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Tight Bounds for The Price of Fairness

Yifeng Cao, Yichuan Ding, Daniel Granot

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.

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 , 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.
Paper Structure (34 sections, 30 theorems, 172 equations, 7 figures, 1 table)

This paper contains 34 sections, 30 theorems, 172 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Fairness
  3. Related Literature
  4. Fairness Notions
  5. Proportional Fairness
  6. Max-min Fairness
  7. The Price of Fairness
  8. Utilitarian Solution
  9. Fairness Solution
  10. The Price of Fairness
  11. Upper Bounds for The Price of Proportional Fairness
  12. Equal Maximum Achievable Utilities
  13. Unequal Maximum Achievable Utilities
  14. Proportional Fairness
  15. Utilitarian Solution
  16. ...and 19 more sections

Key Result

Proposition 1

Suppose $U \subseteq \{\boldsymbol{u}:=(u_1,u_2,\dots,u_n)\, |\, 0 \leq u_i \leq 1, i \in N \}$ is a convex and compact utility set, and $\max\{u_i\, |\,\boldsymbol{u} \in U\}=1$ for $i\in N$. Then there exist $c_i \in [1/n,1],\ i\in N$, such that $U \subseteq U'=\{\boldsymbol{u}\, |\,\sum_{i=1}^n c

Figures (7)

  • Figure 1: Upper bound of the Proportional POF as a function of the number of players.
  • Figure 2: The relative improvement of our upper bound of the proportional POF over the BFT bound.
  • Figure 3: The two upper bounds of the price of proportional fairness as a function of the variance for $n=9$ and for a vector $\boldsymbol{L}$ for which $L_1 \geq L_2=\dots=L_n$.
  • Figure 4: The two bounds for the price of PF when $n=9$ and $L_1=L_2=\dots=L_{n-1}=1, L_n \in (0,1]$.
  • Figure 5: Comparison of two bounds for the price of PF for varying $\sigma_t$
  • ...and 2 more figures

Theorems & Definitions (68)

  • Definition 1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Claim 1
  • Claim 2
  • Proposition 5
  • Claim 3
  • Claim 4
  • ...and 58 more