Intentional Probability: The Impact of Host's Intentions on Winning a Game Show

[N123]

Consider this variation of a well-known problem. As a contestant on a game show, you choose one of three doors. One door has a prize behind it, the other two have nothing. The host knows which one has the prize. The host opens a door and reveals what is behind it. If it has nothing, should you change your answer to the remaining door?
To make it more concrete, let's say you choose door #1, the host opens door #2 and it is empty. Should you switch to door #3?
Here, the advantage of switching depends on (your a-priori knowledge of) how helpful or malicious the host is. For example:
1. Completely helpful: for example, you know that the host will reveal the door with the prize if you didn't choose it. Since he did not, your door has the prize and you should not switch. (If he had, obviously you should choose to the open door.)
2. Completely malicious: you know that the host will always deliberately choose a door that does NOT have a prize. Since your door (#1) initially had a probability of 1/3 and #2 and #3 together had 2/3; and now you know that #2 does not have the prize, switch to door #3 which has double the probability of having the prize.
3. Random: the host picks the door randomly and #2 happens to not have the prize. The probability is 1/2 each that your door (#1) and the remaining door (#3) have the prize. Switch or not, makes no difference.

What is fascinating to me is that these probabilities depend on your knowledge of the host's intentions. Two people with different understanding or ideas about what the host is thinking, will compute different probabilities. Assume that the host is god-like and cannot be questioned or approached. Who is right?

Is this a legitimate line of thinking that can be or has been developed into a branch in mathematics? Perhaps already a part of game theory?

 

[chiro]

Hey N123.

I think if you want to show evidence of intent then you should show evidence that a person is trying to maximize or minimize likelihood given certain information.

Obviously though the likelihood and information functions are arbitrary but the idea itself has a very precise meaning (mathematically speaking - under optimization).

If you can construct a test statistic and do inference then you can put this on a firmer foundation.

 

[Dale]

Run of the mill p-values also depend on intentions:

http://www.indiana.edu/~kruschke/articles/Kruschke2010WIRES.pdf

 

[pbuk]

Is this a legitimate line of thinking that can be or has been developed into a branch in mathematics? Perhaps already a part of game theory?

The Monty Hall problem and its variations are well understood and have been thoroughly discussed here and elsewhere but I hardly think it could be considered a "branch of mathematics".

 

[FactChecker]

That is an aspect of the "Monte Hall" problem that is not often mentioned. The typical answer assumes that Monte Hall will never open the door with the prize. In that case, his action has given you information that changes the probabilities and makes it better for you to switch. If Monte Hall has no knowledge and randomly picks the door to open, then it doesn't help you to switch (it doesn't hurt either, so you might as well switch.)

 

[mathman]

If Monte Hall [icks a door at random, 1/3 of the time he will open the door with the prize.

 

[FactChecker]

If Monte Hall [icks a door at random, 1/3 of the time he will open the door with the prize.

Exactly. But then, the fact that there was no prize displayed does not bias the result toward the door that you did not pick and he did not open. In the standard problem, it is Monte's possibly intentional avoidance ot the third door that creates the bias.

 

[N123]

The Monty Hall problem and its variations are well understood and have been thoroughly discussed here and elsewhere but I hardly think it could be considered a "branch of mathematics".

It's a little more than that. The Monte Hall problem was just an example. I almost regret giving it.
The idea is to mix up random and "non-random" (intentional) events. Intent / free will is a bit controversial, so let's say "events with unpredictable probability distribution."

Dalespam: thanks for the link! Still reading.

 

[FactChecker]

It's a little more than that. The Monte Hall problem was just an example. I almost regret giving it.
The idea is to mix up random and "non-random" (intentional) events. Intent / free will is a bit controversial, so let's say "events with unpredictable probability distribution."

Dalespam: thanks for the link! Still reading.

The Monte Hall problem is an example of the influence of intentional actions on a random event. Monte intentionally avoids opening a door with the prize. Bayesian statistics covers it well. Even if the event has already happened, but the result is unknown, Bayesian statistics applies. The theory of how to adjust probabilities based on new information or on dependent events is already well developed. There is also the well established subject of game theory, which includes studying very sophisticated strategies of opponents in a game.

 

[mathman]

The Monte Hall question can be simply analyzed by examining all possible scenarios. Pick door 1. Three equally probable events, prize behind door 1, prize behind door 2, prize behind door 3.
Case 1 - prize behind door 1, another door is opened - switch and lose.
Case 2 - prize behind door 2, door 3 is opened - switch and win.
Case 3 - prize behind door 3, door 2 is opened - switch and win.

 

[FactChecker]

The Monte Hall question can be simply analyzed by examining all possible scenarios. Pick door 1. Three equally probable events, prize behind door 1, prize behind door 2, prize behind door 3.
Case 1 - prize behind door 1, another door is opened - switch and lose.
Case 2 - prize behind door 2, door 3 is opened - switch and win.
Case 3 - prize behind door 3, door 2 is opened - switch and win.

Yes. Because of the intentional avoidance of opening a door with the prize. Your answer depends on that. I think that is what the OP is referring to. Another possible game strategy is if Monte is a hostile opponent who will only offer a switch when he knows you have picked the prize door. Or he could randomly do that enough to eliminate your advantage of switching. For all I know, Monte might be doing that. (I wonder if anyone has kept records and studied that.) It's a nice example of game theory.

 

[mathman]

The Monte Hall game show on television was as originally described. He opened a door showing it did not have the prize and gave the contestant the option of switching. Other variations in various game shows have been used and require different analyses.

 

[FactChecker]

The Monte Hall game show on television was as originally described. He opened a door showing it did not have the prize and gave the contestant the option of switching. Other variations in various game shows have been used and require different analyses.

I never watched it enough to know. Does he always allow the contestant to switch after opening a door? Or does he only do it sometimes? The second case allows Monte to select which times he offers a switch and can have completely different probabilities, depending on how he makes that selection.

 

[mathman]

I never watched it enough to know. Does he always allow the contestant to switch after opening a door? Or does he only do it sometimes? The second case allows Monte to select which times he offers a switch and can have completely different probabilities, depending on how he makes that selection.

The best of my recollection is the switch option was always given.