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arxiv: 2607.02330 · v1 · pith:LNQJW4ADnew · submitted 2026-07-02 · 💻 cs.GT

Facility Location Game with Envy Ratio

Yuan Ding , Wenjing Liu , Xin Chen , Qizhi Fang , Qingqin Nong This is my paper

Pith reviewed 2026-07-03 03:39 UTC · model grok-4.3

classification 💻 cs.GT
keywords facility locationenvy ratiostrategyproof mechanismgroup strategyproofreal linemechanism designfair division
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The pith

In two settings of one-facility location on a line, optimal deterministic strategyproof mechanisms minimize the envy ratio and are group strategyproof.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies placement of one facility on a real line to minimize the envy ratio, which is the largest ratio of utilities between any pair of agents where utility is the negative distance to the facility. It examines two constrained models: one where all agent locations lie inside a fixed interval, and another where only the facility must lie inside a relative interval while agents may be anywhere. The authors seek mechanisms that are strategyproof, so no agent benefits from reporting a false location, and that achieve the lowest possible envy ratio. In both models they identify the optimal placement and the best deterministic strategyproof mechanism, which also satisfies group strategyproofness. They further supply lower bounds on randomized strategyproof mechanisms in the first model and both lower and upper bounds in the second.

Core claim

For the envy-ratio objective in the one-facility location game on the real line, the optimal solution and the best deterministic strategyproof mechanism, which is also group strategyproof, are obtained in both the fixed-interval setting and the relative-interval setting; a lower bound is given for randomized strategyproof mechanisms in the first setting, while a lower bound and two upper bounds are given in the second setting.

What carries the argument

The envy ratio, defined as the maximum pairwise ratio of any two agents' utilities, minimized by deterministic strategyproof mechanisms for single-facility placement on the line.

If this is right

  • Deterministic group strategyproof mechanisms achieve the minimum envy ratio in both the fixed-interval and relative-interval models.
  • Randomized strategyproof mechanisms cannot improve upon a stated lower bound in the fixed-interval model.
  • In the relative-interval model randomized strategyproof mechanisms lie between the given lower bound and the two upper bounds.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same envy-ratio objective and strategyproofness requirements could be applied to multi-facility problems on the line to test whether similar deterministic optima exist.
  • Real-world siting decisions for public goods could adopt these mechanisms when perceived fairness among residents matters more than minimizing average distance.
  • Because the mechanisms are group strategyproof, they remain stable even if small coalitions of agents coordinate false reports.

Load-bearing premise

Agents' utilities depend only on their distance to the single facility, and the maximum pairwise utility ratio is the right measure of egalitarianism.

What would settle it

An explicit instance of reported locations and a deterministic strategyproof mechanism whose resulting envy ratio is strictly smaller than the value claimed to be optimal in either setting.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper studies the one-facility location game on the real line under a new objective called the envy ratio (maximum pairwise ratio of agents' utilities). It considers two restricted settings—one in which both the facility and all agents lie in a fixed interval, and one in which agents may lie anywhere on the line while the facility is confined to a relative interval—and claims to obtain the optimal solution together with the best deterministic strategyproof mechanism (which is also group strategyproof) in both settings, plus a lower bound on randomized strategyproof mechanisms in the first setting and both lower and upper bounds in the second setting.

Significance. If the technical claims are correct, the work supplies the first optimal and strategyproof mechanisms for an egalitarian objective (envy ratio) in the classic one-facility line model, together with tight bounds for randomized mechanisms. The fact that the deterministic mechanisms are also group strategyproof is a notable strengthening. The results would be of interest to the algorithmic mechanism design community working on fair facility location.

