# On the Three-Person Game Baccara Banque

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

My idea is so sensational that practically nobody will play chemin-de-fer. If I guarantee to take any stake of any size, all the millionaires will want to take part in this fantastic party. The biggest gamblers in the world will come to ruin me. I suggest we start at Deauville.

## 2. Evaluation of the Payoffs

## 3. Correlated Cooperative Equilibrium

**Table 1.**Banker’s strategy in the correlated cooperative equilibrium when $\theta =1/2$. Specifically the table displays Banker’s maximum drawing total as a function of Player 1’s and Player 2’s third-card values. For example, if Player 1’s third card is 7 and Player 2’s third card is 8, the entry 3 signifies that Banker draws on 0–3 and stands on 4–7. 5+ signifies that Banker draws on 0–5, mixes on 6, and stands on 7. The table is symmetric in Player 1 and Player 2. Similar entries are shaded similarly for readability.

Player 1’s Third-Card Value | Player 2’s Third-Card Value (10 = Stand, 11 = Natural) | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | |

0 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 3 | 3 | 3 | 5 | 3 |

1 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 3 | 3 | 5 | 3 |

2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 4 | 3 | 3 | 5 | 4 |

3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 4 | 4 | 3 | 5 | 4 |

4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 4 | 4 | 5 | 5 |

5 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | 5 | 4 | 5 | 5 |

6 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 5 | 4 | 5+ | 6 |

7 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 3 | 3 | 6 | 6 |

8 | 3 | 3 | 3 | 4 | 4 | 5 | 5 | 3 | 2 | 3 | 5 | 2 |

9 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 3 | 3 | 3 | 5 | 3 |

10 | 5 | 5 | 5 | 5 | 5 | 5 | 5+ | 6 | 5 | 5 | 5 | 5 |

11 | 3 | 3 | 4 | 4 | 5 | 5 | 6 | 6 | 2 | 3 | 5 |

_{*}, 1/2].

_{0}< θ

_{1}< θ

_{2}< … θ

_{109}= θ

_{8}< θ

_{110}= 1/2 such that the correlated cooperative equilibrium is a rational function of θ on interval i, namely (θ

_{i−1}, θ

_{i}), for i = 1, 2, …, 110. Each θ

_{i}is a root of a polynomial of degree 4 or less. At the boundary points, discontinuities occur in Banker’s strategy.

_{11}(θ) = 0; for intervals 42–46, 102–103, and 107–110, p

_{00}(θ) = 0; for internals 61-66, p

_{10}(θ) = 0; and for all remaining intervals the Players have all strategies active. In all cases, despite the correlated cooperative equilibrium being nonunique in mixed strategies, it is unique in behavioral strategies. (This can be proved algebraically.) However, there are exceptions. At each boundary point, the solutions from both adjacent intervals apply, so there is nonuniqueness of Banker behavioral strategies at the 109 such θ.

**Figure 1.**The graph of the value of the game to the Players (or minus the value to Banker), assuming the correlated cooperative equilibrium.

## 4. Independent Cooperative Equilibrium

**Proposition 1**Given $n\ge 2$, let

**A**be the payoff matrix for a $4\times n$ matrix game, with the additional constraint that the row player is required to use a mixed strategy of the form

**Proof.**Equation (21) is by definition. The value of the unconstrained game is, by the minimax theorem,

_{4}as over Δ

_{2}× Δ

_{2}because the latter contains the extreme points of the former (namely (0, 0, 0, 1), (0, 0, 1, 0), (0, 1, 0, 0), (1, 0, 0, 0).

**Remark 1.**A mixed strategy

**p**for the row player that achieves the maximum in Equation (21) is called a maximin strategy, and it assures the row player of an expected gain of at least v. A mixed strategy

**q**for the column player that achieves the minimum in the center or on the right side of Equation (23) is called a minimax strategy, and it assures the column player of an expected loss of at most $\stackrel{\_}{v}$.

