A Method for Multi-Leader–Multi-Follower Games by Smoothing the Followers’ Response Function
The multi-leader–multi-follower game (MLMFG) involves two or more leaders and followers and serves as a generalization of the Stackelberg game and the single-leader–multi-follower game. Although MLMFG covers wide range of real-world applications, its research is still sparse. Notably, fundamental so...
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| Vydáno v: | Journal of optimization theory and applications Ročník 203; číslo 1; s. 305 - 335 |
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01.10.2024
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| Abstract | The multi-leader–multi-follower game (MLMFG) involves two or more leaders and followers and serves as a generalization of the Stackelberg game and the single-leader–multi-follower game. Although MLMFG covers wide range of real-world applications, its research is still sparse. Notably, fundamental solution methods for this class of problems remain insufficiently established. A prevailing approach is to recast the MLMFG as an equilibrium problem with equilibrium constraints (EPEC) and solve it using a solver. Meanwhile, interpreting the solution to the EPEC in the context of MLMFG may be complex due to shared decision variables among all leaders, followers’ strategies that each leader can unilaterally change, but the variables are essentially controlled by followers. To address this issue, we introduce a response function of followers’ noncooperative game that is a function with leaders’ strategies as a variable. Employing this approach allows the MLMFG to be solved as a single-level differentiable variational inequality using a smoothing scheme for the followers’ response function. We also demonstrate that the sequence of solutions to the smoothed variational inequality converges to a stationary equilibrium of the MLMFG. Finally, we illustrate the behavior of the smoothing method by numerical experiments. |
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| AbstractList | The multi-leader–multi-follower game (MLMFG) involves two or more leaders and followers and serves as a generalization of the Stackelberg game and the single-leader–multi-follower game. Although MLMFG covers wide range of real-world applications, its research is still sparse. Notably, fundamental solution methods for this class of problems remain insufficiently established. A prevailing approach is to recast the MLMFG as an equilibrium problem with equilibrium constraints (EPEC) and solve it using a solver. Meanwhile, interpreting the solution to the EPEC in the context of MLMFG may be complex due to shared decision variables among all leaders, followers’ strategies that each leader can unilaterally change, but the variables are essentially controlled by followers. To address this issue, we introduce a response function of followers’ noncooperative game that is a function with leaders’ strategies as a variable. Employing this approach allows the MLMFG to be solved as a single-level differentiable variational inequality using a smoothing scheme for the followers’ response function. We also demonstrate that the sequence of solutions to the smoothed variational inequality converges to a stationary equilibrium of the MLMFG. Finally, we illustrate the behavior of the smoothing method by numerical experiments. |
| Author | Tsuyuguchi, Daisuke Fukuda, Ellen H. Hori, Atsushi |
| Author_xml | – sequence: 1 givenname: Atsushi orcidid: 0000-0002-7020-459X surname: Hori fullname: Hori, Atsushi email: atsushi-hori@st.seikei.ac.jp organization: Faculty of Science and Technology, Seikei University – sequence: 2 givenname: Daisuke surname: Tsuyuguchi fullname: Tsuyuguchi, Daisuke organization: Wakayama Prefectural Board of Education – sequence: 3 givenname: Ellen H. surname: Fukuda fullname: Fukuda, Ellen H. organization: Graduate School of Informatics, Kyoto University |
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| Cites_doi | 10.1007/978-3-030-52119-6_3 10.1109/59.867153 10.1007/s10589-011-9416-0 10.1007/BF02592192 10.1007/s10957-011-9901-8 10.1080/10556780903448052 10.1007/s10957-018-1391-5 10.1137/120863873 10.1007/s10107990015a 10.1287/opre.32.2.390 10.1109/ICC45855.2022.9838425 10.1007/s10287-004-0010-0 10.1007/978-3-642-02431-3 10.1080/10556788.2020.1828412 10.1007/978-1-4757-2825-5 10.1017/CBO9780511614330.005 10.1017/CBO9780511983658 10.1023/B:COAP.0000013057.54647.6d 10.1016/j.future.2020.02.045 10.1007/BF01584847 |
| ContentType | Journal Article |
| Copyright | The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2024. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
