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# Good Deal?

5 November 2006

One of the current popular game shows on TV is the show Deal or No Deal hosted by comedian Howie Mandel. The format is that 26 briefcases are carried onto the stage by 26 beautiful young women. Each case contains a dollar figure ranging from one cent up to a million or more. The contestant chooses one of these cases and it is set aside; this is called the player's case. Then the contestant begins selecting cases and opening them, first six, then five, then four, and so on until just one at a time is opened. At the end of each selecting sequence, a character called the banker calls Howie on his cell phone and offers the contestant a dollar amount to trade in the case he or she selected at the start of the show.

Now I don't know if this banker is a real person, a computer, or a committee, but the offerings have an interesting feature that I'll explore below. Let me show you a rather disappointing game that was played this past September 18th. In the table below I'll list the dollar amounts in each of the 26 briefcases and indicate at which stage each was selected. The initial case and those that never were selected I'll just mark with an x.

 Stages Amount Stages Amount 2 1 cent 1 1000 1 1 1 5,000 3 5 2 10,000 7 10 1 25,000 2 25 1 50,000 x 50 x 75,000 2 75 4 100,000 2 100 x 200,000 4 200 3 300,000 4 300 5 400,000 3 400 5 500,000 x 500 6 1,000,000 1 750 3 2,000,000

Notice that at the start of the game before any case is opened the expected return of the player's case is just the average of all of the figures listed above. For this game that amount is \$179,554.46. If this were a gambling game then that would be the amount the player would have to pay to play if the game were fair. This would mean that anytime the banker offered more than this amount the player could quit with a profit. Of course, this isn't a gambling game and the player is assured of at least a profit of one cent.

After the first round of openings there are 20 cases remaining, so the expected value of the player's case is just the average of the dollar amounts of these 20 cases. Similarly after the second round there are 15 cases remaining, so the expected value of the player's case is just the average of the dollar amounts in these. And so on. So at each step is this the amount offered by the banker? I'll say not. In the table below I give the expected value of the player's case, the amount offered by the banker, and the ratio of the banker's offer to the true expected value.

 Round Value Banker Ratio 0 179,554.46 0 0 1 229,333.25 26,000 0.1134 2 305,097.67 99,000 0.3245 3 206,914.55 73,000 0.3580 4 271,945.00 132,000 0.4854 5 212,593.00 97,000 0.4563 6 55,112.00 41,000 0.7439 7 68,887.50 56,000 0.8129

The player took the bankers offer after the seventh round.

Naturally I don't know how the banker arrives at his amounts, but it does seem clear to me that early on the amounts are intended to keep the player in the game. As the rounds increase the ratio generally increases as well.

The game following this one was a three million dollar game and was very interesting in that the player's case did indeed hold the three million dollars. Also, after the player accepted the banker's offer after the sixth step, Howie had the player specify which cases he would have selected had the game continued and revealed the banker's offer after each step. Here are the facts.

 Stages Amount Stages Amount 1 1 cent 2 1,000 10 1 3 5,000 2 5 5 10,000 1 10 1 25,000 1 25 2 50,000 3 50 4 75,000 3 75 3 100,000 1 100 8 250,000 1 200 2 300,000 5 300 6 500,000 2 400 4 750,000 4 500 7 1,000,000 9 750 x 3,000,000

The sequence of expected values and banker offers shows a rather interesting twist. Here is the table.

 Round Value Banker Ratio 0 233,400.62 0 0 1 302,154.05 32,000 0.1059 2 379,445.07 105,000 0.2767 3 507,868.27 249,000 0.4903 4 549,606.38 400,000 0.6734 5 791,791.33 650,000 0.8209 6* 850,150.20 675,000 0.7940 7 812,687.75 610,000 0.7505 8 1,000,250.33 1,100,000 1.0100 9 1,500,000.50 1,800,000 1.2000 10 3,000,000.00 3,000,000.00 1.0000

*The player accepted the banker's offer on the sixth round.

Here again the early ratios are low and generally increase as the rounds increase. Rounds 8 and 9 are interesting because the ratio exceeds 1 in both cases. My read is that in these situations the banker is either afraid of losing the three million (if the banker knows which case holds the three million) or else is afraid that the player will lose a lot of money and disappoint the audience (if he doesn't know). Who knows? Certainly not me and probably not Howie either. See you next month.

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Donald Catlin

Don Catlin is a retired professor of mathematics and statistics from the University of Massachusetts. His original research area was in Stochastic Estimation applied to submarine navigation problems but has spent the last several years doing gaming analysis for gaming developers and writing about gaming. He is the author of The Lottery Book, The Truth Behind the Numbers published by Bonus books.

#### Books by Donald Catlin:

Lottery Book: The Truth Behind the Numbers
Donald Catlin
Don Catlin is a retired professor of mathematics and statistics from the University of Massachusetts. His original research area was in Stochastic Estimation applied to submarine navigation problems but has spent the last several years doing gaming analysis for gaming developers and writing about gaming. He is the author of The Lottery Book, The Truth Behind the Numbers published by Bonus books.

#### Books by Donald Catlin:

Lottery Book: The Truth Behind the Numbers