A Functional Inequality and a New Class of Probabilities in the <i>N</i>-Person Red-and-Black Game
In this paper, we explore a model of an <i>N</i>-player, non-cooperative stochastic game, drawing inspiration from the discrete formulation of the red-and-black gambling problem, as initially introduced by Dubins and Savage in 1965. We extend upon the work of Pontiggia from 2007, present...
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| Format: | Article |
| Language: | English |
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MDPI AG
2024-03-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/16/3/325 |
| _version_ | 1797239220475002880 |
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| author | Włodzimierz Fechner Maria Słomian |
| author_facet | Włodzimierz Fechner Maria Słomian |
| author_sort | Włodzimierz Fechner |
| collection | DOAJ |
| description | In this paper, we explore a model of an <i>N</i>-player, non-cooperative stochastic game, drawing inspiration from the discrete formulation of the red-and-black gambling problem, as initially introduced by Dubins and Savage in 1965. We extend upon the work of Pontiggia from 2007, presenting a main theorem that broadens the conditions under which bold strategies by all players can achieve a Nash equilibrium. This is obtained through the introduction of a novel functional inequality, which serves as a key analytical tool in our study. This inequality enables us to circumvent the restrictive conditions of super-multiplicativity and super-additivity prevalent in the works of Pontiggia and others. We conclude this paper with a series of illustrative examples that demonstrate the efficacy of our approach, notably highlighting its ability to accommodate a broader spectrum of probability functions than previously recognized in the existing literature. |
| first_indexed | 2024-04-24T17:48:04Z |
| format | Article |
| id | doaj.art-2f223b4ef42c4e0a8f5323e4ac5e9414 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-04-24T17:48:04Z |
| publishDate | 2024-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-2f223b4ef42c4e0a8f5323e4ac5e94142024-03-27T14:05:30ZengMDPI AGSymmetry2073-89942024-03-0116332510.3390/sym16030325A Functional Inequality and a New Class of Probabilities in the <i>N</i>-Person Red-and-Black GameWłodzimierz Fechner0Maria Słomian1Institute of Mathematics, Lodz University of Technology, al. Politechniki 8, 93-590 Łódź, PolandIndependent Researcher, Zygry 71/1, 99-232 Zadzim, PolandIn this paper, we explore a model of an <i>N</i>-player, non-cooperative stochastic game, drawing inspiration from the discrete formulation of the red-and-black gambling problem, as initially introduced by Dubins and Savage in 1965. We extend upon the work of Pontiggia from 2007, presenting a main theorem that broadens the conditions under which bold strategies by all players can achieve a Nash equilibrium. This is obtained through the introduction of a novel functional inequality, which serves as a key analytical tool in our study. This inequality enables us to circumvent the restrictive conditions of super-multiplicativity and super-additivity prevalent in the works of Pontiggia and others. We conclude this paper with a series of illustrative examples that demonstrate the efficacy of our approach, notably highlighting its ability to accommodate a broader spectrum of probability functions than previously recognized in the existing literature.https://www.mdpi.com/2073-8994/16/3/325red-and-black gamestochastic gamebold strategyNash equilibriumfunctional inequality |
| spellingShingle | Włodzimierz Fechner Maria Słomian A Functional Inequality and a New Class of Probabilities in the <i>N</i>-Person Red-and-Black Game Symmetry red-and-black game stochastic game bold strategy Nash equilibrium functional inequality |
| title | A Functional Inequality and a New Class of Probabilities in the <i>N</i>-Person Red-and-Black Game |
| title_full | A Functional Inequality and a New Class of Probabilities in the <i>N</i>-Person Red-and-Black Game |
| title_fullStr | A Functional Inequality and a New Class of Probabilities in the <i>N</i>-Person Red-and-Black Game |
| title_full_unstemmed | A Functional Inequality and a New Class of Probabilities in the <i>N</i>-Person Red-and-Black Game |
| title_short | A Functional Inequality and a New Class of Probabilities in the <i>N</i>-Person Red-and-Black Game |
| title_sort | functional inequality and a new class of probabilities in the i n i person red and black game |
| topic | red-and-black game stochastic game bold strategy Nash equilibrium functional inequality |
| url | https://www.mdpi.com/2073-8994/16/3/325 |
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