"Let's Make a Deal" vs. "Deal Or No Deal"

One of the legal journals I subscribe to has a piece on "cognitive illusions" that starts out with the Monty Hall "paradox":

You are a contestant on a game show with three doors.  Behind one door is $10,000 cash; behind the other two doors are goats.  The host asks you to choose the door you want, and you select door number one.  The host opens door number three, revealing a goat, and then gives you the opportunity to keep what's behind door number one or switch to door number two.  What do you do? 

Most attorneys [and some mathematicians -- AC] would answer that the decision makes no difference, because there is a 50-50 chance that the cash is behind door number one.  That answer is wrong.  From the moment you chose door number one, you had a one-third chance of having selected the door with the cash behind it.  The other two doors, combined, had a two-thirds chance of hiding the cash.  When it is revealed that door number three does not have the cash, then door number two, alone, has a two-thirds probability of hiding the cash.  Thus, when given the choice, you should always switch doors, because you are twice as likely to win the game.

The writer then says the same principle applies to "Deal or No Deal."  The contestant picks one of 26 briefcases.  One of the 26 has $1 million in it.  "As more and more unpicked briefcases are opened without revealing the $1 million, the contestant believes that the odds he or she picked the right briefcase at the start must be increasing, resulting in the rejection of increasingly high-dollar 'deals.'  In fact, the exact opposite is true:  the odds are increasing -- [up to 25/26] -- that one of the unpicked and unopened briefcases actually contains the $1 million, and the contestant will be left with very little." 

This particular cognitive illusion is indeed deceptive because I think the author is wrong.  He's not wrong about the Monty Hall paradox.  That's right, but he leaves out the critical assumption, which is what trips up most people the first time they hear this: For the sake of suspense, Monty Hall never opens the door with the money. 

The author is wrong about "Deal or No Deal."  Contestants on "Deal or No Deal" pick briefcases at random and (almost always) hit the $1 million.  Suppose a contestant gets down to two briefcases with the $1 million still on the board.  Does he have a 25/26 chance of winning the million if he trades?   If Howie had been picking briefcases, and we knew that Howie would never pick the $1 million briefcase, then yes.  If the contestant had been picking at random, then no.

Update: I've done a more rigorous analysis using Bayes' theorem.

Comments

[Don Johnson]

Fun one. Would make a fascinating short course. Consider this a laymen's attempt because statistics are surprisingly not intuitive but I feel like a try.

He had 1/26 (3.85%) chance of picking $1M originally. So odds are 25/26 (96.15%) that $1M is still in play. Assuming it is still in play, then the odds are 1/25 (4.00%)that he will leave $1M in play after picking 24 cases.

What this means to me is we'll very rarely see someone with two cases and $1M still in play. To me the question so switching is a balance of those three ratios. Chances were GOOD he left it in play(96.15%). Chances are BAD he leaves it in play after selecting 24 cases (4%). 4% is bad, but still better than the 3.85% chance he got it on the first try.

So that's almost coin toss, but the numbers say switch. Ofcourse, I suspect I'd be taking the banker's offer if the low case wasn't close to $1M.

Figuring out when to take the offer would be another fascinating study. The insurance company probably knows this stuff well. How long the show is on the air was probably something they threw in there too. The basis of the show's mathematics alone probably filled up many pages and was some math major's dream job of a lifetime.

[AC]

If the $1M briefcase is among the 25 you did not pick, then you are right that there is a 1/25 chance that it will be the last one picked out of that group. But that's if the briefcase is in that group of 25. The probability is 25/26 that it is. Both events have to occur, for which the probability is (25/26)*(1/25), or 1/26 -- the same probability that you picked the $1 million briefcase.

I got excited the first time I saw DorND, but then I decided the show is really just psychology. The banker calculates the contestant's expected winnings and offers that number minus a discount. He must offer a discount because contestants (and the audience) are risk averse; they'd just take the deal if you offered them the expected value. So the trick is to offer very large discounts early on to keep contestants playing, and then to shrink the discount as the game goes on to increase the tension. A perfect discount is one that causes the contestant to agonize over the deal. Whether the contestant takes the deal is just a matter of her taste for risk, and not a right or wrong decision.

[Don Johnson]

Chances he picked $1M originally are 1/26. Chances it is one of the last two is 2/26. Both unlikely, but picking $1M first was more unlikely. Doesn't that mean switch? If so, the odds only slightly increase by switching and are not improved as significantly as with the Monty Hall scenario.

And yes, the "banker" is just a pre-calculated spreadsheet. Notice at the end of the show when the game is over and they quickly run through the last un-picked cases. The "offers" pop up automatically. The offers just represent some pre-determined odds of mimimizing losses. The house never really wins, but the show makes money. And they certainly won't average losses of $1M per show, but the excitement of $1M being there at the start of every contestant sells commercials.

[Don Johnson]

"Both events have to occur, for which the probability is (25/26)*(1/25), or 1/26 -- the same probability that you picked the $1 million briefcase."

That just means ending up with two cases when $1M was still on the table is just as unlikely as picking $1M in the first place. Doesn't really address the moment you find yourself standing there with two cases.

The more I think about it if you are lucky enough to be standing there with two unopenned cases, then chances ARE 25/26 it was the one you didn't first select.

As far as the offers, I'm not sure there is a discount as much as the average winnings is just lower than you would think intuitively. Statistics are like that.

Doesn't a legal journal know enough to contact an expert before publishing anyway!?

[AC]

"The more I think about it if you are lucky enough to be standing there with two unopenned cases, then chances ARE 25/26 it [$1 million] was the one you didn't first select."

Here's a thought experiment that shows (I think) that this is wrong:

Let's play the game with 26 contestants. Instead of putting amounts on the board, we'll put the numbers 1 through 26. A briefcase is picked out random, and each each contestant picks out a different number. The contestant who picks the number in the briefcase wins $1M.

