Decision situations in which the chance of occurrence of each state of nature is known or can be estimated are defined as decisions made under risk. Such decision-making problems are also known as probabilistic or stochastic decision problems.

The most important advantage of the decision maker to have known of the possibility of a state of nature is that the decision maker can calculate how much risk to the gain/cost it expects to achieve according to the decision alternative it adopts.

 These approaches are Expected Value (EV) (or Expected Monetary Value (EMV)), Expected Opportunity Loss (EOL), and the Maximum Probability.

The Expected Value (EV) is an anticipated value for a given investment at some point in the future.

In decision making at risk, the expected value criterion is commonly used when comparing alternatives, based on maximizing expected profit or minimizing expected loss. The expected value criterion attempts to find the expected profit maximized or the expected cost minimized. The profit or cost that will arise from each alternative is handled by certain possibilities. 

When the decision’s consequences involve only money, we can calculate the Expected Monetary Value (EMV).

Expected opportunity loss, demonstrate the average additional amount the investor would have achieved by making the right decision instead of a wrong one. The basic idea of this criterion is that people try to minimize their regret or opportunity loss. In another saying, an opportunity loss (or regret) can be explained as the loss incurred due to failure to select the best alternative available.

In this criterion, the decision maker confronts the various possible states of nature in a decision under risk and; he or she chooses the alternative that is best for the most likely state of nature, rather than calculating in all states of nature. With another saying, it states that the decision maker should ignore all possible events except the one most likely to occur, and should select the best possible result (maximum gain or minimum loss) in the given circumstances.

In decision theory, the expected value of perfect information is the maximum amount of price you would be willing to pay for additional information about a decision problem. 

Since there is always a possibility that the decision turns out to be wrong, there is always some degree of uncertainty about the decision. The expected value of perfect information analysis tries to measure the expected cost of that uncertainty because the perfect information can eliminate the possibility of making the wrong decision. In other words, the expected value of perfect information is used to place an upper limit on what you should pay for information that will aid in making a better decision.

EVPI = |EVwP| - EVwoPI|

where

EVwPI = Expected Value with Perfect Information, is the expected payoff if perfect information about the states of nature is known

EVwoPI = Expected Value without Perfect Information, is the expected value without perfect information which is the best expected payoff without additional information about the states of nature

The decision tree is a graphical technique that represents all the elements in the decision problem with various geometric symbols. It provides all the elements and details of the decision problem, as well as graphically, and performs the expected value calculations on the tree and provides the solution of the problem at the same time.

The decision trees can help a decision maker to develop a clear view of the structure of a problem and make it easier to determine the possible scenarios which can result if a particular course of action is chosen.

A decision tree is composed of some components as; branches, decision nodes, chance nodes, and payoffs.

Branch: The line connecting the nodes on a decision tree is called a branch. A branch is a single strategy that connects either two nodes or a node and an outcome.

Decision node: The decision node, represented in a square shape on the decision tree is a point from which two or more branches emerge. Each branch from a decision node represents a possible alternative to be chosen by the decision-maker.

Chance node: A chance node, represented in a circle shape on the decision tree, indicates that one of a finite number of states of nature is expected to occur at this point in the process.

Payoff: In decision analysis payoff refers to the consequence resulting from a specific combination of a decision alternative and a state of nature.

Decision trees are visually helpful in the choice of the best decision alternative to the decision maker. It does not need to over calculations and decision trees are suitable for both continuous and discrete variables. On the other side, when the decision alternatives and the number of states of nature increase, to draw a decision tree can be a complex structure.

Bayes’ theorem is used as a normative tool, which tells how it should be revised our probability assessments when new information becomes available. In other words, Bayes’ theorem is basically a process of revising the known possibilities of an event under the light of new knowledge. Bayes’ theorem provides a way for combining prior probabilities with probabilities obtained by other sources; revised or posterior probabilities.

firstly the prior probability distribution of the parameters to be estimated according to the subjective and objective information obtained is determined. Then, the distribution of the parameters after the information is determined according to the additional information obtained from the sample and the type of movement which gives a maximum profit or minimum cost value according to this distribution is determined.

With the formula used in Bayes’s theorem, while the result of a given event is certain, the possibility of the causes of this result is investigated. From the other perspective, in Bayes’ formula, the cause and result are displaced.

In decision making problems, decision branches are used to represent alternatives (strategies) and chance branches are used to show the states of nature (events). The chance branch is labeled with a probability which represents the decision maker’s estimate of the probability that a particular branch will be followed.