In operations research and social choice, the proportional-fair (PF) rule is a rule saying that, among all possible alternatives, one should pick an alternative that cannot be improved, where "improvement" is measured by the sum of relative improvements possible for each individual agent.
It aims to provide a compromise between the utilitarian rule - which emphasizes overall system efficiency, and the egalitarian rule - which emphasizes individual fairness.
The rule was first presented in the context of rate control in communication networks.
[1] However, it is a general social choice rule and can also be used, for example, in resource allocation.
Society wishes to choose a single state from
be a finite set, representing a collection of individuals.
be a utility function, describing the amount of happiness an individual i derives from each possible state.
A social choice rule is a mechanism which uses the data
, represents the relative gain of agent i when switching from x to y.
:That rule ignores the current utility of the individuals.
:This rule ignores the total efficiency of the system.
The proportional-fair rule aims to balance between these two extremes.
In particular, if the utility of some individual in x is 0, and there is another state y in which his utility is larger than 0, then the PF rule would prefer state y, as the relative improvement of individual y is infinite (it is divided by 0).
When the utility sets are convex, a proportional-fair solution always exists.
Moreover, it maximizes the product of utilities (also known as the Nash welfare).
[3] When the utility sets are not convex, a proportional-fair solution is not guaranteed to exist.