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 =
(.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.