# 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

**Lemma**

**2.**

## 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.**

## References

- Selvin, S. A problem in probability. Am. Stat.
**1975**, 29, 67. [Google Scholar] - Selvin, S. On the Monty Hall problem. Am. Stat.
**1975**, 29, 134. [Google Scholar] - Nalebuff, B. Puzzles. J. Econ. Perspect.
**1987**, 1, 157–163. [Google Scholar] [CrossRef] - Gardner, M. The 2nd Scientific American Book of Mathematical Puzzles and Diversions; Simon and Schuster: New York, NY, USA, 1961. [Google Scholar]
- Vos Savant, M. Ask Marilyn. Parade Magazine, 9 September 1990; 15. [Google Scholar]
- Vos Savant, M. Ask Marilyn. Parade Magazine, 2 December 1990; 25. [Google Scholar]
- Vos Savant, M. Marilyn vos Savant’s reply. Am. Stat.
**1991**, 45, 347–348. [Google Scholar] - Tierney, J. Behind Monty Hall’s doors: Puzzle, debate and answer. The New York Times, 21 July 1991; 1. [Google Scholar]
- Franco-Watkins, A.M.; Derks, P.L.; Dougherty, M.R.P. Reasoning in the Monty Hall problem: Examining choice behaviour and probability judgements. Think. Reason.
**2003**, 9, 67–90. [Google Scholar] [CrossRef] - Friedman, D. Monty Hall’s three doors: Construction and deconstruction of choice anomaly. Am. Econ. Rev.
**1998**, 88, 933–946. [Google Scholar] - Petrocelli, J.V.; Harris, A.K. Learning inhibition in the Monty Hall Problem: The role of dysfunctional counterfactual prescriptions. Personal. Soc. Psychol. Bull.
**2011**, 37, 1297–1311. [Google Scholar] [CrossRef] [PubMed] - Saenen, L.; Heyvaert, M.; van Dooren, W.; Onghena, P. Inhibitory control in a notorious brain teaser: The Monty Hall dilemma. Think. Reason.
**2015**, 21, 176–192. [Google Scholar] [CrossRef] - Saenen, L.; van Dooren, W.; Onghena, P. A randomized Monty Hall experiment: The positive effect of conditional frequency feedback. ZDM Math. Educ.
**2015**, 47, 837. [Google Scholar] [CrossRef] - Tubau, E.; Aguilar-Lleyda, D.; Johnson, E.D. Reasoning and choice in the Monty Hall Dilemma (MHD): Implications for improving Bayesian reasoning. Front. Psychol.
**2015**, 6, 353. [Google Scholar] [CrossRef] [PubMed][Green Version] - Lecoutre, M.P. Cognitive models and problem spaces in ‘purely random’ situations. Educ. Stud. Math.
**1992**, 23, 557–568. [Google Scholar] [CrossRef] - Granberg, D.; Dorr, N. Further exploration of two stage decision making in the Monty Hall dilemma. Am. J. Psychol.
**1998**, 111, 561–579. [Google Scholar] [CrossRef] - Herbranson, W.T.; Schroeder, J. Are birds smarter than mathematicians ? Pigeons (Columba livia) perform optimally on a version of the Monty Hall Dilemma. J. Comp. Psychol.
**2010**, 12, 1–13. [Google Scholar] [CrossRef] [PubMed] - Granberg, D.; Brown, T.A. The Monty Hall dilemma. Personal. Soc. Psychol. Bull.
**1995**, 21, 711–723. [Google Scholar] - Fernandez, L.; Piron, R. Should she switch ? A game-theoretic analysis of the Monty Hall problem. Math. Mag.
**1999**, 72, 214–217. [Google Scholar] [CrossRef] - Mueser, P.; Granberg, D. The Monty Hall Dilemma Revisited: Understanding the Interaction of Problem Definition and Decision Making; Mimeo, The University of Missouri: Columbia, MO, USA, 1999. [Google Scholar]
- Bailey, H. Monty Hall uses a mixed strategy. Math. Mag.
**2000**, 73, 135–141. [Google Scholar] [CrossRef] - Gill, R.D. The Monty Hall problem is not a probability puzzle (It’s a challenge in mathematical modelling). Stat. Neerl.
**2011**, 65, 58–71. [Google Scholar] [CrossRef] - Fudenberg, D.; Tirole, J. Game Theory; MIT Press: Cambridge, MA, USA, 1991. [Google Scholar]
- Page, S.E. Let’s Make a Deal. Econ. Lett.
**1998**, 61, 175–180. [Google Scholar] [CrossRef] - Segal, L. Letters to the editor. The New York Times. 16 August 1991. Available online: http://www.nytimes.com/1991/08/11/opinion/l-suppose-you-had-100-doors-with-goats-behind-99-of-them-624691.html (accessed on 25 July 2017).
- Chen, W.; Wang, J.T. Epiphany Learning for Bayesian Updating: Overcoming the Generalized Monty Hall Problem; Mimeo, National Taiwan University: Taipei, Taiwan, 2010; Available online: http://homepage.ntu.edu.tw/~josephw/EpiphanyMonty_20101207.pdf (accessed on 2 June 2017).
- Miller, J.B.; Sanjurjo, A. A Bridge from Monty Hall to the (Anti-)Hot Hand: Restricted Choice, Selection Bias, and Empirical Practice; Working Paper; IGIER: Milan, Italy, 2015; Available online: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2709837 (accessed on 5 June 2017).

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: ${S}_{A}=\left\{s,k\right\}\times \left\{{k}^{\prime}\right\}$. 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: ${p}_{1}=1-{\sum}_{i=2}^{6}{p}_{i}$. |

© 2017 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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