Bilevel optimization for the regulation of shared resources under uncertainty

Luigi Greco; Lina Mallozzi

doi:10.55976/dma.420261635100-109

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Bilevel optimization for the regulation of shared resources under uncertainty

Luigi Greco ¹ , Lina Mallozzi ² ^*

1. Department of Mathematics and Applications R. Caccioppoli, University of Naples Federico II, Complesso Monte S. Angelo V. Cinthia, Napoli, 80126, Italy
2. Department of Mathematics and Applications R. Caccioppoli, University of Naples Federico II, P.le Tecchio 80, Napoli, 80125, Italy

Correspondence: mallozzi@unina.it

DOI: https://doi.org/10.55976/dma.420261635100-109

Received

13 July 2026
Revised

24 August 2026
Accepted

08 September 2026
Published

21 September 2026

Leader-follower model Stackelberg-Nash equilibrium Correspondence selections Potential games

Abstract

<h5>1. Introduction</h5>
<p>In a non-cooperative game setting, the Common-Pool Resource (CPR) problem models a scenario in which individual players make extraction decisions independently without binding agreements, as is the case with shared groundwater and aquifers, fisheries, pastures, etc. This strategic context leads directly to over exploitation, a phenomenon widely known as the Tragedy of the Commons <a href="#reference_list_0">[1]</a>. A common-pool resource possesses two distinct economic characteristics: rivalry, since extraction by one user directly reduces the amount of the resource available to all others, and low excludability, since it is difficult or costly to prevent individuals from accessing and exploiting the resource (see <a href="#reference_list_1">[2-4]</a>). Experimental evidence has documented strategic destruction of commonpool resources <a href="#reference_list_1">[2]</a>, while recent groundwater studies examine strategic extraction and groundwater exploitation <a href="#reference_list_4">[5, 6]</a>. Recent experimental work further considers how ecological uncertainty and institutional arrangements affect the management of local commons <a href="#reference_list_6">[7-9]</a>.</p>
<p><em>Please download the PDF from the right hand panel to access the full text.</em></p>
<p> </p>
<h5>Acknowledgements</h5>
<p>The authors are members of the National Group for Mathematical Analysis, Probability and their Applications (GNAMPA) of the Istituto Nazionale di Alta Matematica(INdAM).</p>
<h5>Author contributions</h5>
<p>Luigi Greco: Conceptualization, Methodology, Formal analysis, Writing, review and editing. Lina Mallozzi: Conceptualization, Methodology, Formal analysis, Writing, review and editing. Both authors approved the final manuscript.</p>
<h5>Funding</h5>
<p>The research of L. Mallozzi was funded under the “GAMEL” Research Funding Program-MI–DDR–00423-CUP: F53C25001420001-Mission Innovation 2.0. by the Ministry of the Environment and Energy Security (MASE).</p>
<h5>Data availability</h5>
<p>No datasets were generated or analyzed in this study. The numerical examples are fully specified in the manuscript.</p>
<h5>Code availability</h5>
<p>No external software package or proprietary code was required for the analytical results. The numerical calculations reported in the manuscript can be reproduced directly from the displayed formulas.</p>

References

[1]Hardin G. The tragedy of the commons. Science. 1968; 162(3859): 1243-1248. doi:10.1126/science.162.3859.1243.

[2]Walker JM, Gardner R. Probabilistic destruction of common-pool resources: Experimental evidence. The Economic Journal. 1992; 102(414): 1149-1161. doi:10.2307/2234382.

[3]Gordon HS. The economic theory of a common property resource: The fishery. Journal of Political Economy. 1954; 62(2): 124-142. doi:10.1086/257497.

[4]Ostrom E. The challenge of common-pool resources. Environment: Science and Policy for Sustainable Development. 2008; 50(4): 8-21. doi:10.3200/ENVT.50.4.8-21.

[5]Koch CM, Nax HH. Groundwater usage and strategic complements: Part I (instrumental variables). Games. 2022; 13(5): 67. doi:10.3390/g13050067.

[6]Biancardi M, Iannucci G, Villani G. Groundwater exploitation and illegal behaviors in a differential game. Dynamic Games and Applications. 2022;12(3): 996-1009. doi:10.1007/s13235-022-00436-0.

