Valfunktion

Valfunktion is a term used in several Scandinavian languages to denote a function that assigns a numerical value to elements of a domain, capturing utility, payoff, or value judgments. In its most general form, a valfunktion is a mapping V from a set X to the real numbers, sometimes representing states, outcomes, or decisions in a problem.

In economics and decision theory, the value function expresses preferences or utilities. It often serves as the objective to be maximized or minimized. In dynamic programming and optimal decision problems, the value function satisfies equations such as V(s) = max_a [ U(a) + beta sum_{s'} P(s'|s,a) V(s') ], linking immediate payoff to future value. Value functions may be monotone and, depending on the model, concave to reflect risk attitudes.

In reinforcement learning, the state-value function V^π(s) or the action-value function Q^π(s,a) quantify the expected cumulative reward when following a policy π. V^π(s) = E[ sum_{t=0}^∞ gamma^t R_{t+1} | S_0 = s ], while Q^π(s,a) extends this to the choice of actions. These functions guide policy evaluation and improvement.

Beyond economics and ML, value functions appear in optimal control (satisfying the Hamilton–Jacobi–Bellman equation) and in certain financial dynamic-programming formulations for pricing and portfolio optimization. The term is commonly translated as value function, with regional variants such as värdefunktion or valfunktion in Nordic literature.

See also: value function in reinforcement learning, utility function, Bellman equation, Hamilton–Jacobi–Bellman equation, Q-function.