[Paper Review] Dominance in the Monty Hall Problem

Alexander Gnedin|arXiv (Cornell University)|Jun 4, 2011

Optimization and Search Problems8 references3 citations

TL;DR

This paper establishes that always-switching strategies weakly dominate all other strategies in the Monty Hall problem, regardless of prior beliefs about the host's behavior or the prize location. The dominance result allows for optimal decision-making without assuming a prior distribution, providing a foundation for both Bayesian and minimax approaches.

ABSTRACT

Elementary decision-theoretic analysis of the Monty Hall dilemma shows that the problem has dominance. This makes possible to discard nonswitching strategies, without making any assumptions on the prior distribution of factors out of control of the decision maker. A path to the Bayesian and the minimax decision-making environments is then straightforward.

Motivation & Objective

• To establish a decision-theoretic foundation for the Monty Hall problem that does not rely on prior distributions.
• To demonstrate that the always-switching strategy dominates all other strategies in terms of win probability.
• To extend the dominance result to Bayesian and minimax decision-making frameworks.
• To formalize the concept of 'strategy' and 'situation' in the context of the Monty Hall problem using decision-theoretic tools.
• To show that the minimax winning probability is (n−1)/n for n ≥ 3 doors, achieved when both the player and prize are uniformly distributed.

Proposed method

• Formalizing a strategy as a pair (x, aₓ), where x is the initially chosen door and aₓ is a function mapping the revealed door y to an action (match or switch).
• Defining the payoff function W(θ, x, aₓ) as an indicator of success: 1 if the action matches the true winning door θ.
• Proving weak dominance by showing that for any non-always-switching strategy, there exists an always-switching strategy that performs at least as well in all scenarios.
• Using the admissibility condition that dₜ(x) ≠ θ when x = θ and dₜ(x) = θ when x ≠ θ to ensure logical consistency of the host’s behavior.
• Applying the dominance result to the Bayesian setting by minimizing pₓ over x, where pₓ is the probability that the prize is not at x.
• Deriving the minimax value v = (n−1)/n by assuming uniform priors on the prize location and showing that any always-switching strategy achieves this bound.

Experimental results

Research questions

• RQ1 Can the 'switch or not' decision in the Monty Hall problem be justified without assuming a prior distribution over the prize location?
• RQ2 Is there a strategy that dominates all others in the Monty Hall problem regardless of the host's behavior or the prior distribution?
• RQ3 What is the minimax probability of winning in the generalized Monty Hall problem with n ≥ 3 doors?
• RQ4 How does the dominance result extend to Bayesian and zero-sum game-theoretic frameworks?
• RQ5 Under what conditions is the always-switching strategy optimal in terms of minimizing expected loss or maximizing worst-case win probability?

Key findings

• The class of always-switching strategies weakly dominates all other strategies in the Monty Hall problem, regardless of the host's behavior or prior beliefs.
• For any strategy that sometimes matches (does not switch), there exists an always-switching strategy that performs at least as well in every possible scenario.
• In the Bayesian setting, the optimal strategy is (θ*, switch), where θ* minimizes the prior probability pₜ, and the win probability is 1 − pₓ.
• The minimax winning probability is (n−1)/n, achieved when the prize is uniformly distributed over n ≥ 3 doors and the player uses any always-switching strategy.
• In the zero-sum game formulation, every saddle-point solution involves a uniformly distributed player choice X and an always-switching strategy, with the game value v = (n−1)/n.
• The value v = (n−1)/n is achieved regardless of the distribution of the revealed door Y, as long as the host's behavior is admissible.