Monty Hall Variant (1 Viewer)

[seanieg89]

Many of you will have heard of the Monty Hall problem, presented below:

1. There are three briefcases, two are empty and one contains one million dollars. Monty knows which one contains the prize.
2. The contestant chooses a case but does not open it.
3. Monty opens an empty case out of the two remaining cases (if they are both empty, he chooses the case to open at random.)
4. Monty offers you the choice to swap you case for the last unopened case.

Is it in your best interest to do so? What is your expected prize if you do/don't switch cases?

(If you do not know the answer, I will not spoil it here...view it as part A of the question.)


Now consider the following variant:

Monty has decided he is losing too much money by always offering this swap. He decides to only offer it some of the time.

Monty offers you a swap with probability p if your first choice is correct and offers you a swap with probability q if your first choice is incorrect.

Suppose you as the contestant decide to accept the swap with probability r.


For a given pair (p,q), which r maximises your expected prize and what is this expected prize in terms of p and q?

Hence find all optimal choices of p and q for Monty if his first priority is minimising the maximum expected prize of contestants (as he is paranoid about his assistants leaking the values of p and q) and his second priority is offering the switch as often as possible (as this is after all a game show and the swap adds entertainment). Remember that the first priority trumps the second priority though!

For these optimal choices of (p,q), what is the expected prize of the contestant?

 

[obliviousninja]

21

 

[braintic]

There is one more requirement for the original Monty Hall problem - the contestant knows Monty's system.

 

[seanieg89]

There is one more requirement for the original Monty Hall problem - the contestant knows Monty's system.

Of course, I meant the question to read as if you (the reader) are the contestant, and hence know everything that has been written.

 

[nerdasdasd]

Of course, I meant the question to read as if you (the reader) are the contestant, and hence know everything that has been written.

Hmm are the probabilities, 50%, 50%, and 0?....

 

[seanieg89]

Forgot about this question. Got around to looking at it and it turned out to be rather straightforward:

If we let the prize be 1 for simplicity, the expectation for a given (p,q,r) is

E=1/3 + r(2q/3 - p/3)

by straightforward calculation.

If 2q > p then the optimal r is 1, which improves expectation from a 1/3. Otherwise the best expectation the contestant can attain is 1/3.

So we have to maximise the frequency of swaps (ie maximise p/3 + 2q/3) subject to the constraint 2q =< p.

This maximum is 2/3. achieved by the unique strategy (p,q)=(1,0.5).


In other words, if Monty offers the swap every time the contestants original choice is correct, and half of the time that it isn't, then he makes his show as "entertaining" as possible without offering the contestant any mathematical edge.