The Monty Hall Problem, Part 4

Hopefully, we all agree that we should switch when faced with the problem given in the basic formulation of the Monty Hall problem. However, in the alternative formulation given in my first entry, there's one major difference. In the original game, the host was obliged to reveal a second door after watching you picking one. In my version of the game, I hadn't made any such commitment. So, why would I give you a chance to change your mind? Possibly out of generosity, sure, but most probably because I knew you had made the right choice and wanted you to switch to an empty cup.

Put in game-theory terms, switching is a dominated strategy. Your strategy choices are to switch when given the opportunity or to never switch (switch / stay). I have more strategy choices than you do. This is a full payoff matrix of the game for all possible strategy choices, with your choices represented as columns and my choices as rows. The outcome values are the probabilities of you winning the bill.


My strategy choice "Yes-No", for example, means that I offer you an opportunity to switch if you choose the right cup initially, but I don't offer you that opportunity if you choose the wrong one. So the first Yes or No refers to whether I offer you that opportunity when you choose the right cup, and the second one to the case when you choose the wrong one.

Notice that all of my strategy choices but "Yes-No" (offering the opportunity to switch only when you've picked the right cup) are dominated. This means that they can never lead to better results, only worse, depending on your strategy choice. So there is no reason for me to choose any of those strategies. Thus removing those strategy choices, we get a much simpler payoff matrix:


It should now be obvious that switching is a dominated strategy. So in a game-theory sense, switching is a bad strategy. However, game theory isn't everything. Maybe you have a "read" on me, making you believe that I want you to have the bill. Maybe you think that I intended to always give you the switching opportunity as in the original Monty Hall problem. So there may be reasons to deviate from game-theory optimal play. But lacking such guidance, you're probably better off resorting to game theory, in this case guaranteeing you a 1/3 chance to win the prize.


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