# The Monty Hall Problem as a Bayesian Game

## Abstract

**:**

## 1. Introduction

## 2. The Model

- Reveal a goat behind one of the unselected doors, what we shall call “reveal” $\left(r\right)$.
- Not reveal a goat $\left(h\right)$.

- Switch $\left(s\right)$.
- Not switch (or keep) $\left(k\right)$.

**Incomplete Information**:

- Case 1:
- This is what we will call the
**Sympathetic Case**: Monty and Amy’s preferences are aligned in the sense that all else equal Monty would prefer that Amy be correct rather than incorrect. - Case 2:
- We call this the
**Antipathetic Case**: Here, all else equal, Monty would prefer that Amy be incorrect. Their preferences are not aligned.

#### 2.1. Case 1, Sympathetic

#### 2.2. Case 2, Antipathetic

## 3. Uncertainty about Monty’s Motives

**Lemma**

**3.**

**Lemma**

**4.**

#### Generalizing to n Doors

**Lemma**

**5.**

**Lemma**

**6.**

**Corollary**

**1.**

**Theorem**

**1.**

**Lemma**

**7.**

**Corollary**

**2.**

**Proof.**

**Theorem**

**2.**

## 4. Conclusions

## Acknowledgments

## Conflicts of Interest

## Appendix A.

#### Appendix A.1. Lemma 1 Proof

**Proof.**

#### Appendix A.2. Lemma 2 Proof

**Proof.**

#### Appendix A.3. Corollary 1 Proof

**Proof.**

#### Appendix A.4. Theorem 2 Proof

**Proof.**

## Appendix B. Generalization of the Model

#### Appendix B.1. Pure Strategy Equilibria

**Lemma**

**A1.**

**Theorem**

**A1.**

**Lemma**

**A2.**

**Proof.**

**Theorem**

**A2.**

#### Appendix B.2. Mixed Strategy Equilibria

**Lemma**

**A3.**

**Theorem**

**A3.**

**Corollary**

**A1.**

#### Appendix B.3. Theorem A3 Proof

**Lemma**

**A4.**

**Proof.**

**Lemma**

**A5.**

**Proof.**

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1. | |

2. | See, for example, Fudenberg and Tirole (1991) [23]. |

3. | Since a strategy for a player must specify an action at every information set encountered by a player, Amy’s strategy should, strictly speaking, be: ${S}_{A}=\left(\right)open="\{"\; close="\}">s,k$. Our simplification of this; however, is clearer and does not affect the analysis. |

4. | In [21], Bailey references a humorous letter to the New York Times editor [25] by the American author Lore Segal concerning this very assumption. Segal writes, “Your front-page article 21 July on the Monty Hall puzzle controversy neglects to mention one of the behind-the-door options: to prefer the goat to the auto. The goat is a delightful animal, although parking might be a problem”. |

5. | Presumably, the tension engendered by Amy’s decision on whether or not to switch is attractive to the audience of the show, and Monty recognizes this. |

6. | For derivation of this, see Appendix A.1. |

7. | For derivation of this, see Appendix A.2. |

8. | For derivation of this, see Appendix A.3. |

9. | In [24] Monty always reveals the goats, though it is unclear whether the subjects know that this was a mandatory action. |

10. | A reformulation of Bayes’ Rule in odds form. |

11. | Each pair separated by semicolons refers to a type ${p}_{i}$’s strategy. These can be obtained by reading off the top rows of Figure A1. |

12. | ${p}_{4}$ and ${p}_{6}$ need not be listed, as they are fully determined by their respective submeasures, which we have listed. Of course, one of our listed measures is redundant, e.g., we could write ${p}_{1}$ in terms of the other measures: ${p}_{1}=1-{\sum}_{i=2}^{6}{p}_{i}$. |

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**MDPI and ACS Style**

Whitmeyer, M.
The Monty Hall Problem as a Bayesian Game. *Games* **2017**, *8*, 31.
https://doi.org/10.3390/g8030031

**AMA Style**

Whitmeyer M.
The Monty Hall Problem as a Bayesian Game. *Games*. 2017; 8(3):31.
https://doi.org/10.3390/g8030031

**Chicago/Turabian Style**

Whitmeyer, Mark.
2017. "The Monty Hall Problem as a Bayesian Game" *Games* 8, no. 3: 31.
https://doi.org/10.3390/g8030031