We Don't Need No Stinkin' Bayesian Up-Dating!
1 April 2009The Classic Monty Hall Problem
Andy is a contestant in a game. In this game, each contestant makes a choice amongst three tags. Each tag is committed to an outcome, with the commitment concealed from each contestant. Two outcomes are undesirable; one is desirable. Nothing reveals a pattern to assignments.
After Andy makes his choice, it is revealed to him that a specific tag that he did not choose is committed to an undesirable outcome. Andy is offered a chance to change his selection. Should he change?
Three Contestants
Andy, Barb, and Pat are contestants in a game. In this game, each contestant makes an independent choice amongst three tags. Each tag is committed to an outcome, with the commitment concealed from each contestant. Two outcomes are undesirable; one is desirable. Nothing reveals a pattern to assignments. In the event that multiple players select the same tag, outcomes are duplicated.
After all contestants make their choices, it is revealed that Andy, Barb, and Pat have selected tags each different from those of the other two contestants. And it is revealed that Pat's tag is associated with an undesirable outcome. Andy and Barb are each offerd a chance to change their selections. What should each do?
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I'd be interested to see the circumstances under which they would each want to exchange places.
It could be any or all of the following reasons: a) They both being firm believers that the grass is always greener; b) They both having reason to believe that their luck is at least as bad as Pat's; c) Their ending up in an "'After you,' and 'I insist,'" loop until Monty had a nervous breakdown which may or may not have lead directly to c) Pat's prize involving a chainsaw wielding madman.
Given the interest of probability theorists in problems of this sort, and the pronouncements of some of those theorists, there is every possibility that one of them might be running-around with a chain-saw.
In any case, a fair share of those who would switch would be people dogmatically persuaded by the standard analysis of the classic Monty Hall Problem. The three-player problem above is constructed such that Andy is in the same epistemological position as in the classic problem, and Barb's position is isomorphic to that of Andy.