Abstract
The Monty Hall problem is a classic puzzle that, in addition to intriguing the general public, has stimulated research into the foundations of reasoning about uncertainty. A key insight to understanding the Monty Hall problem is to realize that the specification of the behavior of the host (i.e. Monty) of the game is fundamental. Here we go one step further and reason, in Bayesian way, in terms of epistemic uncertainty about the behavior of host, assuming subjective probabilities.
We also consider several generalizations of the classic Monty Hall problem considering different priors for the doors, several doors instead of three, and different ways the host can choose which door to open when several are possible. We show that in these generalized versions, the player faces a sequential decision problem, since the choice of the first door is key. We provide a general solution for the most general case using decision trees and determine the optimal policy.
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Notes
- 1.
A possible extension of this work could investigate the use of Bayesian hierarchical models, adopting prior distributions on the mixture’s parameters.
- 2.
Indeed the original statement of the MHP concerns the specific decision of what to do when offered the possibility of switching.
- 3.
Rosenhouse [11] also addresses the case where the car is not placed behind the doors with equal probability, but assuming the fixed “always open” protocol for Monty.
- 4.
In this generalized model, switching occurs when the choice at node F is a different door from the one chosen at the root S.
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Manfredotti, C., Viappiani, P. (2021). A Bayesian Interpretation of the Monty Hall Problem with Epistemic Uncertainty. In: Torra, V., Narukawa, Y. (eds) Modeling Decisions for Artificial Intelligence. MDAI 2021. Lecture Notes in Computer Science(), vol 12898. Springer, Cham. https://doi.org/10.1007/978-3-030-85529-1_8
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