On the structure of two-person, finite, zero-sum games
Abstract
The algebraical and topological structure of the set 𝓖 (resp. 𝓖1) of all n×m matrix games with saddle points (resp. with unique saddle point) in the space Rn×m of all such games is studied. It has been shown that 𝓖 is a closed cone with vertex zero and includes the origin. Moreover, it is neither convex nor dense subset of Rn×m. The set 𝓖1 is a non-convex cone which does not include the origin. It is neither closed nor open.
The concept of “reserve of non-saddlexity” has been also introduced.
References
2. G.N. Dyubin, Concerning the set of the games over unit square with the unique solution, Dokl. Akad. Nauk, SSSR, 184, No. 2 (1969), 26-269 (in Russian).
3. J. Klamka, Estimation of controllability and observability by canonical Jordan form, Podstawy sterowania, vol. 4, No. 4 (1974), 349-369 (in Polish).
4. V.W. Kreps, On maximality of some linear spaces of continuous functions with saddle points, in: Achievements of game theory, Vilnius 1973, 47-50 (in Russian).
5. K. Kuratowski, An introduction to set theory and topology, Warszawa 1966 (in Polish).
6. N.Y. Naumowa, On the existence of solutions for cooperative games, in: Game theory, Erewan 1973, 253 (in Russian).
7. A.I. Sobolev, On dimension of linear space of matrices with saddle points, in: Achievements of game theory, Vilnius 1973, 66-68 (in Russian).
8. N.N. Vorobyev, Present-day state of game theory, in: Game theory, Erewan 1973, 5-57 (in Russian).
9. N.N. Vorobyev, Scientific results of conference, in: Achievements of game theory, Vilnius 1973, 7-13 (in Russian).
Instytut Matematyki, Uniwersytet Śląski w Katowicach Poland
The Copyright Holders of the submitted text are the Author and the Journal. The Reader is granted the right to use the pdf documents under the provisions of the Creative Commons 4.0 International License: Attribution (CC BY). The user can copy and redistribute the material in any medium or format and remix, transform, and build upon the material for any purpose.
- License
This journal provides immediate open access to its content under the Creative Commons BY 4.0 license (http://creativecommons.org/licenses/by/4.0/). Authors who publish with this journal retain all copyrights and agree to the terms of the above-mentioned CC BY 4.0 license. - Author’s Warranties
The author warrants that the article is original, written by stated author/s, has not been published before, contains no unlawful statements, does not infringe the rights of others, is subject to copyright that is vested exclusively in the author and free of any third party rights, and that any necessary written permissions to quote from other sources have been obtained by the author/s. - User Rights
Under the Creative Commons Attribution license, the users are free to share (copy, distribute and transmit the contribution) and adapt (remix, transform, and build upon the material) the article for any purpose, provided they attribute the contribution in the manner specified by the author or licensor. - Co-Authorship
If the article was prepared jointly with other authors, the signatory of this form warrants that he/she has been authorized by all co-authors to sign this agreement on their behalf, and agrees to inform his/her co-authors of the terms of this agreement.
