Decision-making is the cognitive process of selecting a course of action from among multiple alternatives based on preferences and information.Decision making is a cognitive process of choosing among alternatives, studied in psychology, neuroscience, economics and mathematics. Classical theories (expected utility) assume a rational agent maximizing benefit. However, experiments have revealed heuristics (simplified rules) and biases: people are subject to the framing effect, avoid losses more than they seek gains, and are time-inconsistent. Modern models include quantum-like formalisms to describe context dependence and violations of the law of total probability.
Decision analysis began with Pascal and Fermat in the 17th century in correspondence about gambling problems. In the 20th century, von Neumann and Morgenstern formalized expected utility theory. Later, Herbert Simon introduced the concept of bounded rationality. Daniel Kahneman and Amos Tversky experimentally demonstrated systematic biases, for which Kahneman won the Nobel Prize. David Deutsch contributed to quantum computing, which provided tools for quantum-like decision models.
Classical approach: a utility function U(x) assigns a number to each outcome, and the action maximizing expected utility E[U(x)] = ∑ P_i U(x_i) is chosen. Prospect theory replaces this with weighted value: V = ∑ π(p_i) v(x_i), where π(p) is the decision weight, v(x) is the value function, concave for gains and convex for losses. In the quantum model, the state |ψ⟩ in Hilbert space evolves unitarily until a 'question' which is described by a projector, and the probability of the answer is given by the squared modulus of the amplitude. The principle of superposition and Born's rule are applied.
It is used everywhere: from everyday purchases to financial management and artificial intelligence strategies. By learning to model this process, we create smart assistants capable of advising or making decisions.
If people always acted rationally, lotteries would not exist because the expected value of a ticket is always less than its price.
| a_0, a_1, ..., a_{n- | variable | First row entries of the circulant matrix |
| n | variable | Matrix size (positive integer) |
| det | operator | Determinant of a square matrix — determinant |
| Circ | operator | Constructs a circulant matrix from its first row — circulant matrix |



















