{"title":"A Game-Theoretic Analysis of Baccara Chemin de Fer, II","authors":"Stewart N. Ethier, Jiyeon Lee","doi":"10.3390/g14050063","DOIUrl":null,"url":null,"abstract":"In a previous paper, we considered several models of the parlor game baccara chemin de fer, including Model B2 (a 2×2484 matrix game) and Model B3 (a 25×2484 matrix game), both of which depend on a positive-integer parameter d, the number of decks. The key to solving the game under Model B2 was what we called Foster’s algorithm, which applies to additive 2×2n matrix games. Here “additive” means that the payoffs are additive in the n binary choices that comprise a player II pure strategy. In the present paper, we consider analogous models of the casino game baccara chemin de fer that take into account the 100α percent commission on Banker (player II) wins, where 0≤α≤1/10. Thus, the game now depends not just on the discrete parameter d but also on a continuous parameter α. Moreover, the game is no longer zero sum. To find all Nash equilibria under Model B2, we generalize Foster’s algorithm to additive 2×2n 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.","PeriodicalId":35065,"journal":{"name":"Games","volume":"36 1","pages":"0"},"PeriodicalIF":0.6000,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Games","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3390/g14050063","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"ECONOMICS","Score":null,"Total":0}
引用次数: 0
Abstract
In a previous paper, we considered several models of the parlor game baccara chemin de fer, including Model B2 (a 2×2484 matrix game) and Model B3 (a 25×2484 matrix game), both of which depend on a positive-integer parameter d, the number of decks. The key to solving the game under Model B2 was what we called Foster’s algorithm, which applies to additive 2×2n matrix games. Here “additive” means that the payoffs are additive in the n binary choices that comprise a player II pure strategy. In the present paper, we consider analogous models of the casino game baccara chemin de fer that take into account the 100α percent commission on Banker (player II) wins, where 0≤α≤1/10. Thus, the game now depends not just on the discrete parameter d but also on a continuous parameter α. Moreover, the game is no longer zero sum. To find all Nash equilibria under Model B2, we generalize Foster’s algorithm to additive 2×2n 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.
GamesDecision Sciences-Statistics, Probability and Uncertainty
CiteScore
1.60
自引率
11.10%
发文量
65
审稿时长
11 weeks
期刊介绍:
Games (ISSN 2073-4336) is an international, peer-reviewed, quick-refereeing open access journal (free for readers), which provides an advanced forum for studies related to strategic interaction, game theory and its applications, and decision making. The aim is to provide an interdisciplinary forum for all behavioral sciences and related fields, including economics, psychology, political science, mathematics, computer science, and biology (including animal behavior). To guarantee a rapid refereeing and editorial process, Games follows standard publication practices in the natural sciences.