The Monty Hall Problem is not a Probability Puzzle (it's a challenge in mathematical modelling)

TL;DR

This paper challenges the common probabilistic explanation of the Monty Hall problem, emphasizing the importance of game theory and mathematical modeling over simple probability calculations.

Contribution

It argues that the Monty Hall problem is better understood through game theory and mathematical modeling rather than traditional probability reasoning.

Findings

The classic probability solution is solution-driven and relies on assumptions.

Game theory provides a more robust explanation for the problem.

Switching doors maximizes the player's advantage based on minimax principles.

Abstract

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, ``Do you want to pick door No. 2?'' Is it to your advantage to switch your choice? The answer is ``yes'' but the literature offers many reasons why this is the correct answer. The present paper argues that the most common reasoning found in introductory statistics texts, depending on making a number of ``obvious'' or ``natural'' assumptions and then computing a conditional probability, is a classical example of solution driven science. The best reason to switch is to be found in von Neumann's minimax theorem from game theory, rather than in Bayes' theorem.