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| Keywords | Equilibrium problem with equilibrium constraints Smoothing approximation Nash equilibrium problem Bilevel optimization 91A65 91A10 90C33 Multi-leader–follower game |
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| References | FacchineiFJiangHQiLA smoothing method for mathematical programs with equilibrium constraintsMath. Program.199985107134168936610.1007/s10107990015a ChenXFukushimaMA smoothing method for a mathematical program with P-matrix linear complementarity constraintsComput. Optim. Appl.200427223256203571410.1023/B:COAP.0000013057.54647.6d LeyfferSMunsonTSolving multi-leader-common-follower gamesOptim. Methods Softw.201025601623272415810.1080/10556780903448052 RockafellarRTWetsRJVariational Analysis1998New YorkSpringer10.1007/978-3-642-02431-3 AubinJPMathematical Method of Game and Economic Theory1979AmsterdamNorth-Holland Lyu, T., Xu, H., Han, Z.: Multi-leader multi-follower Stackelberg game based resource allocation in multi-access edge computing. In: ICC 2022—IEEE International Conference on Communications, Seoul, Republic of Korea, pp. 4306–4311 (2022) HuMFukushimaMSmoothing approach to Nash equilibrium formulations for a class of equilibrium problems with shared complementarity constraintsComput. Optim. Appl.201252415437292578010.1007/s10589-011-9416-0 JiangSLiXWuJMulti-leader multi-follower Stackelberg game in mobile blockchain miningIEEE Trans. Mob.20202120582071 HuMFukushimaMMulti-leader-follower games: models, methods and applicationsJ. Oper. Res. Soc. Jpn.2015581233363211 HertyMSteffensenSThünenASolving quadratic multi-leader-follower games by smoothing the follower’s best responseOptim. Method Softw.202237772799448860310.1080/10556788.2020.1828412 KanzowCJiangHA continuation method for (strongly) monotone variational inequalitiesMath. Program.199881103125161776010.1007/BF01584847 LucaTDFacchineiFKanzowCA semismooth equation approach to the solution of nonlinear complementarity problemsMath. Program.199675407439142217910.1007/BF02592192 SuC-LEquilibrium Problems with Equilibrium Constraints: Stationarities, Algorithms, and Applications2005StanfordStanford University FerrisMCDirksePMeerausAKehoeTJSrinivasanTNWhalleyJMathematical programs with equilibrium constraints: automatic reformulation and solution via constrained optimizationFrontiers in Applied General Equilibrium Modeling2005CambridgeCambridge University Press679310.1017/CBO9780511614330.005 AusselDSvenssonADempeSZemkohoAAshort state of the art on multi-leader–follower gamesBilevel Optimization—Advances and Next Challenges2020BerlinSpringer537610.1007/978-3-030-52119-6_3 OutrataJKočvaraMZoweJNonsmooth Approach to Optimization Problems with Equilibrium Constraints1998New YorkSpringer10.1007/978-1-4757-2825-5 BertsekasDPConvex Analysis and Optimization2003NashuaAthena Scientific HuMFukushimaMVariational inequality formulation for multi-leader-follower gamesJ. Optim. Theory Appl.2012151455473285122510.1007/s10957-011-9901-8 ChenYLiZYangBNaiKLiKA Stackelberg game approach to multiple resources allocation and pricing in mobile edge computingFuture Gener. Comput. Syst.202010827328710.1016/j.future.2020.02.045 Pang, J.-S., Fukushima, M.: Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games. Comput. Manag. Sci. 2, 21–56 (2006). Erratum. ibid. 6, 373–375 (2005) SheraliHDA multiple leader Stackelberg model and analysisOper. Res.19843239040474775010.1287/opre.32.2.390 ClarkeFHOptimization and Nonsmooth Analysis1983New YorkWiley HobbsBMetzlerCPangJStrategic gaming analysis for electric power networks: an MPEC approachIEEE Trans. Power Syst.20001563864510.1109/59.867153 HuMFukushimaMExistence, uniqueness, and computation of robust Nash equilibria in a class of multi-leader-follower gamesSIAM J. Optim.201323894916304966010.1137/120863873 HoriAFukushimaMGauss-Seidel method for multi-leader-follower gamesJ. Optim. Theory Appl.2019180651670390797810.1007/s10957-018-1391-5 FacchineiFPangJ-SFinite-Dimensional Variational Inequalities and Complementarity Problems2003New YorkSpringer XiongZKangJNiyatoDWangPPoorHVCloud/edge computing service management in blockchaing networks: multi-leader multi-follower game-based ADMM for pricingIEEE Trans. Serv. Comput.201913356367 LuoZ-QPangJ-SRalphDMathematical Programs with Equilibrium Constraints1996CambridgeCambridge University Press10.1017/CBO9780511983658 2506_CR24 2506_CR22 Y Chen (2506_CR5) 2020; 108 Z-Q Luo (2506_CR21) 1996 S Leyffer (2506_CR19) 2010; 25 M Hu (2506_CR15) 2013; 23 J Outrata (2506_CR23) 1998 HD Sherali (2506_CR26) 1984; 32 M Hu (2506_CR16) 2015; 58 S Jiang (2506_CR17) 2020; 21 M Hu (2506_CR14) 2012; 52 FH Clarke (2506_CR6) 1983 JP Aubin (2506_CR1) 1979 DP Bertsekas (2506_CR3) 2003 Z Xiong (2506_CR28) 2019; 13 A Hori (2506_CR12) 2019; 180 F Facchinei (2506_CR7) 1999; 85 X Chen (2506_CR4) 2004; 27 M Hu (2506_CR13) 2012; 151 MC Ferris (2506_CR9) 2005 C Kanzow (2506_CR18) 1998; 81 F Facchinei (2506_CR8) 2003 RT Rockafellar (2506_CR25) 1998 C-L Su (2506_CR27) 2005 D Aussel (2506_CR2) 2020 M Herty (2506_CR10) 2022; 37 B Hobbs (2506_CR11) 2000; 15 TD Luca (2506_CR20) 1996; 75 |
