Decision-making under risk

Decision making under risk is the process of choosing the best alternative in a situation where the outcome depends not only on the actions of the decision-maker (DM) but also on random factors whose probability is known or can be reasonably estimated. Such situations are characterized by incomplete certainty: the possible outcomes are known, but the result of a specific choice is not fully predetermined, but only with a certain probability.

Characteristics of a Risk Situation

The conditions of a risk situation apply to problems in which:

• the possible states of nature (scenarios) are known and can be enumerated;
• a numerical probability of its occurrence can be assigned to each state;
• the outcome of a choice is a random variable that depends on the combined effect of the decision and the resulting state of nature.

Unlike complete certainty, where the outcome is unambiguous, and uncertainty, where probabilities are unknown, risk situations allow for the use of numerical assessments based on probabilistic assumptions.

Key Criteria for Choice under Risk

• Expected Value Criterion (Mathematical Expectation) — selects the alternative with the highest weighted average value of the outcome.
• Savage's Criterion (Minimax Regret) — considers not the absolute payoff, but the maximum possible "loss" or regret from not choosing the optimal decision for each state of nature.
• Hurwicz's Criterion — combines pessimism and optimism.
• Entropy Criterion — takes into account the degree of uncertainty in the probability distribution and aims to reduce the information risk of the choice.
• Germeier's Criterion — focuses on minimizing losses, especially in problems where the consequences of an error are critical.

The choice of criterion depends on the behavioral characteristics of the DM: risk appetite, level of responsibility, and the goal of the analysis (e.g., maximizing income, minimizing losses, or ensuring the stability of the result).

Subjective Probability Assessment

Subjective probability assessments are formed by the DM based on experience, judgment, and expert information. This is particularly relevant in situations where statistical data is unavailable or the probability of events cannot be objectively measured. Subjective probabilities are used in a similar way to objective ones but require careful verification and consensus within the decision-making process.

Tools and Representation of Decisions

• Payoff Matrices — present the estimated outcomes for all combinations of "alternative × state of nature";
• Decision Trees — illustrate sequential choices and the probabilities at each stage;

• Utility Functions — allow for consideration not only of the outcome but also the DM's subjective attitude towards it.