fghjgfu comments on Open Thread February 25 - March 3 - Less Wrong

8 Post author: Coscott 25 February 2014 04:57AM

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Comment author: [deleted] 25 February 2014 06:20:23PM *  6 points [-]

A little bit of How An Algorithm Feels From Inside:

Why is the Monty Hall problem so horribly unintuitive? Why does it feel like there's an equal probability to pick the correct door (1/2+1/2) when actually there's not (1/3+2/3)?

Here are the relevant bits from the Wikipedia article:

Out of 228 subjects in one study, only 13% chose to switch (Granberg and Brown, 1995:713). In her book The Power of Logical Thinking, vos Savant (1996, p. 15) quotes cognitive psychologist Massimo Piattelli-Palmarini as saying "... no other statistical puzzle comes so close to fooling all the people all the time" and "that even Nobel physicists systematically give the wrong answer, and that they insist on it, and they are ready to berate in print those who propose the right answer." Interestingly, pigeons make mistakes and learn from mistakes, and experiments show that they rapidly learn to always switch, unlike humans (Herbranson and Schroeder, 2010).

[...]

Although these issues are mathematically significant, even when controlling for these factors nearly all people still think each of the two unopened doors has an equal probability and conclude switching does not matter (Mueser and Granberg, 1999). This "equal probability" assumption is a deeply rooted intuition (Falk 1992:202). People strongly tend to think probability is evenly distributed across as many unknowns as are present, whether it is or not (Fox and Levav, 2004:637). Indeed, if a player believes that sticking and switching are equally successful and therefore equally often decides to switch as to stay, they will win 50% of the time, reinforcing their original belief. Missing the unequal chances of those two doors, and in not considering that (1/3+2/3) / 2 gives a chance of 50%, similar to "the little green woman" example (Marc C. Steinbach, 2000).

The problem continues to attract the attention of cognitive psychologists. The typical behaviour of the majority, i.e., not switching, may be explained by phenomena known in the psychological literature as: 1) the endowment effect (Kahneman et al., 1991); people tend to overvalue the winning probability of the already chosen – already "owned" – door; 2) the status quo bias (Samuelson and Zeckhauser, 1988); people prefer to stick with the choice of door they have already made. Experimental evidence confirms that these are plausible explanations which do not depend on probability intuition (Morone and Fiore, 2007).

Those bias listed in the last paragraph maybe explain why people choose not to switch the door, but what explains the "equal probability" intuition? Do you have any insight on this?

Comment author: gwern 25 February 2014 06:55:41PM 7 points [-]

Another datapoint is the counterintuitiveness of searching a desk: with each drawer you open looking for something, the probability of finding it in the next drawer increases, but your probability of ever finding it decreases. The difference seems to whipsaw people; see http://www.gwern.net/docs/statistics/1994-falk

Comment author: [deleted] 04 March 2014 07:52:52PM *  1 point [-]

A bit late, but I think this part of your article was most relevant to the Monty Hall problem:

Our impression is that subjects’ conservatism, as revealed by the prevalence of the constancy assumptions, is a consequence of their external attribution of uncertainty (Kahneman & Tversky, 1982). The parameters L0 and/or S0 are apparently perceived as properties that belong to the desk, like color, size and texture. Subjects think of these parameters in terms of “the probabilities of the desk”, whereas the Bayesian view would imply expressions like “my probability of the target event”. Thus, subjects fail to incorporate the additional knowledge they acquire when given successive search results.

People probably don't distinguish between their personal probability of the target event and the probabilities of the doors. It feels like the probability of there being a car behind the doors is a parameter that belongs to those doors or to the car - however you want to phrase it. Since you're only given information about what's behind the doors, and that information can't actually change the reality of what's behind the doors then it feels like the probability can't change just because of that.

Comment author: lmm 25 February 2014 11:09:27PM 2 points [-]

I think the monty hall problem very closely resembles a more natural one in which the probability is 1/2; namely, that where the host is your opponent and chose whether to offer you the chance to switch. So evolutionarily-optimized instincts tell us the probability is 1/2.

Comment author: DanielLC 27 February 2014 11:14:18PM 2 points [-]

I'd say it's that it closely resembles the one where the host has no idea which door has the car in it, and picks a door at random.

Comment author: Coscott 26 February 2014 01:17:31AM 2 points [-]

I do not think this is correct. First, the host should only offer you the chance to switch if you are winning, so the chance should be 0. Second, this example seems too contrived to be something that we would have evolved a good instinct about.

Comment author: drethelin 26 February 2014 06:07:04AM 0 points [-]

Unless they're trying to trick you. The problem collapses to a yes or no question of whether one of you is able to guess the level the other one of you is thinking on

Comment author: Coscott 26 February 2014 06:10:46AM 2 points [-]

Um, no, the only Nash equilibria are where you never accept the deal. If you ever accept it at all, then they will only offer it when it hurts you.

Comment author: [deleted] 26 February 2014 03:57:12PM 0 points [-]

I'd probably broaden this beyond 1/2 - I think the base case is the host gives you a chance to gamble with a question or test of skill, and the result is purely dependent on the player. The swap-box scenario is then an extreme case of that where the result depends less and less on the skill of the player, eventually reaching 50% chance of winning. I wouldn't say evolutionary-optimised, but maybe familiarity with the game-show tropes being somewhere along this scale.

Monty Hall is then a twist on this extreme case, which pattern-matches to the more common 50% case with no allowance for the effect of the host's knowledge.

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