Constructing risk functions in a decision problem under probabilistic uncertainty

Lecture

The concept of risk plays an important role in comparing random variables.

Definition

The risk R(?) of the random variable ? is defined as its deviation from the expected value Constructing risk functions in a decision problem under probabilistic uncertainty

Note!

The risk of a random variable is itself a random variable.

The following values can be chosen as the expected value of a random variable:

1) some predetermined constant: M = const;
2) the mathematical expectation ?: M = E(?);
3) the median ?: M = med(^) .

Definition

A measure or risk function ρ: Ψ* —? 9? is defined as the mathematical expectation of the quantity g(R(i))): Constructing risk functions in a decision problem under probabilistic uncertainty

where g(x) — is a certain function called the loss function.

A number of requirements are imposed on the loss function g(x):

• the loss function at zero equals zero: ?(0) = 0;
• it is continuous;
• it is convex (concave up).

Note that satisfying all the requirements is not mandatory, although it is desirable. The following can be given as examples of commonly used loss functions:

1) uniform: ?(x) = |x|;
2) power: g(x) = x(,y 1 < p < ∞;

3) «with a flat bottom» Constructing risk functions in a decision problem under probabilistic uncertainty

i.e., in this case it is assumed that in some neighborhood [-b, b] of the expected value the risk equals zero.

Other types of loss functions also exist. The choice of a specific form of the function is dictated by the problem being solved.

An important special case of a risk measure is the variance, which computes the square of the mean deviation from the mean value of the random variable:

Constructing risk functions in a decision problem under probabilistic uncertainty

Such a measure is obtained for the expected value M = ?(%) and the power loss function ?(x) = x2.

Note!

Just as with the utility function, the risk measure is a function of x: Constructing risk functions in a decision problem under probabilistic uncertainty

Example of determining the value of the risk function. Consider a random variable p of discrete type such that P{p = -1} = 0.3, P{r = 0} = 0.3, P{x] = 1} = 0.4. Let us take as the expected value the mean value of the random variable M = E{r|} = -0.3 + 0.4 = 0.1, and as the loss function the quadratic function; then the risk measure will represent the variance of the random variable p and will equal

Constructing risk functions in a decision problem under probabilistic uncertainty

If we take M = 0 as the expected value, and the uniform function as the loss function, then the risk measure will equal

Constructing risk functions in a decision problem under probabilistic uncertainty

Just as with the utility function, the risk measure generates preference relations on the set of distribution functions T as follows:

Constructing risk functions in a decision problem under probabilistic uncertainty

i.e., the random variable ^ is no less preferable than c; if and only if ρ(^) < ρ(<;).

Example: suppose the random variable ?, takes the values -1 and 1 with probabilities 0.5, and the random variable <; — the values -5 and 5 with probabilities 0.5. Then if we choose the variance as the risk measure, we obtain ?, r|, since Constructing risk functions in a decision problem under probabilistic uncertainty