A branch-and-prune algorithm for discrete Nash equilibrium problems
We present a branch-and-prune procedure for discrete Nash equilibrium problems with a convex description of each player's strategy set. The derived pruning criterion does not require player convexity, but only strict convexity of some player's objective function in a single variable. If sa...
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| Vydáno v: | Computational optimization and applications Ročník 86; číslo 2; s. 491 - 519 |
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| Hlavní autoři: | , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
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New York, NY
Springer US
01.11.2023
Springer Nature B.V |
| Témata: | |
| ISSN: | 1573-2894, 0926-6003, 1573-2894 |
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| Abstract | We present a branch-and-prune procedure for discrete Nash equilibrium problems with a convex description of each player's strategy set. The derived pruning criterion does not require player convexity, but only strict convexity of some player's objective function in a single variable. If satisfied, it prunes choices for this variable by stating activity of certain constraints. This results in a synchronous branching and pruning method. An algorithmic implementation and numerical tests are presented for randomly generated instances with convex polyhedral strategy sets and convex quadratic as well as non-convex quadratic objective functions. |
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| AbstractList | We present a branch-and-prune procedure for discrete Nash equilibrium problems with a convex description of each player’s strategy set. The derived pruning criterion does not require player convexity, but only strict convexity of some player’s objective function in a single variable. If satisfied, it prunes choices for this variable by stating activity of certain constraints. This results in a synchronous branching and pruning method. An algorithmic implementation and numerical tests are presented for randomly generated instances with convex polyhedral strategy sets and convex quadratic as well as non-convex quadratic objective functions. |
| Author | Stein, Oliver Schwarze, Stefan |
| Author_xml | – sequence: 1 givenname: Stefan surname: Schwarze fullname: Schwarze, Stefan – sequence: 2 givenname: Oliver surname: Stein fullname: Stein, Oliver |
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| Cites_doi | 10.1287/opre.1110.0964 10.1006/game.1996.0044 10.1016/j.ejor.2022.03.048 10.1073/pnas.36.1.48 10.1137/15M1052445 10.1007/s10589-010-9331-9 10.1007/s10288-014-0256-5 10.1017/S0962492913000032 10.1007/s10898-011-9727-9 10.2307/1969529 10.1080/02331934.2018.1545125 10.1007/s10107-007-0160-2 |
| ContentType | Journal Article |
| Copyright | The Author(s) 2023 The Author(s) 2023. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License. |
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| References | Pappalardo, Mastroeni, Passacantando (CR13) 2014; 12 CR2 Facchinei, Piccialli, Sciandrone (CR7) 2011; 50 CR8 Monderer, Shapley (CR10) 1996; 14 Sagratella (CR15) 2019; 68 Belotti, Kirches, Leyffer, Linderoth, Luedtke, Mahajan (CR1) 2013; 22 Facchinei, Fischer, Piccialli (CR5) 2009; 117 Dreves, Kanzow, Stein (CR4) 2012; 53 Köppe, Ryan, Queyranne (CR9) 2011; 59 Carvalho, Lodi, Pedroso (CR3) 2022; 303 Nash (CR12) 1951; 54 Nash (CR11) 1950; 36 Sagratella (CR14) 2016; 26 Facchinei, Kanzow (CR6) 2007; 5 JF Nash (500_CR12) 1951; 54 F Facchinei (500_CR5) 2009; 117 M Pappalardo (500_CR13) 2014; 12 A Dreves (500_CR4) 2012; 53 S Sagratella (500_CR15) 2019; 68 M Carvalho (500_CR3) 2022; 303 F Facchinei (500_CR7) 2011; 50 M Köppe (500_CR9) 2011; 59 F Facchinei (500_CR6) 2007; 5 P Belotti (500_CR1) 2013; 22 JF Nash (500_CR11) 1950; 36 500_CR2 S Sagratella (500_CR14) 2016; 26 500_CR8 D Monderer (500_CR10) 1996; 14 |
