Thursday, December 22, 2005

Thoughts About "Deal or No Deal"

At Annie's request, I have been sucked into a very enjoyable TV show. The show is "Deal or No Deal," which purportedly gives you a chance to win $1,000,000 without offering any trivia questions, and no other problematic catches.

The game works sort of like Monty Hall problem, but without the problem that the host knows what's behind the doors. In "Deal or No Deal," there are 26 briefcases with various dollar amounts from One penny to One Million Dollars. There are thus two divisions of 13 cases: those of relatively large value, and those of relatively small.

Of course, watch the show for an hour, and you'll realize quite quickly (if you don't get caught up in the hype) that the game is not actually about winning $1 million. The game gets interesting because there are only 25 briefcases that you can choose to eliminate...one case is chosen by the player right away, and that case stays with the player throughout the game. After each round of elimination picks, to whittle down the cases to purportedly win $1 million, a "banker" calls host Howie Mandel, with a counter offer of money to lure the contestant away from winning $1 million...the offer goes up or down depending upon how many large values there are on the board.

Here's what I've figured out with my non-technical math skills:

1.) The Odds are never what the host says they are. If you've got 5 total cases (4 out on the floor, and 1 sitting next to you), and one has the $1 million dollars, the host will say "you only have a 20% chance of revealing the $1 million." This is to lure the contestant into playing further...I mean, if I want to figure out whether or not I have $1 million, the chance that 4 out of 5 times I will reveal something other than $1 million looks like very good odds...

In fact, the odds are worse. As Annie first pointed out to me...you either have the $1 million or you don't. So, if $1 million is still listed as a possible value to win, and 5 cases remain (1 next to you, 4 on the floor), your odds are not 1 in 5. Either...

(a) you have the $1 million. Your odds of revealing it in the 4 cases on the floor are zero.

or

(b) you don't have the $1 million next to you. That means that you have a 25% chance of revealing the $1 million, not 20%. The odds get worse as the cases disappear:

Chances of revealing $1 million when $1 million is an available value
4 cases total, 3 on the floor: Host says 25%; actual=zero or 33%
3 cases total, 2 on the floor: Host says 33%; actual=zero or 50%
2 cases total, 1 on the floor: Host says 50%; actual=zero or 100%

(2) Thus, the game leads down to a major assumption: (a) whether or not the $1 million is in your case. Given that only 1 in every 26 cases has $1 million, the odds are greatly against $1 million being in the case.

For the purpose of playing the game, one should not assume that one has the $1 million. If that assumption is made while playing, the player will have a very difficult time making a good deal.

Here's an interesting way to narrow the terms on which the game is played, in order to make a good deal:

(A) Divide the 26 cases into 2 categories: (x)=smaller values; (y)=larger values. The goal is to end up with a larger value, and if not a larger value, a really good deal.

Thus, every time the player gets into a situation with five or less cases remaining, this division is very helpful.

Take an instance from tonight's game...

Situation #1: 5 cases remain; 4 on the floor, 1 next to contestant.
VALUES: (y1) 1,000,000; (y2) 500,000; (x1) 300; (x2) 200; (x3) 50

The host says that there is a 60% chance that the contestant reveals something other than 500,000 or 1,000,000, or (y) values. This is not true....

The (x) division helps solve this puzzle....

Assuming (any y) is in your case, that means that the 4 cases contain (x1), (x2), (x3), and (other y). This is a favorable situation for the contestant.

But, if (any x) is in your case, that means that the 4 cases contain (other x(a)) , (other x(b)), (y1) and (y2). In this case, the odds jump from 60% chance of revealing an (x) value to 50%.

Thus, the entire game is based upon one major assumption: is your case and (x) value or a (y) value. It is advantageous to assume that your value is small, or an (x) value.

Why?

This gives you enormous leverage in handling the deals from the banker. Rather than worrying about what you lose from taking the deal (if your case has $1 million), you end up thinking about what you gain if the value in your case is small (an (x) value).

So, in conclusion, it seems that rather than winning $1 million, the object of the game is...

(a) Keep larger numbers on the board longer (random task, cannot be rationally controlled) in hopes of keeping banker's offers high
(b) Always assume a small value is in your case.
(c) Make a good deal based on the following two strategies (one random, one rational)

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