After 24 briefcases are opened the two remaining numbers are, say, 1 and 2. (Contestants 3 through 26 are out of the game.)

It doesn't matter to the contestant who picked the number 1 that the others are out of the game. Instead, he reasons reasons as above and says, "There was only a 1/26 chance that the briefcase first picked had the 1, therefore there is a 25/26 chance that the "other" briefcase has the 1."

But the contestant who picked 2 is in the same boat. She reasons that there is a 25/26 chance that the 2 is in the "other" briefcase.

Both can't be true. In fact, symmetry implies the probability is exactly the same -- i.e., a 50-50 chance the 1 is in either briefcase.

Anyway, that's the simplest verbal explanation I've been able to come up with.

I've noticed too that at the end they just flash the offers up there without any pause. I've wondered why they do that since it makes the delays during the game seem kind of hokey.

I've also suspected, though, that they adjust the offers based on the contestant's popularity and taste for risk. I remember one contestant who was really popular with the audience; you could tell no one wanted her to lose. She was down to 3 briefcases with 1 big number left. The producers were apparently worried that she'd pick again and lose because they offered her more than the expected value. The only time I've seen that happen out of the 10 or so episodes I"ve ssen.

[M1EK]

You guys are hurting my head. But thanks very much for not taking the bogus "expected value" argument so many jump to so quickly (no, straight-up expected value is not a good way to treat this game given that you get to play it one and only one time).

[M1EK]

Too quick on the draw. What I meant by that is the argument pushed by so many that the contestant should evaluate all offers based purely on expected value (the banker, of course, should do so). A guaranteed $100,000 is, in fact, worth quite a bit more than a 20% chance of $500,000 and a 80% chance of $100, if you have no cash in the bank.

[AC]

If anything, I think contestants are too aggressive sometimes, overestimating the upside and underestimating the downside.

Example:

Four briefcases left, $500K and 3 little ones. Expected winnings: $125K (ignoring the 3 little ones as immaterial). Most contestants pass on this because they recognize there's a 1/4 chance they'll get nothing.

Now consider this one: Four briefcases left -- $400K, $100K, and 2 little ones. Expected winnings: $125,000. I haven't kept a formal count, but this situation comes up a lot, and many contestants take another pick, figuring they have the $100K as insurance.

I think there's a better way of framing the choice, though. Suppose you were to pick another briefcase in the first example. If you get a low number, your expected winnings go up to $500,000/3 = $167,000 -- a gain of just $42,000. Of course, if you hit the $500K, you get nothing. So playing another round gives you a 75% chance of a $42,000 gain and a 25% chance of a $125,000 loss. (I'm ignoring the banker's discounts for now.) Lots of people understandably turn that down.

But things aren't really that different in the other example. If you pick one of the two low-dollar briefcases, your winnings go up to $167,000, again a $42,000 gain.

If you pick the $100,000, your winnings go up to $133,000, an $8,000 gain. And if you pick the $500K, your winnings go down to $33,000, a loss of $92,000.

so here's how they compare side by side. Each of these has a 25% chance in the first example:

$42K, $42K, $42K, -$125K.

Each of these has a 25% chance in the second example:

$42K, $42K, $8K, -$92K.

I'm not sure why you'd play game #2 if you're not willing to play game #1. You're trading a 25% chance of a good payout (from $42K to $8K) in order to cut a huge loss by a (relatively) small amount. I'm not saying that some people wouldn't make that choice, but not me. I prefer game #1 over game #2 (but would probably pass on either, depending on the discount).

(PS, I ignored the banker's discounts because they shouldn't really matter if you're comparing the relative appeal of games #1 and #2.)

[Don Johnson]

The players "1" and "2" scenario doesn't work because both of their cases were always in play. It is a 50/50 at that point.

I don't know if switching on DorND really gives you a 25/26 chance, but it isn't worse than 50/50 if you switch and I think it is probably better than that if you are lucky enough to end up with 2 cases and $1M still in play.

Chances are 4% that if you didn't pick $1M originally that you'll find yourself in a 2 case scenario with $1M in play. But chances were even worse you picked the $1M first. So what I'm saying is that if you end up with 2 cases, chances are that it was the other case. SWITCH!!

I like this stuff. A real statistician would make us feel foolish. I took a statistics course in college and it is far out.

[AC]

Let me try again this way. Briefcases contain the numbers 1 to 26. One briefcase is selected at random. We'll play with two contestants.
Player 1 guesses the briefcase has a 1; player 2 guesses 2. If either wins he gets the $1m.

So they play they game a bunch of times. In most games, both the 1 and 2 pop up before the final two briefcases, but in game #333, the game comes down to two briefcases -- and only "1" and "2" are still on the board.

If contestant 1 should switch, then so should contest 2. But the probability of switching can't be better for both. In fact, the probability must be equal.

Now, if you'll excuse me, I must rearrange my Battlestar Gallactica DVDs.

[Don Johnson]

Very funny last sentence. I'm totally nerding out over this. Your example is bad because both cases stay in play. In that scenario each case has the same probability.

However, when you pull one case from a large sample, then separate it completely while going through the rest of the cases, that represents two different samples. I vaguely remember this from statistics. You just can't treat the two last cases the same when one case was pulled out of the sample at 1/26 and then is re-introduced as a choice. Before you try to compute this, remember having two cases with $1M still in play will be a very rare occurence.

[Don Johnson]

"Before you try to compute this, remember having two cases with $1M still in play will be a very rare occurence."

It is even rarer that when you have two cases left and $1M in play, that the $1M was the first case you picked. More often, in this very rare two case scenario, it is the case still on the board.