A Game-Theoretic Analysis of <i>Baccara Chemin de Fer</i>, II
In a previous paper, we considered several models of the parlor game <i>baccara chemin de fer</i>, including Model B2 (a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn>&l...
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MDPI AG
2023-09-01
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| Series: | Games |
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| Online Access: | https://www.mdpi.com/2073-4336/14/5/63 |
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| author | Stewart N. Ethier Jiyeon Lee |
| author_facet | Stewart N. Ethier Jiyeon Lee |
| author_sort | Stewart N. Ethier |
| collection | DOAJ |
| description | In a previous paper, we considered several models of the parlor game <i>baccara chemin de fer</i>, including Model B2 (a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mo>×</mo><msup><mn>2</mn><mn>484</mn></msup></mrow></semantics></math></inline-formula> matrix game) and Model B3 (a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mn>2</mn><mn>5</mn></msup><mo>×</mo><msup><mn>2</mn><mn>484</mn></msup></mrow></semantics></math></inline-formula> matrix game), both of which depend on a positive-integer parameter <i>d</i>, the number of decks. The key to solving the game under Model B2 was what we called Foster’s algorithm, which applies to additive <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mo>×</mo><msup><mn>2</mn><mi>n</mi></msup></mrow></semantics></math></inline-formula> matrix games. Here “additive” means that the payoffs are additive in the <i>n</i> binary choices that comprise a player II pure strategy. In the present paper, we consider analogous models of the casino game <i>baccara chemin de fer</i> that take into account the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>100</mn><mspace width="0.166667em"></mspace><mi>α</mi></mrow></semantics></math></inline-formula> percent commission on Banker (player II) wins, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>α</mi><mo>≤</mo><mn>1</mn><mo>/</mo><mn>10</mn></mrow></semantics></math></inline-formula>. Thus, the game now depends not just on the discrete parameter <i>d</i> but also on a continuous parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>. Moreover, the game is no longer zero sum. To find all Nash equilibria under Model B2, we generalize Foster’s algorithm to additive <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mo>×</mo><msup><mn>2</mn><mi>n</mi></msup></mrow></semantics></math></inline-formula> bimatrix games. We find that, with rare exceptions, the Nash equilibrium is unique. We also obtain a Nash equilibrium under Model B3, based on Model B2 results, but here we are unable to prove uniqueness. |
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| issn | 2073-4336 |
| language | English |
| last_indexed | 2024-03-11T10:14:52Z |
| publishDate | 2023-09-01 |
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| series | Games |
| spelling | doaj.art-854a139e4b2a4eb8947d8d743251dfb42023-11-16T10:24:40ZengMDPI AGGames2073-43362023-09-011456310.3390/g14050063A Game-Theoretic Analysis of <i>Baccara Chemin de Fer</i>, IIStewart N. Ethier0Jiyeon Lee1Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, UT 84112, USADepartment of Statistics, Yeungnam University, 280 Daehak-Ro, Gyeongsan 38541, Gyeongbuk, Republic of KoreaIn a previous paper, we considered several models of the parlor game <i>baccara chemin de fer</i>, including Model B2 (a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mo>×</mo><msup><mn>2</mn><mn>484</mn></msup></mrow></semantics></math></inline-formula> matrix game) and Model B3 (a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mn>2</mn><mn>5</mn></msup><mo>×</mo><msup><mn>2</mn><mn>484</mn></msup></mrow></semantics></math></inline-formula> matrix game), both of which depend on a positive-integer parameter <i>d</i>, the number of decks. The key to solving the game under Model B2 was what we called Foster’s algorithm, which applies to additive <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mo>×</mo><msup><mn>2</mn><mi>n</mi></msup></mrow></semantics></math></inline-formula> matrix games. Here “additive” means that the payoffs are additive in the <i>n</i> binary choices that comprise a player II pure strategy. In the present paper, we consider analogous models of the casino game <i>baccara chemin de fer</i> that take into account the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>100</mn><mspace width="0.166667em"></mspace><mi>α</mi></mrow></semantics></math></inline-formula> percent commission on Banker (player II) wins, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>α</mi><mo>≤</mo><mn>1</mn><mo>/</mo><mn>10</mn></mrow></semantics></math></inline-formula>. Thus, the game now depends not just on the discrete parameter <i>d</i> but also on a continuous parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>. Moreover, the game is no longer zero sum. To find all Nash equilibria under Model B2, we generalize Foster’s algorithm to additive <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mo>×</mo><msup><mn>2</mn><mi>n</mi></msup></mrow></semantics></math></inline-formula> bimatrix games. We find that, with rare exceptions, the Nash equilibrium is unique. We also obtain a Nash equilibrium under Model B3, based on Model B2 results, but here we are unable to prove uniqueness.https://www.mdpi.com/2073-4336/14/5/63<i>baccara</i><i>chemin de fer</i>sampling without replacementbimatrix gamebest responseNash equilibrium |
| spellingShingle | Stewart N. Ethier Jiyeon Lee A Game-Theoretic Analysis of <i>Baccara Chemin de Fer</i>, II Games <i>baccara</i> <i>chemin de fer</i> sampling without replacement bimatrix game best response Nash equilibrium |
| title | A Game-Theoretic Analysis of <i>Baccara Chemin de Fer</i>, II |
| title_full | A Game-Theoretic Analysis of <i>Baccara Chemin de Fer</i>, II |
| title_fullStr | A Game-Theoretic Analysis of <i>Baccara Chemin de Fer</i>, II |
| title_full_unstemmed | A Game-Theoretic Analysis of <i>Baccara Chemin de Fer</i>, II |
| title_short | A Game-Theoretic Analysis of <i>Baccara Chemin de Fer</i>, II |
| title_sort | game theoretic analysis of i baccara chemin de fer i ii |
| topic | <i>baccara</i> <i>chemin de fer</i> sampling without replacement bimatrix game best response Nash equilibrium |
| url | https://www.mdpi.com/2073-4336/14/5/63 |
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