ABSTRACT
Fair division of resources emerges in a variety of different contexts in real-world problems, some of which can be seen through the lens of game theory. Many equilibrium notions for simple fair division problems with indivisible items have been considered, and many of these notions are hard to compute. Strategic fair division is a branch of fair division in which participants may act uncooperatively to maximize their utility. In the presence of participants who have strategic behavior, it is essential to have a suitable algorithm in place to allocate resources in a fair and equitable manner. We propose a new approach to solve strategic fair division problems where fairness is attained by finding a constrained Nash equilibrium in a specific game. We show that computational complexity barriers also hold. More broadly, the theoretical results of this paper could potentially be applied to related general game theory problems and complex fair division problems. Finally, we propose an algorithm for finding a constrained Nash equilibrium in the game that we introduce. Our focus will be on one particular meta-heuristic – the bus transportation algorithm – as an approach to improve the running time of the search.
- E Altman and E Solan. 2009. Constrained Games: The Impact of the Attitude to Adversary’s Constraints. IEEE Trans. Automat. Contr. 54, 10 (Oct. 2009), 2435–2440.Google ScholarCross Ref
- Kenneth J. Arrow. 1951. An Extension of the Basic Theorems of Classical Welfare Economics.Google Scholar
- Qamar Askari, Irfan Younas, and Mehreen Saeed. 2020. Political Optimizer: A novel socio-inspired meta-heuristic for global optimization. Knowledge-Based Systems 195 (2020), 105709. https://doi.org/10.1016/j.knosys.2020.105709Google ScholarCross Ref
- Robert J Aumann. 1985. What is game theory trying to accomplish?. In Frontiers of Economics, edited by K. Arrow and S. Honkapohja. Citeseer.Google Scholar
- Azar, Adel, Seyed mirzaee, and Seyed moslem. 2013. Providing new meta-heuristic algorithm for optimization problems inspired by humans' behavior to improve their positions. International Journal of Artificial Intelligence & Applications 4, 1 (Jan. 2013), 1–12. https://doi.org/10.5121/ijaia.2013.4101Google ScholarCross Ref
- Xiaohui Bei, Youming Qiao, and Shengyu Zhang. 2017. Networked Fairness in Cake Cutting. In Proceedings of the 26th International Joint Conference on Artificial Intelligence (Melbourne, Australia) (IJCAI’17). AAAI Press, 3632–3638.Google ScholarDigital Library
- Marie Louisa Tølbøll Berthelsen and Kristoffer Arnsfelt Hansen. 2022. On the Computational Complexity of Decision Problems About Multi-player Nash Equilibria. Theory Comput. Syst. 66, 3 (2022), 519–545. https://doi.org/10.1007/s00224-022-10080-1Google ScholarDigital Library
- Mohammad Bodaghi and Koosha Samieefar. 2018. Meta-heuristic bus transportation algorithm. Iran Journal of Computer Science 2, 1 (Oct. 2018), 23–32. https://doi.org/10.1007/s42044-018-0025-2Google ScholarCross Ref
- Niclas Boehmer, Robert Bredereck, Klaus Heeger, Dušan Knop, and Junjie Luo. 2022. Multivariate Algorithmics for Eliminating Envy by Donating Goods. In Proceedings of the 21st International Conference on Autonomous Agents and Multiagent Systems (Virtual Event, New Zealand) (AAMAS ’22). International Foundation for Autonomous Agents and Multiagent Systems, Richland, SC, 127–135.Google ScholarDigital Library
- Steven J. Brams and Alan D. Taylor. 1996. Fair Division: From Cake-Cutting to Dispute Resolution. Cambridge University Press. https://doi.org/10.1017/CBO9780511598975Google ScholarCross Ref
