return — a random variable
To solve the problem of constructing an optimal investment portfolio, it is necessary to construct a decision rule that determines for which set of capital shares y1 y2,yn, such that y1, + y2 + ... yn = 1, the result (portfolio return) will be the best. Let us denote
Let us consider several approaches to solving this problem. First, let us consider extremal optimization by utility.
By definition, the utility of the portfolio return is the number
where u — is some non-decreasing function. Let us set u(t) = t for all real numbers 
. Then the utility will be the mathematical expectation of the portfolio return
The most preferable will be that set of capital shares y * = = {y1*, y2*,..., yn*}, for which the average value of the portfolio return will be maximal
Note that by the properties of mathematical expectation
E(Y) = y1 E(ξ1) + y2 E{ξ 2 ) + ... + yn-E( ξ n).
Thus, to construct an optimal investment portfolio using this approach, it is not necessary to know the distribution laws of the returns of the securities included in the portfolio. It is sufficient to estimate only their average values. However, this approach has a number of drawbacks, since it evaluates only average values and does not take into account possible deviations of the returns from them, although such deviations can be quite significant.
To determine the utility of the portfolio return, other types of function u> can also be chosen, for example, u(t) = 1 - exp(-at) , where a > 0 — is some given parameter. Then the most preferable will be such a portfolio 
, that
A drawback of this approach is the complexity of its application. In particular, it is necessary to determine the exact distribution law of the vector 
of security returns.
Let us consider another way of constructing a decision rule for solving the problem of finding the optimal investment portfolio — extremal optimization by risk measure.
The most common measure of risk is the variance of the random variable
Then the most preferable portfolio will be the one 
for which the mean deviation of the investment portfolio's return from its mean value is minimal: 
To find the variance of the portfolio's return, it is necessary to know the variances of the returns of the securities included in it, as well as their pairwise covariances, i.e. 
for all 
. It is not necessary to know the distribution laws of the securities' returns themselves. This simplifies the solution of this problem.
In general, different approaches to the solution can yield different and even opposite decisions. The choice of a specific method of solution remains with the decision maker .
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