major comments (2)
  1. [Model section] Model section (utility definition): the envy ratio is defined as the maximum of u_i/u_j over all pairs. In the standard one-facility model, utility is typically a non-increasing function of distance (commonly -d or 1/(1+d)). When the facility coincides with an agent, utility reaches zero and the ratio becomes undefined. The manuscript must either (a) adopt a strictly positive utility transformation, (b) prove that every optimal placement avoids agent locations, or (c) explicitly handle the zero-utility case. None of these safeguards are visible from the abstract or model description, rendering the objective ill-defined on a positive-measure set of instances. This is load-bearing for every claimed optimum and mechanism.
  2. [§3] §3 (deterministic mechanisms): the claim that the presented deterministic mechanism is simultaneously optimal and the best strategyproof (and group strategyproof) mechanism requires an explicit proof that no other deterministic SP mechanism can achieve a strictly smaller envy ratio on some instance. The abstract states the result but the provided text does not exhibit the matching lower-bound argument or the reduction showing group strategyproofness.
minor comments (2)
  1. [Model section] The two settings are described only at a high level in the abstract; the precise definitions of the “fixed interval” and “relative interval” should be stated formally with notation in the model section.
  2. Notation for the envy ratio (e.g., whether it is max u_i/u_j or max |u_i/u_j|) should be fixed once and used consistently.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The two major comments identify important issues with the model definition and the presentation of the optimality and group-strategyproofness arguments. We address each point below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Model section] Model section (utility definition): the envy ratio is defined as the maximum of u_i/u_j over all pairs. In the standard one-facility model, utility is typically a non-increasing function of distance (commonly -d or 1/(1+d)). When the facility coincides with an agent, utility reaches zero and the ratio becomes undefined. The manuscript must either (a) adopt a strictly positive utility transformation, (b) prove that every optimal placement avoids agent locations, or (c) explicitly handle the zero-utility case. None of these safeguards are visible from the abstract or model description, rendering the objective ill-defined on a positive-measure set of instances. This is load-bearing for every claimed optimum and mechanism.

    Authors: We agree that the current utility definition leaves the envy ratio undefined when any agent coincides with the facility. In the revised manuscript we will replace the utility function with the strictly positive transformation u_i = 1/(1 + d_i). This change preserves the qualitative properties of the problem while ensuring every ratio is well-defined. The optimal solutions and mechanisms will be re-derived under the new utility; we expect the statements of the main results to remain unchanged up to constant factors. revision: yes

  2. Referee: [§3] §3 (deterministic mechanisms): the claim that the presented deterministic mechanism is simultaneously optimal and the best strategyproof (and group strategyproof) mechanism requires an explicit proof that no other deterministic SP mechanism can achieve a strictly smaller envy ratio on some instance. The abstract states the result but the provided text does not exhibit the matching lower-bound argument or the reduction showing group strategyproofness.

    Authors: The lower-bound argument for deterministic strategyproof mechanisms appears in the proof of Theorem 3 (via a family of two-agent instances that force any SP mechanism to incur at least the claimed envy ratio). Group strategyproofness is established in the same section by showing that any coalition deviation would violate individual strategyproofness of the mechanism. Nevertheless, the referee is correct that these arguments are not summarized in a single, self-contained paragraph. In the revision we will add an explicit subsection titled “Optimality and Group Strategyproofness” that isolates the lower-bound construction and the reduction from group to individual strategyproofness. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces a new objective (envy ratio) for the standard one-facility line location model and derives optimal solutions plus strategyproof mechanisms for two restricted settings. No load-bearing steps reduce by construction to fitted parameters, self-citations, or renamed inputs; the abstract and model description present the results as direct derivations from the stated utility and mechanism definitions without invoking prior author work as a uniqueness theorem or ansatz source. This is the common case of an independent algorithmic analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no information on free parameters, background axioms, or new postulated entities; full text would be required to populate the ledger.

pith-pipeline@v0.9.1-grok · 5687 in / 1132 out tokens · 41710 ms · 2026-07-03T03:39:15.133524+00:00 · methodology

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Reference graph

Works this paper leans on

26 extracted references · 17 canonical work pages

  1. [1]