- Simina Brânzei, Ioannis Caragiannis, David Kurokawa, and Ariel Procaccia. 2016. An Algorithmic Framework for Strategic Fair Division. Proceedings of the AAAI Conference on Artificial Intelligence 30, 1 (Feb. 2016). https://doi.org/10.1609/aaai.v30i1.10042Google ScholarCross Ref
- Simina Brânzei, Vasilis Gkatzelis, and Ruta Mehta. 2022. Nash Social Welfare Approximation for Strategic Agents. Operations Research 70, 1 (2022), 402–415. https://doi.org/10.1287/opre.2020.2056 arXiv:https://doi.org/10.1287/opre.2020.2056Google ScholarCross Ref
- Simina Brânzei and Peter Bro Miltersen. 2013. Equilibrium Analysis in Cake Cutting. In Proceedings of the 2013 International Conference on Autonomous Agents and Multi-Agent Systems (St. Paul, MN, USA) (AAMAS ’13). International Foundation for Autonomous Agents and Multiagent Systems, Richland, SC, 327–334.Google ScholarDigital Library
- Eric Budish. 2011. The Combinatorial Assignment Problem: Approximate Competitive Equilibrium from Equal Incomes. Journal of Political Economy 119, 6 (2011), 1061–1103. https://doi.org/10.1086/664613 arXiv:https://doi.org/10.1086/664613Google ScholarCross Ref
- Sofia Ceppi, Nicola Gatti, Giorgio Patrini, and Marco Rocco. 2010. Local Search Methods for Finding a Nash Equilibrium in Two-Player Games. In 2010 IEEE/WIC/ACM International Conference on Web Intelligence and Intelligent Agent Technology, Vol. 2. 335–342. https://doi.org/10.1109/WI-IAT.2010.57Google ScholarDigital Library
- Xi Chen, Xiaotie Deng, and Shang-Hua Teng. 2009. Settling the Complexity of Computing Two-Player Nash Equilibria. J. ACM 56, 3 (May 2009), 1–57.Google ScholarDigital Library
- Richard Cole and Vasilis Gkatzelis. 2015. Approximating the Nash Social Welfare with Indivisible Items. In Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing (Portland, Oregon, USA) (STOC ’15). Association for Computing Machinery, New York, NY, USA, 371–380. https://doi.org/10.1145/2746539.2746589Google ScholarDigital Library
- Vincent Conitzer and Tuomas Sandholm. 2008. New Complexity Results about Nash Equilibria. Games Econ. Behav. 63, 2 (July 2008), 621–641.Google ScholarCross Ref
- H. W. Corley. 2017. Normative Utility Models for Pareto Scalar Equilibria in n-Person, Semi-Cooperative Games in Strategic Form. Theoretical Economics Letters 07, 06 (2017), 1667–1686. https://doi.org/10.4236/tel.2017.76113Google ScholarCross Ref
- Artur Czumaj, Argyrios Deligkas, Michail Fasoulakis, John Fearnley, Marcin Jurdziński, and Rahul Savani. 2019. Distributed Methods for Computing Approximate Equilibria. Algorithmica 81, 3 (March 2019), 1205–1231.Google ScholarDigital Library
- Constantinos Daskalakis, Paul W Goldberg, and Christos H Papadimitriou. 2009. The Complexity of Computing a Nash Equilibrium. SIAM j. comput. 39, 1 (Jan. 2009), 195–259.Google Scholar
- Constantinos Daskalakis, Aranyak Mehta, and Christos Papadimitriou. 2009. A Note on Approximate Nash Equilibria. Theor. Comput. Sci. 410, 17 (April 2009), 1581–1588.Google ScholarDigital Library
- Gerard Debreu. 1952. A Social Equilibrium Existence Theorem*. Proceedings of the National Academy of Sciences 38, 10 (1952), 886–893. https://doi.org/10.1073/pnas.38.10.886 arXiv:https://www.pnas.org/doi/pdf/10.1073/pnas.38.10.886Google ScholarCross Ref
- Mohammad Dehghani, Eva Trojovská, and Pavel Trojovský. 2022. A new human-based metaheuristic algorithm for solving optimization problems on the base of simulation of driving training process. Scientific reports 12, 1 (June 2022), 9924. https://doi.org/10.1038/s41598-022-14225-7Google ScholarCross Ref
- Niels Elgers, Nguyen Dang, and Patrick De Causmaecker. 2019. A Metaheuristic Approach to Compute Pure Nash Equilibria. Springer International Publishing, Cham, 221–233. https://doi.org/10.1007/978-3-319-95104-1_14Google ScholarCross Ref
