Complete Lattice Using Lexicographical Order and Its Application in Non-Cooperative Ordered Game

Abstract

One application of lattices in optimization is defining the equilibrium set of ordered games, typically using the usual (coordinate-wise) order, which is incomplete in Rn. This incompleteness makes some strategies incomparable, requiring a special game concept. Using a complete order, like the lexicographic order, results in a complete lattice. This study explores the properties of a complete lattice with lexicographic order for noncooperative games and provides a Python algorithm to determine the Nash equilibrium of a supermodular game.

References

S. Penmatsa and A. T. Chronopoulos, “Game theoretic static load balancing for distributed systems,” Journal of Parallel and Distributed Computing, vol. 71(4), pp. 537–555, 2011.

D. Grosu and A. T. Chronopoulos, “Noncooperative static load balancing for distributed systems,” Journal of Parallel and Distributed Computing, vol. 65(9), pp. 1022–1034, 2005.

M. Y. Leung et al., “Load balancing in distributed systems: An approach using cooperative games,” IEEE 16th International Parallel and Distributed Processing Symposium (IPDPS 2002), Fort Lauderdale, Florida, pp. 8–11, 15-19 April 2002.

D. Grosu and A. T. Chronopoulos, “A game-theoritic model and algorithm for load balancing in distributed systems,” IEEE 16th International Parallel and Distributed Processing Symposium (IPDPS 2002), Fort Lauderdale, Florida, pp. 146–153, 2002.

J. Bilbao, Cooperative games on combinatorial structures. Springer Science+Business Media, 2012, vol. 26.

M. Van der Merwe, Non-cooperative games on networks. Ph.D dissertation, Stellenbosch University, 2012.

F. R. Csori, Structural and computational aspects of simple and influence games. Tesis, Universitat Politecnica de Catalunya, 2014.

D. M. Topkis, “Minimizing a submodular function on a lattice,” Operations Research, vol. 26(2), pp. 305–321, 1978.

P. Milgrom and C. Shannon, “Monotone comparative statics,” Econometrica, vol. 62(1), pp. 157–180, 1994.

G. Cachon, “Stock wars: Inventory competition in a two-echelon supply chain with multiple retailers,” Oper. Res., vol. 49(5), pp. 658–674, 2001. doi: 10.1287/opre.49.5.658.10611

D. M. Topkis, Supermodularity and Complementary. Princeton University Press, 1998.

G. Cachon and S. Netessine, Game Theory in Supply Chain Analysis, Handbook of Quantitative Supply Chain Analysis: Modelling in the E-Business Era. Springer Science+Business Media, 2024.

R. Setiawan, Salmah., I. Endrayanto, and Indarsih., “Analysis of the n-person noncooperative supermodular multiobjective games,” Oper. Res. Lett., vol. 51, pp. 278–284, 2023. doi: 10.1016/j.orl.2023.03.007

X. Vives, “Nash equilibrium with strategic complementarities,” J.Math.Econ., vol. 19, pp. 305–321, 1990.

G. Rota-Graziosi, “The supermodularity of the tax competition game,” J.Math.Econ., vol. 83, pp. 25–35, 2019.

G. Koshevoy, T. Suzuki, and D. Talman, “Supermodular ntu-games,” Oper. Res. Lett., vol. 44, pp. 446–450, 2016. doi: 10.1016/j.orl.2016.04.007
Published
2025-07-31
How to Cite
SETIAWAN, Rubono; KURDHI, Nughthoh Arfawi. Complete Lattice Using Lexicographical Order and Its Application in Non-Cooperative Ordered Game. Yugoslav Journal of Operations Research, [S.l.], july 2025. ISSN 2334-6043. Available at: <https://yujor.fon.bg.ac.rs/index.php/yujor/article/view/1361>. Date accessed: 07 aug. 2025. doi: https://doi.org/10.2298/YJOR240815030S.
Section
Research Articles

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