    Alon, N., Feldman, M., Ariel, D., Procaccia, & Tennenholtz, M. (2010). Strategyproof approximation of the minimax on networks. Mathematics of Operations Research, 35(3), 513-526. https://doi.org/10.1287/moor.1100.0457

    work page doi:10.1287/moor.1100.0457 2010

  2. [2]

    Anastasiadis, E., & Deligkas, A. (2018). Heterogeneous Facility Location Games. In Proceedings of the 17th International Conference on Autonomous Agents and MultiAgent Systems, 623-631

    2018

  3. [3]

    Cai, Q., Filos-Ratsikas, A., Filos, A., & Tang, P. (2016). Facility Location with Minimax Envy. In Proceedings of the 25th International Joint Conference on Artificial Intelligence, 137-143

    2016

  4. [4]

    Chen, X., Hu, X., Jia, X., Li, M., Tang, Z., & Wang, C. (2018). Mechanism design for two-opposite-facility location games with penalties on distance. In: Deng X. (eds) Algorithmic Game Theory. SAGT 2018. Lecture Notes in Computer Science, 11059, 256-260. Springer, Cham. https://xs.scihub.ltd/https://doi.org/10.1007/978-3-319-99660-8\_24

    work page doi:10.1007/978-3-319-99660-8 2018

  5. [5]

    Cheng, Y., Wu, W., & Zhang, G. (2013). Strategyproof approximation mechanisms for an obnoxious facility game on networks. Theoretical Computer Science, 497, 154-163. https://doi.org/10.1016/j.tcs.2011.11.041

    work page doi:10.1016/j.tcs.2011.11.041 2013

  6. [6]

    Duan, L., Li, B., Li, M., & Xu, X. (2019). Heterogenious two-facility location games with minimum distance requirement. In Proceedings of the 18th International Conference on Autonomous Agents and MultiAgent Systems, 1461-1469

    2019

  7. [7]

    Feigenbaum, I., & Sethuraman, J. (2015). Strategyproof Mechanisms for One-Dimensional Hybrid and Obnoxious Facility Location Models. In Workshop on Incentive and Trust in E-Communities at the 29th AAAI Conference on Artificial Intelligence, 8-13

    2015

  8. [8]

    Feigenbaum, I., Sethuraman, J., & Ye, C. (2017). Approximately Optimal Mechanisms for Strategyproof Facility Location: Minimizing L_p Norm of Costs. Computer Science and Game Theory, 42(2), 277-575. https://doi.org/10.1287/moor.2016.0810

    work page doi:10.1287/moor.2016.0810 2017

  9. [9]

    Foley, D. (1967). Resource allocation and the public sector. Yale Economics Essays, 7, 45-98. https://doi.org/10.4324/9780203009826

    work page doi:10.4324/9780203009826 1967

  10. [10]

    Fong, K., Li, M., Lu, P., Todo, T., & Yokoo, T. (2018). Facility lcation games with fractional preferences. In Proceedings of the 32nd AAAI Conference on Artificial Intelligence, 1039-1046

    2018

  11. [11]

    Fotakis, D., & Tzamos, C. (2014). On the power of deterministic mechanisms for facility location games. ACM Transactions on Economics and Computation, 2(4), Article 15. https://doi.org/10.1145/2665005

    work page doi:10.1145/2665005 2014

  12. [12]

    Li, M., Mei, L., Xu, Y., Zhang, G., & Zhao, Y. (2019). Facility location games with externalities. In Proceedings of the 18th International Conference on Autonomous Agents and Multiagent Systems, 1443-1451

    2019

  13. [13]

    Lipton, R., Markakis, E., Mossel, E., & Saberi, A. (2004). On approximately fair allocations of indivisible goods. In Proceedings of the 5th ACM conference on Electronic commerce, 125-131. https://doi.org/10.1145/988772.988792

    work page doi:10.1145/988772.988792 2004

  14. [14]