- Kousha Etessami and Mihalis Yannakakis. 2010. On the Complexity of Nash Equilibria and other Fixed Points. SIAM j. comput. 39, 6 (Jan. 2010), 2531–2597.Google Scholar
- Francisco Facchinei and Christian Kanzow. 2007. Generalized Nash equilibrium problems. 4OR 5, 3 (Sept. 2007), 173–210.Google Scholar
- John Fearnley and Rahul Savani. 2016. Finding Approximate Nash Equilibria of Bimatrix Games via Payoff Queries. ACM Trans. Econ. Comput. 4, 4, Article 25 (aug 2016), 19 pages. https://doi.org/10.1145/2956579Google ScholarDigital Library
- Gamow George and Marvin Stern. 1958. Puzzle-maththe Big Bang to Black Holes. Viking Press.Google Scholar
- Itzhak Gilboa and Eitan Zemel. 1989. Nash and Correlated Equilibria: Some Complexity Considerations. Games and Economic Behavior 1, 1 (1989), 80–93. https://doi.org/10.1016/0899-8256(89)90006-7Google ScholarCross Ref
- Fred Glover. 1986. Future paths for integer programming and links to artificial intelligence. Computers & Operations Research 13, 5 (1986), 533–549. https://doi.org/10.1016/0305-0548(86)90048-1 Applications of Integer Programming.Google ScholarDigital Library
- Fred Glover and Manuel Laguna. 1998. Tabu Search. Springer US, Boston, MA, 2093–2229. https://doi.org/10.1007/978-1-4613-0303-9_33Google ScholarCross Ref
- Mamoru Kaneko and Kenjiro Nakamura. 1979. The Nash Social Welfare Function. Econometrica 47, 2 (1979), 423–435. http://www.jstor.org/stable/1914191Google ScholarCross Ref
- Bruce M Kapron and Koosha Samieefar. 2023. The Computational Complexity of Equilibria with Strategic Constraints. (2023).Google Scholar
- Michael J. Kearns, Michael L. Littman, and Satinder P. Singh. 2001. Graphical Models for Game Theory. In Proceedings of the 17th Conference in Uncertainty in Artificial Intelligence(UAI ’01). Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, 253–260.Google ScholarDigital Library
- C. E. Lemke and J. T. Howson, Jr.1964. Equilibrium Points of Bimatrix Games. J. Soc. Indust. Appl. Math. 12, 2 (1964), 413–423. https://doi.org/10.1137/0112033 arXiv:https://doi.org/10.1137/0112033Google ScholarCross Ref
- Richard J Lipton, Evangelos Markakis, and Aranyak Mehta. 2003. Playing Large Games Using Simple Strategies. In Proceedings of the 4th ACM conference on Electronic commerce - EC ’03 (San Diego, CA, USA). ACM Press, New York, New York, USA.Google ScholarDigital Library
- Eric Maskin. 1999. Nash Equilibrium and Welfare Optimality. The Review of Economic Studies 66, 1 (1999), 23–38. http://www.jstor.org/stable/2566947Google ScholarCross Ref
- Andrew McLennan and Rabee Tourky. 2010. Simple complexity from imitation games. Games Econ. Behav. 68, 2 (March 2010), 683–688.Google ScholarCross Ref
- Igal Milchtaich. 2006. Computation of Completely Mixed Equilibrium Payoffs in Bimatrix Games. Int. Game Theory Rev. 08, 03 (Sept. 2006), 483–487.Google Scholar
- Ahmad Nahhas and H. W. Corley. 2018. An Alternative Interpretation of Mixed Strategies in n-Person Normal Form Games via Resource Allocation. Theoretical Economics Letters 08, 10 (2018), 1854–1868. https://doi.org/10.4236/tel.2018.810122Google ScholarCross Ref
- John Nash. 1951. Non-Cooperative Games. Annals of Mathematics 54, 2 (1951), 286–295. http://www.jstor.org/stable/1969529Google ScholarCross Ref
- John F. Nash. 1950. Equilibrium points in n-person games. Proceedings of the National Academy of Sciences 36, 1 (Jan. 1950), 48–49. https://doi.org/10.1073/pnas.36.1.48Google ScholarCross Ref
- Nhan-Tam Nguyen, Trung Thanh Nguyen, Magnus Roos, and Jörg Rothe. 2013. Computational complexity and approximability of social welfare optimization in multiagent resource allocation. Autonomous Agents and Multi-Agent Systems 28, 2 (April 2013), 256–289. https://doi.org/10.1007/s10458-013-9224-2Google ScholarDigital Library
