Tight Bounds for The Price of Fairness

Abstract: A central decision maker (CDM), who seeks an efficient allocation of scarce resources among a finite number of players, often has to incorporate fairness criteria to avoid unfair outcomes. Indeed, the Price of Fairness (POF), a term coined in the seminal work by Bertsimas et al. (2011), refers to the efficiency loss due to the incorporation of fairness criteria into the allocation method. Quantifying the POF would help the CDM strike an appropriate balance between efficiency and fairness. In this paper we improve upon existing results in the literature, by providing tight bounds for the POF for the proportional fairness criterion for any $n$, when the maximum achievable utilities of the players are equal or are not equal. Further, while Bertsimas et al. (2011) have already derived a tight bound for the max-min fairness criterion for the case that all players have equal maximum achievable utilities, we also provide a tight bound in scenarios where these utilities are not equal. For both criteria, we characterize the conditions where the POF reaches its peak and provide the supremum bounds of our bounds over all maximum achievable utility vectors, which are shown to be asymptotically strictly smaller than the supremum of the Bertsimas et al. (2011) bounds. Finally, we investigate the sensitivity of our bounds and the bounds in Bertsimas et al. (2011) for the POF to the variability of the maximum achievable utilities.