ok, everyone's heard of the monty hall problem. if you haven't, there's a good bit of info on wikipedia.

today in a finance class i took a test with a bonus question as follows:

"you're on the show deal or no deal. you start with 20 cases. you choose 1. you play until there are 2 cases left, one for $1,000,000 and one for $0.01. Should you switch and why?"

i answered that it doesn't matter, each case has a 50% probability of being $1,000,000. after the test i argued with my professor for a good 15 minutes about how the monty hall problem has no application in this case. he insisted that switching gives a better chance for $1,000,000.

your thoughts?

You are absolutely right. The MHP is predicated upon the fact that the host knows something the doors/cases/whatever beforehand, and his opening a door for you gives you more information. In DOND, no one knows what is behind the cases, especially not the player, and the player is the one opening cases! So you're 50/50.

It is possible your prof has no idea how the show works.

This has already been discussed in some threads:

http://forumserver.twoplustwo.com/47...related-76428/
http://forumserver.twoplustwo.com/47...o-deal-134892/
http://forumserver.twoplustwo.com/47...aradox-306039/

the key point in Monty Hall is the host knows what's in every box and thus only reveals one that has a goat, he never reveals the car. If he doesn't know where the car is, he'd randomly open the car 1/3 of the time and thus there'd be no difference between staying with your pick or switching (think of it as you picking door 1, the host picks door 2. The times a car is behind door 2, 1/3, he'll open, but if he knows it's there that 1/3 of the time he'll go and open door 3 thus weighing the odds for switching)

Other arguments against it:
a. That question is flawed to begin with. You don't switch suitcases, you take what the banker gives you or open your suitcase.

b. If this were monty hall, then odds of staying with your suitcase would be 1/20, but odds of the switch would be 19/20, which means the EV of that suitcase would be about $950,000 which is ridiculous equity. The banker would then be 95% certain that the suitcase that's out there holds the money and wouldn't be offering people $200k and $300k deals for the suitcases they're holding now, he'd be giving them $30-$40k deals because 95% of the time they'd be opening a penny. Even in the event of $100k and $1m suitcase, he'd be giving like $120k odds where the true EV would be $145k. These deals are entirely unreasonable for that stage because late-game deals that exceed $200k are common.

c. Imagine an extremely wealthy friend prop-bets you that if you pick the penny suitcase, he will pay you $100 million. Now you're rooting to pick the penny. This fact alone has no bearing on the picking of previous suitcases, and thus doesn't affect the previous state of the game. If this follows monty hall, you should be very delighted because you'll pick the penny 19/20 times if you switch. This is a contradiction and so this can't be monty hall.

d. Run a simulation. Run it a hundred million times and only use the situations where a penny and a million remained, switch every time and see your EV. If it's monty hall, your EV will be ~$950k, if it's not your EV will be ~$500k. A very drastic and noticeable difference.

thanks. i really thought he was off base. he even went as far as to say he "probably knows more about game theory than me." maybe i should have challenged him to hu4rollz? lol

and the thought of coding a simulation in matlab crossed my mind, but i don't have matlab installed so i didn't get around to it.

The real trick will be to parlay this into respect and not contempt.

I might if this guy wasn't just a PhD student who's probably not more than 3 years older than me.

I'm not going to parlay it into anything other than 5 extra points on my test at this point.

You can ask him how the Deal/No Deal probabilities are any different than:
You open 28 cases, and discover that .01 and $1M are left.
Which of the remaining two cases do you chose?

You don't have to do any fancy simulations.
You can just consider the $.01, $.01, $1M case:

There are three equally likely scenarios in Deal/No Deal in this case:
Initially pick the 1M case (1/3 chance), and then pick a .01 case -> net 1/3 chance.
or
Initially pick a $.01 case (2/3 chance), and then pick the other .01 case (1/2 chance) -> net 1/3 chance.
or
Initially pick a $.01 case (2/3 chance), and then pick the $1M case (1/2 chance) -> net 1/3 chance.

The third possibility is eliminated when the case is opened to reveal $.01, which leaves two equally likely scenarios.

lol Dwarrior, probably impossible. I want to throw a brick in your prof's face.