Facility Location Game with Envy Ratio

Abstract: We study the one-facility location game on a real line with a new objective called envy ratio. The envy ratio, which is adopted from fair division and represents the egalitarianism, is defined as the maximum over the ratios between any two agents' utilities. We are interested in strategyproof or group strategyproof mechanisms that can minimize the envy ratio objective. We consider the model in two settings that can capture natural scenarios: the facility location and all the agents' locations are restricted on a fixed interval; every agent's location can be any point on the real line but the facility location is restricted on a relative interval. In both settings, we obtain the optimal solution and the best deterministic strategyproof mechanism which is also group strategyproof. In the first setting, we provide a lower bound for randomized strategyproof mechanisms. In the second setting, we give a lower bound and two upper bounds for randomized strategyproof mechanisms.

• The paper introduces the envy ratio as a new fairness measure for evaluating facility location outcomes in one-dimensional spaces.
• It establishes tight bounds for both deterministic and randomized strategyproof mechanisms under fixed and relative interval settings.
• The results highlight the trade-off between fairness and incentive compatibility, providing actionable insights for public facility allocation.

Facility Location Games with Envy Ratio: Strategyproof Mechanisms and Fairness Bounds

Introduction

This paper introduces and investigates a new fairness criterion—envy ratio—for the classical one-facility location game on the real line, focusing on strategyproof and group strategyproof mechanisms without payments. The envy ratio, motivated by the fair division literature, measures egalitarianism by taking the maximum ratio of utilities across any two agents, shifting attention from utilitarian or max-min objectives to a more direct form of fairness. The problem is analyzed in two primary settings: (1) all agent and facility locations are constrained within a fixed interval, and (2) only the facility is restricted to an interval relative to agent locations.

Model, Objective, and Mechanism Design

The setting considers $n$ agents with private locations on $\mathbb{R}$, with cost defined as the absolute distance to a single facility. The mechanism seeks to locate this facility to optimize a fairness objective, while ensuring incentive compatibility (strategyproofness or group strategyproofness). The primary objective is to minimize the envy ratio:

$$ER(y, \mathbf{x}) = \max_{i\neq j} \frac{u(y, x_i)}{u(y, x_j)}$$

where $u(y, x_i)$ is agent $i$'s utility with the facility at $y$. The envy ratio generalizes the notion of minimizing envy from fair division to the spatial resource allocation domain.

Approximate mechanism design without monetary transfers is required, as traditional mechanisms relying on payments are often infeasible in scenarios such as public goods provision or political decision aggregation.

Main Results

Fixed Interval Setting

When both agent and facility locations are restricted to $[0, L]$, the envy ratio uses utility $u(y, x_i) = L - |x_i - y|$. The following results are established:

• Optimal Solution: The facility location at the midpoint of agents' span, $mid(\mathbf{x})$, uniquely minimizes the envy ratio over all deterministic (possibly non-strategyproof) choices.
• Strategyproofness Barrier: $mid(\mathbf{x})$ is not strategyproof; agents can manipulate their reports to shift the facility to their own locations.
• Deterministic Mechanisms:
  • Any deterministic strategyproof mechanism has a minimum approximation ratio of 2.
  • Locating the facility at the midpoint of the interval $\mathbb{R}$0 achieves this bound and is group strategyproof.
  • No improvement is possible within deterministic, strategyproof mechanisms.
• Randomized Mechanisms:
  • The best possible (lower bound) approximation ratio for any randomized strategyproof mechanism is proven to be at least $\mathbb{R}$1.
  • The randomized strategyproof mechanism achieving this lower bound remains an open question.

These results contrast with prior results for minimizing social or maximum cost—highlighting the stringency of the envy ratio objective.