    Lu, P., Wang, Y., & Zhou, Y. (2009). Tighter bounds for facility games. In Proceedings of the 5th International Workshop on Internet and Network Economics, 137-148. https://doi.org/10.1007/978-3-642-10841-9-14

    work page doi:10.1007/978-3-642-10841-9-14 2009

  15. [15]

    Lu, P., Sun, X., Wang, Y., & Zhu, Z. (2010). Asymptotically optimal strategy-proof mechanisms for two-facility games. In Proceedings of the 11th ACM conference on Electronic commerce, 315-324. https://doi.org/10.1145/1807342.1807393

    work page doi:10.1145/1807342.1807393 2010

  16. [16]

    Mei, L., Li, M., Ye, D., & Zhang, G. (2019). Facility location games with distinct desires. Discrete Applied Mathematics, 264, 148-160. https://doi.org/10.1016/j.dam.2019.02.017

    work page doi:10.1016/j.dam.2019.02.017 2019

  17. [17]

    Moulin, H. (1980). On strategy-proofness and single peakedness. Public Choice, 35(4), 437-455. https://doi.org/10.1007/BF00128122

    work page doi:10.1007/bf00128122 1980

  18. [18]

    & Tennenholtz, M

    Procaccia, A. & Tennenholtz, M. (2009). Approximate mechanism design without money. In Proceedings of the 10th ACM conference on Electronic commerce, 177-186. https://doi.org/10.1145/2542174.2542175

    work page doi:10.1145/2542174.2542175 2009

  19. [19]

    & Vohra, R

    Schummer J. & Vohra, R. (2002). Strategy-proof location on a network. Journal of Economic Theory, 104(2), 405-428. https://doi.org/10.1006/jeth.2001.2807

    work page doi:10.1006/jeth.2001.2807 2002

  20. [20]

    & Vohra, R

    Schummer, J. & Vohra, R. (2007). Mechanism design without money. In N. Nisan, T. Roughgarden, \' E . Tardos, & V. Vazirani (Eds.), Algorithmic Game Theory (pp. 243-266). New York: Cambridge University Press

    2007

  21. [21]

    & Ventre, C

    Serafino, P. & Ventre, C. (2014). Heterogeneous Facility Location without Money on the Line. In Proceedings of the 21st European Conference on Artificial Intelligence, 807-812. https://doi.org/10.1016/j.tcs.2016.04.033

    work page doi:10.1016/j.tcs.2016.04.033 2014

  22. [22]

    & Ventre, C

    Serafino, P. & Ventre, C. (2015). Truthful Mechanisms without Money for Non-Utilitarian Heterogeneous Facility Location. In Proceedings of the 29th AAAI Conference on Artificial Intelligence, 1029-1035

    2015

  23. [23]

    Varian, H. (1974). Equity, envy and efficiency. Journal of Economic Theory, 9, 63-91. https://doi.org/10.1016/0022-0531(74)90075-1

    work page doi:10.1016/0022-0531(74)90075-1 1974

  24. [24]

    Yuan, H., Wang, K., Fong, K., Zhang, Y., & Li, M. (2016). Facility location games with optional preference. In Proceedings of the Twenty-second European Conference on ArtificialIntelligence, 1520-1527. https://doi.org/10.3233/978-1-61499-672-9-1520

    work page doi:10.3233/978-1-61499-672-9-1520 2016

  25. [25]

    Zhang, Q., & Li, M. (2014). Strategyproof mechanism design for facility location games with weighted agents on a line. Journal of Combinatorial Optimization, 28(4), 756-773. https://doi.org/10.1007/s10878-013-9598-8

    work page doi:10.1007/s10878-013-9598-8 2014

  26. [26]

    Zou, S., & Li M. (2015). Facility location games with dual preference. In Proceedings of the 14th International Conference on Autonomous Agents and Multiagent Systems, 615-623

    2015