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Analyzing Incentives and Fairness in Ordered Weighted Average for Facility Location Games

Kento Yoshida, Kei Kimura, Taiki Todo, Makoto Yokoo

Abstract

Facility location games provide an abstract model of mechanism design. In such games, a mechanism takes a profile of $n$ single-peaked preferences over an interval as an input and determines the location of a facility on the interval. In this paper, we restrict our attention to distance-based single-peaked preferences and focus on a well-known class of parameterized mechanisms called ordered weighted average methods, which is proposed by Yager in 1988 and contains several practical implementations such as the standard average and the Olympic average. We comprehensively analyze their performance in terms of both incentives and fairness. More specifically, we provide necessary and sufficient conditions on their parameters to achieve strategy-proofness, non-obvious manipulability, individual fair share, and proportional fairness, respectively.

Analyzing Incentives and Fairness in Ordered Weighted Average for Facility Location Games

Abstract

Facility location games provide an abstract model of mechanism design. In such games, a mechanism takes a profile of single-peaked preferences over an interval as an input and determines the location of a facility on the interval. In this paper, we restrict our attention to distance-based single-peaked preferences and focus on a well-known class of parameterized mechanisms called ordered weighted average methods, which is proposed by Yager in 1988 and contains several practical implementations such as the standard average and the Olympic average. We comprehensively analyze their performance in terms of both incentives and fairness. More specifically, we provide necessary and sufficient conditions on their parameters to achieve strategy-proofness, non-obvious manipulability, individual fair share, and proportional fairness, respectively.

Paper Structure

This paper contains 9 sections, 10 theorems, 21 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Model
  4. Incentive Properties
  5. Fairness Properties
  6. Warm-Up: Strategy-Proof OWAs
  7. Not-Obviously-Manipulable OWAs
  8. Proportionality-Based Fairness
  9. Discussions and Concluding Remarks

Key Result

Theorem 1

A mechanism satisfies PF and SP if and only if it is a uniform phantom mechanism, i.e., the AGMVS whose $n-1$ parameters are set as $\beta_{l} = l/n$ for each $l \in \{1, \ldots, n-1\}$.

Figures (6)

  • Figure 1: An example of facility location games. Agents' reported locations are represented as circles. A mechanism determines where to locate a facility, represented as a square.
  • Figure 2: A beneficial manipulation in OWA (except for order statistics), presented in the proof of Proposition \ref{['prp:SP']}.
  • Figure 3: The profiles used in the proof of the only if part of Theorem \ref{['thm:NOM']}. The top indicates the original input, the middle indicates the worst-case for truth telling $x_{i}$, and the bottom indicates the worst-case for manipulation $x'_{i} = 0$. E.g., parameters $w_{1} = 0.4$ and $x_{i} = 0.2$ works for this example.
  • Figure 4: Example \ref{['ex:OA-SA']} illustrates the key difference between the Olympic average (OA) and the standard average (SA). The second top figure shows a true input. The top figure shows that the Olympic average satisfies NOM, while the bottom two figures show that the standard average violates NOM-W.
  • Figure 5: Agent 1 is at 0 and other four agents 2, ..., 5 are at 1. The top figure shows that the median mechanism violates IFS, the middle figure shows that the center mechanism violates PF, and the bottom figure shows that the standard average mechanism satisfies PF.
  • ...and 1 more figures

Theorems & Definitions (28)

  • Definition 1: Pareto Efficiency
  • Definition 2: Anonymity
  • Definition 3: Generalized Median Voter Schemes moulin:PC:1980
  • Definition 4: Anonymous GMVS moulin:PC:1980
  • Definition 5: Ordered Weighted Average Yager:SMC:1988
  • Definition 6: Strategy-Proofness
  • Definition 7: Non-Obvious Manipulability (NOM)
  • Definition 8: Individual Fair Share
  • Definition 9: Unanimous Fair Share
  • Definition 10: Proportional Fairness
  • ...and 18 more