| References_xml | – volume: 59 start-page: 1445 issue: 6 year: 2011 end-page: 1460 ident: CR9 article-title: Rational generating functions and integer programming games publication-title: Oper. Res. doi: 10.1287/opre.1110.0964 – volume: 14 start-page: 124 issue: 1 year: 1996 end-page: 143 ident: CR10 article-title: Potential games publication-title: Games Econom. Behav. doi: 10.1006/game.1996.0044 – ident: CR2 – volume: 303 start-page: 1057 issue: 3 year: 2022 end-page: 1070 ident: CR3 article-title: Computing equilibria for integer programming games publication-title: Eur. J. Oper. Res. doi: 10.1016/j.ejor.2022.03.048 – volume: 36 start-page: 48 issue: 1 year: 1950 end-page: 49 ident: CR11 article-title: Equilibrium points in n-person games publication-title: Proc. Natl. Acad. Sci. doi: 10.1073/pnas.36.1.48 – volume: 26 start-page: 2190 issue: 4 year: 2016 end-page: 2218 ident: CR14 article-title: Computing all solutions of Nash equilibrium problems with discrete strategy sets publication-title: SIAM J. Optim. doi: 10.1137/15M1052445 – volume: 5 start-page: 173 issue: 3 year: 2007 end-page: 210 ident: CR6 article-title: Generalized Nash equilibrium problems publication-title: Ann. Oper. Res – volume: 50 start-page: 237 issue: 2 year: 2011 end-page: 262 ident: CR7 article-title: Decomposition algorithms for generalized potential games publication-title: Comput. Optim. Appl. doi: 10.1007/s10589-010-9331-9 – volume: 12 start-page: 1 issue: 1 year: 2014 end-page: 33 ident: CR13 article-title: Merit functions: a bridge between optimization and equilibria publication-title: 4OR doi: 10.1007/s10288-014-0256-5 – volume: 22 start-page: 1 year: 2013 end-page: 131 ident: CR1 article-title: Mixed-integer nonlinear optimization publication-title: Acta Numer. doi: 10.1017/S0962492913000032 – volume: 53 start-page: 587 issue: 4 year: 2012 end-page: 614 ident: CR4 article-title: Nonsmooth optimization reformulations of player convex generalized Nash equilibrium problems publication-title: J. Glob. Optim. doi: 10.1007/s10898-011-9727-9 – ident: CR8 – volume: 54 start-page: 286 issue: 2 year: 1951 end-page: 295 ident: CR12 article-title: Non-cooperative games publication-title: Ann. Math. doi: 10.2307/1969529 – volume: 68 start-page: 197 issue: 1 year: 2019 end-page: 226 ident: CR15 article-title: On generalized Nash equilibrium problems with linear coupling constraints and mixed-integer variables publication-title: Optimization doi: 10.1080/02331934.2018.1545125 – volume: 117 start-page: 163 issue: 1–2 year: 2009 end-page: 194 ident: CR5 article-title: Generalized Nash equilibrium problems and Newton methods publication-title: Math. Program. doi: 10.1007/s10107-007-0160-2 – volume: 22 start-page: 1 year: 2013 ident: 500_CR1 publication-title: Acta Numer. doi: 10.1017/S0962492913000032 – volume: 12 start-page: 1 issue: 1 year: 2014 ident: 500_CR13 publication-title: 4OR doi: 10.1007/s10288-014-0256-5 – volume: 59 start-page: 1445 issue: 6 year: 2011 ident: 500_CR9 publication-title: Oper. Res. doi: 10.1287/opre.1110.0964 – volume: 54 start-page: 286 issue: 2 year: 1951 ident: 500_CR12 publication-title: Ann. Math. doi: 10.2307/1969529 – volume: 53 start-page: 587 issue: 4 year: 2012 ident: 500_CR4 publication-title: J. Glob. Optim. doi: 10.1007/s10898-011-9727-9 – volume: 14 start-page: 124 issue: 1 year: 1996 ident: 500_CR10 publication-title: Games Econom. Behav. doi: 10.1006/game.1996.0044 – volume: 68 start-page: 197 issue: 1 year: 2019 ident: 500_CR15 publication-title: Optimization doi: 10.1080/02331934.2018.1545125 – volume: 26 start-page: 2190 issue: 4 year: 2016 ident: 500_CR14 publication-title: SIAM J. Optim. doi: 10.1137/15M1052445 – volume: 5 start-page: 173 issue: 3 year: 2007 ident: 500_CR6 publication-title: Ann. Oper. Res – volume: 50 start-page: 237 issue: 2 year: 2011 ident: 500_CR7 publication-title: Comput. Optim. Appl. doi: 10.1007/s10589-010-9331-9 – volume: 303 start-page: 1057 issue: 3 year: 2022 ident: 500_CR3 publication-title: Eur. J. Oper. Res. doi: 10.1016/j.ejor.2022.03.048 – ident: 500_CR2 – volume: 117 start-page: 163 issue: 1–2 year: 2009 ident: 500_CR5 publication-title: Math. Program. doi: 10.1007/s10107-007-0160-2 – ident: 500_CR8 – volume: 36 start-page: 48 issue: 1 year: 1950 ident: 500_CR11 publication-title: Proc. Natl. Acad. Sci. doi: 10.1073/pnas.36.1.48 |
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| SubjectTerms | Algorithms Branch and bound Branch and prune Convex and Discrete Geometry Convexity Discrete game Equilibrium Game theory Integer programming Management Science Mathematical analysis Mathematics Mathematics and Statistics Nash equilibrium problem Operations Research Operations Research/Decision Theory Optimization Statistics Variables |
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