On Strong-Feasibilities of Equivalence-Completions

The notion of completion has been proposed by Francez et al. to transform a non-equivalence-robust fairness notion to an equivalence-robust one while maintaining several properties of the source. However, a completion may not preserve strong-feasibility|a necessary and suucient condition for a completion to be imple-mentable. In this paper, we study the system requirement for a completion to be strongly-feasible, and determine the strongest implementable completion for every given fairness notion. Moreover, for most systems we obtain a fairness notion, which we refer to as SG + , such that SG + is the strongest fairness notion that is both im-plementable and equivalence-robust. Finally, we show that, if equivalence-robustness is dropped, then in general it is impossible to deene a fairness notion that is im-plementable and stronger than all other implementable fairness notions. This implies plenty of leeway in the design of fairness notions suitable for various applications .