2015/12/28 by Kuize Zhang, Zhang, Kuize
Biochemistry, Genetics and Molecular Biology · Decision Sciences · Mathematics · Medicine · #15A09 #91A10 #91A70 #FOS: Mathematics #Game Theory and Applications #Gene Regulatory Network Analysis #Mathematical and Theoretical Epidemiology and Ecology Models #Optimization and Control (math.OC) #math.OC #msc:15A09 #msc:91A10 #msc:91A70
paper · pdf · doi:10.48550/arxiv.1512.08319
arxiv created 2015/12/28 · openalex publication_date 2015/12/28 · arxiv updated 2015/12/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The space of finite games can be decomposed into three orthogonal subspaces [5], which are the subspaces of pure potential games, nonstrategic games and pure harmonic games. The orthogonal projections onto these subspaces are represented as the Moore-Penrose inverses of the corresponding linear operators (i.e., matrices) [5]. Although the representation is compact and nice, no analytic method is given to calculate Moore- Penrose inverses of these linear operators. Hence using their results, one cannot verify whether a finite game belongs to one of these subspaces. In this paper, jumping over calculating Moore-Penrose inverses of these linear operators directly, via using group inverses, in the framework of the semitensor product of matrices, we give explicit polynomial representation for these orthogonal projections and for potential functions of potential games. Using our results, one not only can determine whether a finite game belongs to one of these subspaces, but also can find an arbitrary finite game belonging to one of them. Besides, we give formal definitions for these types of games by using their payoff functions. Based on these results, more properties of finite games are revealed.