Relative Interval Setting

Here, only the facility is constrained (by $\mathbb{R}$2) to be within $\mathbb{R}$3, accommodating applications that require the facility to stay relatively close to all agents. The utility is normalized as $\mathbb{R}$4. Main findings include:

• Optimal Solution: As in the fixed interval setting, $\mathbb{R}$5 minimizes the envy ratio but is not strategyproof.
• Deterministic Mechanisms:
  • Any deterministic strategyproof mechanism has an approximation ratio of at least $\mathbb{R}$6.
  • Always placing the facility at the leftmost or rightmost reported agent is group strategyproof and matches this lower bound on approximation ratio.
• Randomized Mechanisms:
  • For $\mathbb{R}$7, the lower bound for strategyproof randomized mechanisms is $\mathbb{R}$8.
  • A class of mechanisms generalizing Left-Right-Middle (LRM) approaches, where the facility is placed at $\mathbb{R}$9 with probability $$ER(y, \mathbf{x}) = \max_{i\neq j} \frac{u(y, x_i)}{u(y, x_j)}$$0 and at each boundary point with probability $$ER(y, \mathbf{x}) = \max_{i\neq j} \frac{u(y, x_i)}{u(y, x_j)}$$1, achieves approximation ratio $$ER(y, \mathbf{x}) = \max_{i\neq j} \frac{u(y, x_i)}{u(y, x_j)}$$2 for $$ER(y, \mathbf{x}) = \max_{i\neq j} \frac{u(y, x_i)}{u(y, x_j)}$$3. The mechanism with $$ER(y, \mathbf{x}) = \max_{i\neq j} \frac{u(y, x_i)}{u(y, x_j)}$$4 is group strategyproof and achieves $$ER(y, \mathbf{x}) = \max_{i\neq j} \frac{u(y, x_i)}{u(y, x_j)}$$5.
  • An alternative mechanism parameterized by $$ER(y, \mathbf{x}) = \max_{i\neq j} \frac{u(y, x_i)}{u(y, x_j)}$$6 (from the fair division literature) yields an improved approximation ratio $$ER(y, \mathbf{x}) = \max_{i\neq j} \frac{u(y, x_i)}{u(y, x_j)}$$7 when $$ER(y, \mathbf{x}) = \max_{i\neq j} \frac{u(y, x_i)}{u(y, x_j)}$$8.
• Tightness: For deterministic strategyproof mechanisms, the bounds are tight. For randomized mechanisms, there remains a gap between constructive upper bounds and proven lower bounds.

Theoretical and Practical Implications

These results delineate the limits of fairness achievable by strategyproof mechanisms in facility location games under the envy ratio objective. The absolute minimization of the envy ratio is generally incompatible with incentive compatibility, necessitating approximation.

• Theoretical significance:
  • Provides tight characterizations for deterministic mechanisms under this notion of fairness, adding to the mechanism design literature without payments.
  • Establishes new lower bounds for randomization, tightening our understanding of what is attainable.
  • The envy ratio, being sensitive to the relative advantage between any pair of agents, imposes stricter fairness constraints than previously studied objectives (social/worst cost minimization or envy-minimization).
  • Illuminates a trade-off: attaining group strategyproofness requires accepting a provable (and sometimes significant) relaxation in fairness measures.
• Practical relevance:
  • Offers actionable prescriptions for mechanism design in resource and public facility allocation without payments, particularly where egalitarianism is paramount.
  • The parameterizable interval setting (using $$ER(y, \mathbf{x}) = \max_{i\neq j} \frac{u(y, x_i)}{u(y, x_j)}$$9) supplies a means to tailor mechanisms to specific legal or logistical constraints, enhancing real-world applicability.

Future Directions

Potential extensions are clear:

• Multi-facility generalization: Analyzing envy ratio objectives in settings with multiple facilities remains open and likely challenging.
• Generalized metric/topology: Moving beyond one-dimensional spaces to multi-dimensional Euclidean or other metric spaces, or even general networks, would broaden applicability.
• Randomization tightness: Closing the gap between lower and upper bounds for randomized mechanisms is an unresolved problem and might require new construction techniques or impossibility proofs.
• Algorithmic efficiency: Investigating implementability and computational scalability of the proposed mechanisms, especially for large $u(y, x_i)$0 or complex topologies.

Conclusion

This work formalizes and addresses the problem of strategyproof facility location under a novel and direct fairness criterion, the envy ratio. It tightly characterizes optimal deterministic mechanisms and provides new upper and lower bounds for randomized approaches. The paper advances both the fair division and approximate strategyproof mechanism design literature, offering new tools and open questions for how fairness can and cannot be enforced without monetary transfers in collective allocation problems.

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Reference: "Facility Location Game with Envy Ratio" (2607.02330)