Wednesday, September 20, 2006

"Water is wet," physicists estimate

In the game show “Deal or No Deal,” there are 26 cases. One of them contains the million dollar grand prize. The contestant gets one guess as to which one it is.

1 grand prize. Out of 26 cases. What are the odds?

Apparently, TV reporter Alex Strachan doesn’t know. He writes,

“… the program’s producers estimate the player’s chances of winning the top cash prize as one-in-26.”

Not only does Strachan not know the answer, but he thinks it needs to be estimated. And he even feels the need to avoid endorsing the answer by explicitly attributing it to a third party!

No wonder reporters have so much trouble with sabermetrics ...

(“Back with a bang,” Ottawa Citizen, September 16, 2006, page K3. Article not available online as far as I could tell.)


At Wednesday, September 20, 2006 5:55:00 PM, Blogger Arb said...

But 1-in-26 isn't even correct, either -- because most players don't hold out until the end. It's much less than that.

Now if they meant the chance the player picks the case with the top prize, then, yes, that would be 1-in-26. But it's not the chance they *win* it.

At Wednesday, September 20, 2006 6:05:00 PM, Blogger Phil Birnbaum said...

Hey, didn't notice that. I Assumed he meant "picked" rather than "won".

Good catch.

At Thursday, September 21, 2006 10:33:00 AM, Blogger Tangotiger said...

I'm sure the reporter wouldn't even understand the qualification between "pick" and "win".

The show is also silly, because the "banker" never deviates from the pattern. The fair amount to trade would be "total left on the board" divided by "total unopened cases". The banker always offers a huge discount at the beginning,and then slowly inches up, where he hovers around the fair trading price.

What would be more interesting is if he jumps from a discount to a premium after every selection, like a real stock market, making the player wonder to establish how risky a player he is. Offering 10 or 15% on the dollar at the beginning is simply silly. Offer the player 120% after the first round, and that'll tell you how much of a risk-taker the player is.

At Thursday, September 21, 2006 10:45:00 AM, Blogger Phil Birnbaum said...

I'm sure the show's producers would be quite upset if the offers are big enough that the contestants quit early ... they're shooting for high drama.

I think I read that in other countries, the banker *knows* how much the contestant chose, and can strategically tweak the offer in response.

It would be neat to see what algorithm they use. They can't just offer $1 if the contestant chose $1, because future contestants would see the pattern and adjust their behavior.

Interesting game theory problem ...

At Thursday, September 21, 2006 3:07:00 PM, Blogger Tangotiger said...

Algorithm: watch it a few times, you'll see it. 10% the first time. 25% the second time. 50%. 75%. 100%. Something along those lines.

I agree they can't offer 100% each time, but offering 10% off-the-bat is just boring. Why not offer 50% off-the-bat? Then, jump it to 125%. Then, down to 80%. Play with the people's minds.

At Thursday, September 21, 2006 3:15:00 PM, Blogger Phil Birnbaum said...

Yeah, that would be cool. Make the show more interesting, too.

At Friday, September 22, 2006 3:22:00 PM, Blogger Arb said...

I had a student write a paper on the first season of the American version in the spring, specifically trying to reverse-engineer the banker's algorithm.

It's nowhere near as simple as tangotiger's claim. Yes, the percentage offers do go up later in the game. However, you need to look at the whole distribution of values left on the table. you can't just compare it to the actuarially fair expected value; you have to look at the variance too. There's some evidence that the banker's algorithm responds to the variance as well, which it ought to.

There's another reason for starting off with lowball offers, beyond the simple drama of it. The offer pattern allows the show to screen contestants based on their risk preferences. If you offered a generous buyout early, a large number of people would accept it -- but you would have paid "too much" to many of them. With this struture, ratcheting up the percentage, you can pick off the most risk-averse ones early as cheaply as possible, and then go from there.

There have been a couple behavioral economics papers on the European versions, but I don't have a cite on me.

At Monday, September 25, 2006 12:10:00 PM, Blogger Tangotiger said...

arb, I'm sure it's not as "simple" as I make it seem, but, it's a good baseline. The biggest driving force is the mean, and then there's some play with the variance.

I'm sure with a big payout still on the board, the banker will offer say 60% instead of 75% of the mean.

The "weeding out" you cite would be done during the "Testing" period, off-camera. Once the contestants have been selected, now you should use real game theory. Since no one would ever accept a 10% buyout, why in the world would they ever offer it? Clearly, it's to prolong the game. But, offering 50% buyout to start with would be more exciting. And, if they do get contestants accepting a 50% buyout, the show makes money (or whoever their underwriter is).

I've love to see the research paper. Can this be posted, or can you send it to me? (

At Friday, September 29, 2006 6:53:00 PM, Blogger Phil Birnbaum said...

In Las Vegas, there are "Deal or No Deal" slot machines. If certain symbols come up, you get to play a smaller version of the game.

Every "banker's offer" I saw on that slot machine was exactly the expected value.


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