| References_xml | – reference: KanzowCJiangHA continuation method for (strongly) monotone variational inequalitiesMath. Program.199881103125161776010.1007/BF01584847 – reference: SuC-LEquilibrium Problems with Equilibrium Constraints: Stationarities, Algorithms, and Applications2005StanfordStanford University – reference: HuMFukushimaMSmoothing approach to Nash equilibrium formulations for a class of equilibrium problems with shared complementarity constraintsComput. Optim. Appl.201252415437292578010.1007/s10589-011-9416-0 – reference: RockafellarRTWetsRJVariational Analysis1998New YorkSpringer10.1007/978-3-642-02431-3 – reference: FerrisMCDirksePMeerausAKehoeTJSrinivasanTNWhalleyJMathematical programs with equilibrium constraints: automatic reformulation and solution via constrained optimizationFrontiers in Applied General Equilibrium Modeling2005CambridgeCambridge University Press679310.1017/CBO9780511614330.005 – reference: FacchineiFPangJ-SFinite-Dimensional Variational Inequalities and Complementarity Problems2003New YorkSpringer – reference: AubinJPMathematical Method of Game and Economic Theory1979AmsterdamNorth-Holland – reference: SheraliHDA multiple leader Stackelberg model and analysisOper. Res.19843239040474775010.1287/opre.32.2.390 – reference: Lyu, T., Xu, H., Han, Z.: Multi-leader multi-follower Stackelberg game based resource allocation in multi-access edge computing. In: ICC 2022—IEEE International Conference on Communications, Seoul, Republic of Korea, pp. 4306–4311 (2022) – reference: XiongZKangJNiyatoDWangPPoorHVCloud/edge computing service management in blockchaing networks: multi-leader multi-follower game-based ADMM for pricingIEEE Trans. Serv. Comput.201913356367 – reference: ChenXFukushimaMA smoothing method for a mathematical program with P-matrix linear complementarity constraintsComput. Optim. Appl.200427223256203571410.1023/B:COAP.0000013057.54647.6d – reference: BertsekasDPConvex Analysis and Optimization2003NashuaAthena Scientific – reference: HoriAFukushimaMGauss-Seidel method for multi-leader-follower gamesJ. Optim. Theory Appl.2019180651670390797810.1007/s10957-018-1391-5 – reference: AusselDSvenssonADempeSZemkohoAAshort state of the art on multi-leader–follower gamesBilevel Optimization—Advances and Next Challenges2020BerlinSpringer537610.1007/978-3-030-52119-6_3 – reference: JiangSLiXWuJMulti-leader multi-follower Stackelberg game in mobile blockchain miningIEEE Trans. Mob.20202120582071 – reference: LuoZ-QPangJ-SRalphDMathematical Programs with Equilibrium Constraints1996CambridgeCambridge University Press10.1017/CBO9780511983658 – reference: HuMFukushimaMVariational inequality formulation for multi-leader-follower gamesJ. Optim. Theory Appl.2012151455473285122510.1007/s10957-011-9901-8 – reference: HuMFukushimaMExistence, uniqueness, and computation of robust Nash equilibria in a class of multi-leader-follower gamesSIAM J. Optim.201323894916304966010.1137/120863873 – reference: ClarkeFHOptimization and Nonsmooth Analysis1983New YorkWiley – reference: FacchineiFJiangHQiLA smoothing method for mathematical programs with equilibrium constraintsMath. Program.199985107134168936610.1007/s10107990015a – reference: HertyMSteffensenSThünenASolving quadratic multi-leader-follower games by smoothing the follower’s best responseOptim. Method Softw.202237772799448860310.1080/10556788.2020.1828412 – reference: HuMFukushimaMMulti-leader-follower games: models, methods and applicationsJ. Oper. Res. Soc. Jpn.2015581233363211 – reference: LucaTDFacchineiFKanzowCA semismooth equation approach to the solution of nonlinear complementarity problemsMath. 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| SubjectTerms | Applications of Mathematics Calculus of Variations and Optimal Control; Optimization Complex variables Engineering Equilibrium Game theory Mathematics Mathematics and Statistics Operations Research/Decision Theory Optimization Response functions Smoothing Theory of Computation |
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| Title | A Method for Multi-Leader–Multi-Follower Games by Smoothing the Followers’ Response Function |
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