Talk:Introduction to Game Theory/Deal Or No Deal

The two versions are talking about different things
The first one is talking about switching cases, the second version is talking about switching to the banker's offer which is a completely different problem.

Switching cases in Deal or no Deal is NOT equivalent to the Monty Hall problem
The real question we want to answer is the following:

"What is the probability you are holding the $1,000,000 case given that you have two cases to choose from and one of them contains $1,000,000?"

Whenever the word given shows up we're likely dealing with conditional probability, which is certainly the case here. The probability that event A will happen given that event B happens is

$$ \frac{\mathrm{Probability}(A (\mathrm{AND}) B)}{\mathrm{Probability}(B)} $$

In this situation, A represents "the event that you are holding the $1,000,000 case", and B represents "the event that there are two cases left, one of which contains the $1,000,000".

Monty Hall

Consider a version of the Monty Hall problem where you have to pick between 26 cases, where one contains $1,000,000 and the others contain lesser amounts. After you pick your case, Monty opens 24 cases which do not contain $1,000,000 and gives you the option to switch cases.

The probability that you are holding the $1,000,000 case is $$\frac{1}{26}$$. Having two cases left, where one of them contains the $1,000,000 dollars always occurs, so the probability is $$1$$. The above formula shows that the probability that you are holding the $1,000,000 case given that you are choosing between two cases where one contains $1,000,000 to be $$\frac{1}{26}$$.

Deal or no Deal

The probability that you are holding the $1,000,000 case is again $$\frac{1}{26}$$, and again if you pick the $1,000,000 case, you will always end up with the million dollar choice. But the probability that you end up with the million dollar choice in general is only $$\frac{2}{26}$$ in deal or no deal. The above formula shows that the probability that you are holding the $1,000,000 case given that you are choosing between two cases where one contains $1,000,000 to be $$\frac{1}{2}$$.

It's clear that there is no advantage to switching in Deal or no Deal, because the odds that you are already holding the $1,000,000 case are 50:50.

Problems with Version 2
Version 2 is wrong. I propose changing it to something like this:

It is sometimes argued that, "The former seems to ignore the fact that, assuming that at a given point in the game, there are multiple low dollar amounts in the left column and one high dollar amount in the right column, the expected value in dollars approximates one-half of the right column's dollar amount instead of the unchecked briefcases' average, which is because at the start of the game, it is equally likely that a briefcase from the left or right side (below or above the median) is picked by the contestant - a fact which does not change throughout the game. This constitutes a variant of the Monty Hall Problem, in which it is better to switch to the banker's offer that overestimates the potential of the chosen briefcase, if there are many dollar amounts on the right side of the screen and only few on the left."

However this is incorrect for a number of reasons:
 * 1) Monty Hall only works because the host knows which door has a car behind it, and so can deliberately avoid it. In Deal Or No Deal, cases are opened at random.
 * 2) The chance of a case being above or below the median does change as brief cases are opened. The opening of these cases adds extra information, which needs to be factored in, by application of Bayes' Theorem; in which P(A) is the chance of a high money case, and P(B) is the chance of several low money cases being opened at the start.
 * 3) Consider the situation when there are 10 unopened cases, one for a high dollar amount and nine for a low dollar amount. The argument runs that the contestant's case had a 50/50 chance of being a high dollar amount at the start, and so it still has a 50/50 chance of being high. The problem is that this same logic can be applied to all other unopened cases. Since it is impossible for all ten unopened cases to have a 50/50 chance of having a high dollar amount, we can see that the reasoning is wrong.

Peter Ballard 12:23, 17 September 2007 (UTC)

Peter I agree completely. As you say the critical point of the Monty Hall problem is the host's knowledge. I was tempted to say remove this section completely. But given that there is a lot of misunderstanding on this subject, then I like your version. I say put your version in. --AdRiley 14:50, 17 September 2007 (UTC)


 * Thanks. I'm not sure I've expressed point 2 in a way which is completely accurate (I have studied probability, but it was a long time ago), so perhaps someone can improve it. But for me point 3 is what made me sure I was right. Peter Ballard 23:52, 17 September 2007 (UTC)


 * After trying to explain the logic behind Version 2 to my family last night, I've come to the conclusion that the logic is so flawed that it doesn't deserve the dignity of a mention; all that is required is to make the correct reasoning (i.e. the old version 1) a little clearer. Peter Ballard 00:13, 19 September 2007 (UTC)


 * Yes I have just re-read the old version 1 and it isn't clear. I don't like the bit about fixing the chance of the top prize being 1, it isn't necessary.  In fact Deal or no Deal works as you would logically expect, it is Monty Hall that is the more complex to understand and it all comes down to the hosts knowledge of where the prize is.  --AdRiley 12:59, 19 September 2007 (UTC)

This does not appear to be an article about Game Theory.
Game theory is a well-established branch of theoretical statistics, which has existed for many decades. I see very little of Game theory in this.

This article appears to be more about Psychology and how experience influences behaviour in areas of uncertainty. Rolo Tamasi (talk)

Claim that Banker wants to "minimize expected winnings per hour play"
From the article:


 * The expected value [the banker] seeks to minimize is not on a per game basis, as is the case for the contestant, but on a per hour of play basis. A strategy that would cause an average contestant to play for an hour and win $100,000, is more advantageous to the banker than a strategy that would cause an average contestant to play for only a quarter hour and win $50,000.

This appears to misunderstand how television works. The episodes are edited down to fit into the broadcast time. A contestant could play for three hours and NBC could still edit the game into a 20 minute segment.

Another part where the article demonstrates a clear lack of understanding how television works:


 * [...] if contestants always chose this strategy, the game would be boring as it would consist of contestants always declining offers and continuing to open cases until the end (or until the banker's offer exceeds expected value).

Actually MANY, MANY episodes follow exactly this pattern, with each offer falling below the expected value and contestants turning down each and every offer to the end (and usually sticking with their case). If anything, the audience seems to be much more excited to watch a contestant who's willing to "go all the way". Indeed, almost by definition the "most exciting" outcome is that the contestant wins $1,000,000, which requires (unless the banker has a temporary loss of sanity, I suppose) they "go all the way", turning down every offer. This is basically how all gambling-type shows work: human beings find it MORE exciting to keep pressing your luck than to make rational calculations and back out at a reasonable time.

It's still possible to understand the very low offers at the beginning of the game from the perspective that the banker doesn't want a very fast turnover, because the show won't get enough footage. However that definitely doesn't explain why the fraction-of-expected-value given by each offer goes up round by round at a fairly linear rate from about 0.1 to just under 1 (sometimes slightly over) over the course of the 26 rounds.

Fenn B612 (discuss • contribs) 01:02, 10 July 2022 (UTC)