[7]Schill C, Rocha JC. Sustaining local commons in the face of uncertain ecological thresholds: Evidence from a framed field experiment with Colombian small-scale fishers. Ecological Economics. 2023; 207:107695. doi:10.1016/j.ecolecon.2022.107695.

[8]Ahn M, Baldwin E, Girone D. Caution as a response to scientific uncertainty: A groundwater game experiment. International Journal of the Commons. 2024; 18(1): 369-383. doi:10.5334/ijc.1347.

[9]Bayle G, Farolfi S, Lavaine E, Willinger M. Solving conflict over common pool resources through delegation to a stakeholder. Journal of Economic Behavior & Organization. 2024; 227: 106702. doi:10.1016/j.jebo.2024.106702.

[10]Sherali HD, Soyster AL, Murphy FH. Stackelberg-Nash-Cournot equilibria: Characterizations and computations. Operations Research. 1983; 31(2): 253-276. doi:10.1287/opre.31.2.253.

[11]Cominetti R, Dose V, Scarsini M. The price of anarchy in routing games as a function of the demand. Mathematical Programming. 2024; 203: 531-558. doi:10.1007/s10107-021-01701-7.

[12]Jensen MK. Aggregative games and best-reply potentials. Economic Theory. 2010; 43(1): 45-66. doi:10.1007/s00199-008-0419-8.

[13]Borges P, Sagastizábal C, Solodov M. Decomposition algorithms for some deterministic and two-stage stochastic single-leader multi-follower games. Computational Optimization and Applications. 2021;78(3): 675-704. doi:10.1007/s10589-020-00257-0.

[14]Zhang Y, Liu F, Wang Z, Chen Y, Feng S, Wu Q, et al. On Nash-Stackelberg-Nash games under decision-dependent uncertainties: Model and equilibrium. Automatica. 2022; 142: 110401. doi:10.1016/j.automatica.2022.110401.

[15]Beck Y, Ljubić I, Schmidt M. A survey on bilevel optimization under uncertainty. European Journal of Operational Research. 2023; 311(2): 401-426. doi:10.1016/j.ejor.2023.01.008.

[16]DeMiguel V, Xu H. A stochastic multiple-leader Stackelberg model: Analysis, computation, and application. Operations Research. 2009; 57(5): 1220-1235. doi:10.1287/opre.1080.0686.

[17]Banerjee A, Gravel N. Contribution to a public good under subjective uncertainty. Journal of Public Economic Theory. 2020; 22(3): 473-500. doi:10.1111/jpet.12423.

[18]Khan AA, Tammer C, Zălinescu C. Set-Valued Optimization: An Introduction with Applications. Berlin: Springer; 2015.

[19]Liu Y, Xu H, Yang S-J, Zhang J. Distributionally robust equilibrium for continuous games: Nash and Stackelberg models. European Journal of Operational Research. 2018; 265(2): 631-643. doi:10.1016/j.ejor.2017.07.050.

[20]Garagic D, Cruz JB. An approach to fuzzy noncooperative Nash games. Journal of Optimization Theory and Applications. 2003; 118(3): 475-491. doi:10.1023/B:JOTA.0000004867.66302.16.

[21]Liu Z, Wang G, Yang G. Existence of equilibrium solution for leader-follower games with fuzzy goals and parameters. Fuzzy Sets and Systems. 2023; 473: 108731. doi:10.1016/j.fss.2023.108731.

[22]Zhang C-L. Set-valued equilibrium problems based on the concept of null set with applications. Optimization. 2023; 73(2): 443-460. doi:10.1080/02331934.2022.2109969.

[23]Klein E, Thompson AC. Theory of Correspondences. New York: Wiley; 1984.

[24]Monderer D, Shapley LS. Potential games. Games and Economic Behavior. 1996; 14(1): 124-143. doi:10.1006/game.1996.0044.

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Greco, L., & Mallozzi , L. (2026). Bilevel optimization for the regulation of shared resources under uncertainty. Decision Making and Analysis, 4(1), 100–109. https://doi.org/10.55976/dma.420261635100-109

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