**Figure 2.**The graphs of the Players’ strategies ${p}_{1}$ and ${p}_{2}$ in the independent cooperative equilibrium, restricted to $\theta \in (0,1/4]$. (${p}_{1}=9/11$ for $0.241681<\theta <0.432877$ and ${p}_{2}=9/11$ for $0.161238<\theta <0.495084$, approximately.) ${p}_{1}$ has 13 discontinuities on $(0,1/2]$, whereas ${p}_{2}$ has 11 discontinuities.

**Figure 3.**The graph of the difference between the upper and lower value functions, multiplied by ${10}^{5}$, in the independent cooperative equilibrium.

## 5. Nash Equilibrium

_{1}and

**b**(9/11, p

_{2})q

^{⊺}is constant in p

_{2}. This would ensure that p

_{1}= 9/11 (in fact any strategy p

_{1}of Player 1) is a best response to p

_{2}= 9/11 and

**q**; similarly, p

_{2}= 9/11 (in fact any p

_{2}) is a best response to p

_{1}= 9/11 and

**q**. And of course

**q**is automatically a best response to p

_{1}= p

_{2}9/11. A necessary and sufficient condition on

**q**is

_{j}≥ 0 for j = 1, 2, …, 8. Summing the three equations gives q

_{6}+ q

_{7}+ q

_{8}= 1 − q

_{1}− q

_{2}− q

_{3}− q

_{4}− q

_{5}, so any such

**q**is automatically a probability vector.

1. | (0, 15175619=33313280, 15175619=33313280, 0, 0, 0, 0, 1481021=16656640). |

2. | (0, 1=2, 4229827=11421696, 0, 0, 0, 1481021=11421696, 0). |

3. | (0, 4229827=11421696, 1=2, 0, 0, 1481021=11421696, 0, 0). |

4. | (0, 4705731=12373504, 4705731=12373504, 0, 1481021=6186752, 0, 0, 0). |

5. | (0, 3753923=10469888, 3753923=10469888, 1481021=5234944, 0, 0, 0, 0). |

6. | (15175619=21891584, 0, 0, 0, 0, 0, 0, 6715965=21891584). |

7. | (1988135=3331328, 0, 0, 0, 0, 1343193=6662656, 1343193=6662656, 0). |

8. | (1568577=3807232, 0, 0, 0, 2238655=3807232, 0, 0, 0). |

9. | (3753923=10469888, 0, 0, 6715965=10469888, 0, 0, 0, 0). |

10. | (4229827=10945792, 0, 6715965=21891584, 0, 0, 6715965=21891584, 0, 0). |

11. | (4229827=10945792, 6715965=21891584, 0, 0, 0, 0, 6715965=21891584, 0). |

_{1}, r

_{2}, r

_{2}), and there are only two extreme points, namely

**A**and

**B**for Players 1 and 2 (now $4\times 4$), we can argue as above, and this leads to two extreme equilibria, which have the same Banker behavioral strategies. The left endpoint of the interval is

_{1}; it suffices that it be maximized at p

_{1}= 0.

**Figure 4.**The graphs of the Players’ strategies ${p}_{1}$ and ${p}_{2}$ in the Nash equilibrium, restricted to $\theta \in (0,1/4]$. (${p}_{1}={p}_{2}=9/11$ for $\theta >5772/33847\approx 0.170532$.) Both ${p}_{1}$ and ${p}_{2}$ have a unique discontinuity, at ${\theta}_{31}\approx 0.0844782$.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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Ethier, S.N.; Lee, J.
On the Three-Person Game Baccara Banque. *Games* **2015**, *6*, 57-78.
https://doi.org/10.3390/g6020057

**AMA Style**

Ethier SN, Lee J.
On the Three-Person Game Baccara Banque. *Games*. 2015; 6(2):57-78.
https://doi.org/10.3390/g6020057

**Chicago/Turabian Style**

Ethier, Stewart N., and Jiyeon Lee.
2015. "On the Three-Person Game Baccara Banque" *Games* 6, no. 2: 57-78.
https://doi.org/10.3390/g6020057