AUT Journal of Mathematics and Computing

AUT Journal of Mathematics and Computing

Homology groups and decomposition of the game complex

Document Type : Original Article

Authors
Neda Shojaee
Aidin Fallah
Department of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Iran
10.22060/ajmc.2024.20822.1216
Abstract
In this paper, we introduce a novel simplicial complex named Game Complex for finite non-cooperative games in the strategic form. We prove that the number of Nash equilibrium in non-cooperative games with more than two players is the rank of the first homology group of the game complex. Furthermore, we give a decomposition of the game complex.
Keywords
Hodge operator
Nash equilibrium
Game complex
Subjects
[1] O. Candogan, I. Menache, A. Ozdaglar, and P. A. Parrilo, Flows and decompositions of games: harmonic and potential games, Math. Oper. Res., 36 (2011), pp. 474–503.
[2] H. Edelsbrunner and J. L. Harer, Computational topology, American Mathematical Society, Providence, RI, 2010. An introduction.
[3] F. Erhun and P. Keskinocak, Game theory in business applications, Management Science and Engineering, Stanford University, Stanford, (2003).
[4] A. Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002.
[5] A. Hayes, Ultimate guide to game theory: Principles and applications, 2020. https://www.investopedia. com/terms/g/gametheory.asp.
[6] M. Herlihy, D. Kozlov, and S. Rajsbaum, Distributed computing through combinatorial topology, Else- vier/Morgan Kaufmann, Waltham, MA, 2014.
[7] X. Jiang, L.-H. Lim, Y. Yao, and Y. Ye, Statistical ranking and combinatorial Hodge theory, Math. Pro- gram., 127 (2011), pp. 203–244.
[8] J. R. Munkres, Elements of algebraic topology, Addison-Wesley Publishing Company, Menlo Park, CA, 1984.
AUT Journal of Mathematics and Computing
Volume 7, Issue 3
Summer 2026
Pages 283-291