In social choice and operations research, the Nash welfare rule (also called the max-product rule, the Nash-optimal rule or maximum Nash welfare, often abbreviated MNW) is a rule saying that, among all possible alternatives, society should pick the alternative which maximizes the product of the utilities of all individuals in society. The product of utilities is called the Nash social welfare, after John Forbes Nash Jr., whose solution to the cooperative bargaining problem selects the utility profile maximizing the product of the agents' gains. The corresponding social welfare function was axiomatized by Kaneko and Nakamura. The Nash welfare rule is often described as a compromise between the utilitarian rule, which maximizes the sum of utilities and emphasizes aggregate efficiency, and the egalitarian rule, which maximizes the minimum utility and emphasizes the worst-off individual. Because the logarithm of the Nash welfare equals the sum of the logarithms of the utilities, maximizing the Nash welfare rewards increases in the utility of an agent in inverse proportion to that agent's current utility level: raising a poor agent's utility from 1 to 2 doubles the product, whereas raising a rich agent's utility from 10 to 11 increases it by only ten percent. The rule is closely related to the proportional-fair rule used in the analysis of communication networks.
Definition Let X {\displaystyle X} be a set of possible "states of the world" or "alternatives". Society wishes to choose a single state from X {\displaystyle X} . For example, in a single-winner election, X {\displaystyle X} may represent the set of candidates; in a resource allocation setting, X {\displaystyle X} may represent all possible allocations of the resource. Let I {\displaystyle I} be a finite set, representing a collection of individuals. For each i ∈ I {\displaystyle i\in I} , let u i : X → R ≥ 0 {\displaystyle u_{i}:X\to \mathbb {R} _{\geq 0}} be a utility function, describing the amount of happiness that individual i derives from each possible state. The Nash welfare rule selects a state x ∈ X {\displaystyle x\in X} which maximizes the product of utilities:
x Nash ∈ arg max x ∈ X ∏ i ∈ I u i ( x ) . {\displaystyle x^{\text{Nash}}\in \arg \max _{x\in X}\prod _{i\in I}u_{i}(x).}
Equivalently, since the maximizer of a product of positive numbers is also the maximizer of the sum of their logarithms, the rule selects a state maximizing ∑ i ∈ I log u i ( x ) {\displaystyle \sum _{i\in I}\log u_{i}(x)} , or the geometric mean ( ∏ i ∈ I u i ( x ) ) 1 / | I | {\displaystyle \left(\prod _{i\in I}u_{i}(x)\right)^{1/|I|}} . When some agents may inevitably receive zero utility, the rule is commonly refined to first maximize the number of agents with positive utility and then maximize the product of utilities among those agents.
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