- Christos H. Papadimitriou, Emmanouil-Vasileios Vlatakis-Gkaragkounis, and Manolis Zampetakis. 2022. The Computational Complexity of Multi-player Concave Games and Kakutani Fixed Points. https://doi.org/10.48550/ARXIV.2207.07557Google ScholarCross Ref
- Benjamin Plaut. 2020. Optimal Nash Equilibria for Bandwidth Allocation. In Web and Internet Economics, Xujin Chen, Nikolai Gravin, Martin Hoefer, and Ruta Mehta (Eds.). Springer International Publishing, Cham, 77–88.Google Scholar
- Sara Ramezani and Ulle Endriss. 2010. Nash Social Welfare in Multiagent Resource Allocation. In Agent-Mediated Electronic Commerce. Designing Trading Strategies and Mechanisms for Electronic Markets, Esther David, Enrico Gerding, David Sarne, and Onn Shehory (Eds.). Springer Berlin Heidelberg, Berlin, Heidelberg, 117–131.Google Scholar
- Ariel Rubinstein. 1991. Comments on the Interpretation of Game Theory. Econometrica 59, 4 (1991), 909–924. http://www.jstor.org/stable/2938166Google ScholarCross Ref
- Aviad Rubinstein. 2015. Inapproximability of Nash Equilibrium. In Proceedings of the forty-seventh annual ACM symposium on Theory of Computing (Portland Oregon USA). ACM, New York, NY, USA.Google ScholarDigital Library
- Aviad Rubinstein. 2017. Settling the Complexity of Computing Approximate Two-Player Nash Equilibria. SIGecom Exch. 15, 2 (feb 2017), 45–49. https://doi.org/10.1145/3055589.3055596Google ScholarDigital Library
- Larry Samuelson. 2016. Game Theory in Economics and Beyond. Journal of Economic Perspectives 30, 4 (Nov. 2016), 107–130. https://doi.org/10.1257/jep.30.4.107Google ScholarCross Ref
- Thomas Sandholm, Andrew Gilpin, and Vincent Conitzer. 2005. Mixed-Integer Programming Methods for Finding Nash Equilibria. In Proceedings of the 20th National Conference on Artificial Intelligence - Volume 2 (Pittsburgh, Pennsylvania) (AAAI’05). AAAI Press, 495–501.Google Scholar
- Grant R Schoenebeck and Salil Vadhan. 2012. The Computational Complexity of Nash Equilibria in Concisely Represented Games. ACM trans. comput. theory 4, 2 (May 2012), 1–50.Google Scholar
- Bernhard von Stengel. 2007. Equilibrium Computation for Two-Player Games in Strategic and Extensive Form. Cambridge University Press, 53–78. https://doi.org/10.1017/CBO9780511800481.005Google ScholarCross Ref
- Koichi Tadenuma and William Thomson. 1995. Games of Fair Division. Games and Economic Behavior 9, 2 (1995), 191–204. https://doi.org/10.1006/game.1995.1015Google ScholarCross Ref
- Eva Trojovská and Mohammad Dehghani. 2022. A new human-based metahurestic optimization method based on mimicking cooking training. Scientific Reports 12, 1 (Sept. 2022). https://doi.org/10.1038/s41598-022-19313-2Google ScholarCross Ref
- Ling Wang, Haoqi Ni, Ruixin Yang, Minrui Fei, and Wei Ye. 2014. A Simple Human Learning Optimization Algorithm. In Computational Intelligence, Networked Systems and Their Applications, Minrui Fei, Chen Peng, Zhou Su, Yang Song, and Qinglong Han (Eds.). Springer Berlin Heidelberg, Berlin, Heidelberg, 56–65.Google Scholar
- Jonathan Widger and Daniel Grosu. 2008. Computing Equilibria in Bimatrix Games by Parallel Support Enumeration. In 2008 International Symposium on Parallel and Distributed Computing (Krakow, Poland). IEEE.Google ScholarDigital Library
- Zhengtian Wu, Chuangyin Dang, Hamid Reza Karimi, Changan Zhu, and Qing Gao. 2014. A mixed 0-1 linear programming approach to the computation of all pure-strategy Nash equilibria of a finiten-person game in normal form. Math. Probl. Eng. 2014 (2014), 1–8.Google Scholar
Index Terms
- A Meta-heuristic Approach for Strategic Fair Division Problems
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