Utilityn

Utilityn is a generalized multi-attribute utility function used in economics, decision theory, and operations research. It maps a vector of n attributes or goods to a real-valued utility score, aggregating diverse criteria into a single measure for comparison and choice.

Formally, let x be an element of R_+^n representing quantities or attributes. Utilityn is a function U: R_+^n -> R such that U(x) captures an agent’s overall satisfaction. In typical models, U is increasing in each argument, reflecting that more of a desired attribute raises utility, and U is often concave, expressing diminishing marginal utility. The exact shape and properties of Utilityn depend on the context and the decision maker’s preferences.

Common forms of Utilityn include additive separable and multiplicative or CES (constant elasticity of substitution) constructions. Additive forms take U(x) = sum_{i=1}^n w_i u_i(x_i), with positive weights w_i and subutility functions u_i. Multiplicative or CES forms include U(x) = [sum w_i u_i(x_i)^{rho}]^{1/rho}, with special cases leading to Cobb-Douglas when specific parameters are chosen. These forms illustrate different ways to combine attributes into a single scalar.

Applications of Utilityn span modeling consumer preferences across multiple goods, welfare analysis, and multi-criteria decision making in operations research, policy evaluation, and risk assessment. Utilityn provides a framework for comparing options that differ on several attributes while retaining a coherent ordering of preferences.

Remarks: Utilityn is not a single standardized object; definitions and forms vary by domain. The concept emphasizes aggregating multiple criteria into a tractable scalar measure while preserving essential preference structure.