## Saturday, June 30, 2007

### Deal or No Deal Banker's Formula

UPDATED 8:20 p.m.: Another episode of data were added, and the formula was updated. Scatterplot updated.

NBC's Deal or No Deal is one of the most popular programs on television. Even though the show is in summer re-runs, Nielsen Media Research says the show was No. 3 for the week of June 18, 2007.

I was first introduced to the show by former Ohio State master's student Tim Laubacher. Although I am far from a regular viewer, I do find the show interesting.

Here is the basic premise: the contestant begins with 26 cases, each of which represents a monetary amount ranging from \$0.01 to \$1,000,000.

The contestant selects one case, which then becomes "their" cases. The contestant is entitled to whatever monetary amount is inside.

At this point, 25 cases remain on stage. The contestant must "open" six cases (in the first round), which reveals the amount inside each case. As more cases are opened, we have more information about what amount might be inside the contestant's case.

After each round, a silhouetted "banker" makes an offer to buy the case. Host Howie Mandel repeatedly reminds the audience that the banker wants to buy the case for "as little as possible."

If the contestant takes the "deal," then the game is over. If the contestant refuses the deal (i.e., "no deal"), then more cases are opened. If the contestant has bad luck, the opened cases have large amounts, which means that large amount was not in the contestant's case. The opposite is true with small amounts. After each round, a new offer is made. And each round requires the contestant to open fewer cases.

As a statistically oriented person, I immediately wondered how the banker came up with the "offer."

At any moment, we can estimate the expected value of the contestant's case. This is a simple part of probability theory, and it is intuitive to most people. Imagine that there is one case left on stage. The contestant has one case. Now assume that the amounts \$100,000 and \$200,000 remain.

What is the expected value of the case? If you were to play the game in those exact same circumstances many times, the long-term average value of the contestant's case would be \$150,000. That is the expected value.

This expected value is the most logical offer for the banker. However, it does not take long to realize that although the offer is usually close to the expected value, it is not a perfect match.

So last night, I watched two episodes and wrote down the amounts remaining and the offer. This morning I used hierarchical regression to figure out the banker's formula. Although there is still some error in the formula, it is 99% accurate in predicting the bank's offer (see scatterplot comparing offers and predictions below).

According to my (latest) calculations, the formula is:

Banker's offer =

\$12,275.30 +

(.748 * expected value) +

(-2714.74 * number of cases left) +

( -.040 * maximum value left ) +

(.0000006986 * expected value squared ) +

( 32.623 * number of cases left squared ).

Together these values explain 99% of the variance in the banker's offer. Admittedly, this is based upon a small sample of only (now 31) offers. When I get bored enough to chart some more data, I will update the formula.

Anonymous said...

In the words of some McDonald's jingle, "I'm lovin' it!"

Nice work Sam.

- Tim

3:56 PM
IUAngelini said...

I've wondered this myself, but never to this extreme.

I think someone may have more time on their hands than they know what to do with.

1:48 PM
Anonymous said...

Wow, thanks. I was looking for this whole day. I'm doing the game in pascal.
Good Job

11:07 PM
tyler said...

Cited you in a paper, thanks.

7:04 AM
Samuel D. Bradley said...

Thanks for the citation!

12:12 PM
Mets411 said...

Try using an example where only the \$750,000 and \$1,000,000 cases remain. I tried your theory and it gave me a result of over one million. Maybe I did it wrong. Let me know what you got.

1:57 AM
Kurupt55 said...

Formula does not work with extremes, try it with only a few small cases (under \$200) left and it gives you an offer in the thousands and as Mets said only \$750,000 and \$1,000,000 left and it gives you an offer of \$1,156,342

8:29 PM
Kalbintion said...

From a different website's comments, the formula of:

OFFER=AVERAGE*ROUND/10

is fairly accurate, but not exact.
Offer - Bank's offer
Average - Statistical average value of remaining suitcases
Round - the Round number (1 to 9)

3:21 PM
Anonymous said...

Interesting, but its a classic case of ex post forecast or data-fitting. I'm suprised that the professor has not thought to verify the formula by seeing if it works with other programs that are not in the original data set.

8:13 AM
Anonymous said...

Good work. I'm sick of the naive saying 'he takes the average and then subtracts a little'

10:57 PM
Anonymous said...

Thanks sooooo much!!!!!!!

6:40 PM
Anonymous said...

WOW! I SSOOO NEEDED TO KNOW THAT...NOT!!!!!!!!!!!

10:16 AM
Anonymous said...

WOW! Someone doesn't have a life!

10:18 AM
Anonymous said...

Tried using this formula for the "perfect board" - 2 cases remaining and we're talkin' the final offer of the game. The 2 cases are \$750,000 and \$1,000,000 and this formula comes out to \$735,569.5035

9:16 PM
Anonymous said...

In Australia, we have it to \$200,000 (ok our dollar is strong, but not that strong) and I saw the top 2 cases left.

\$100,000 and \$200,000.

The bank offer was \$130,000.

I think we're being given a worse offer, both in the amount and in the variation. Mind you I'd take \$100,000.

12:18 AM
أمين باشا said...

cool thanks

8:53 AM