# Optimal Strategy for Deal or No Deal

When I have watched Deal or No Deal (I try not to make a habit of it) I always do little sums in my head to work out if the banker is offering a good deal. Where odds drop below "evens" it's easy to see it's a bad deal, but what would be the correct mathematical way to decide if you're getting a good deal?

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You might want to explain how the show works – BlueRaja - Danny Pflughoeft Jul 27 '10 at 22:58
I seem to recall that some academic papers have been written on this subject. Economists especially became somewhat interested in the show as a very simple but quantitative experiment in levels of risk aversion. Probably these will be easy to find if you google for them. – Pete L. Clark Dec 8 '10 at 6:04

There are (at least) two factors that mean that simply calculating the average of the remaining options is not enough to describe how someone should play.

1. Someone's utility is not a predictable function of the amount of money that they win. For instance my utility from winning $\$$5 is more than 100 times my utility from winning 5 cents. However, my utility from winning \$$100 million is less than 100 times my utility from winning$\$$1 million. - Thanks Zev for fixing up the  signs. How do you get them to not make it go into LaTeX mode? – bryn May 5 '11 at 11:47 The dollar signs are in \LaTeX; what Zev typed is \$$. – J. M. Aug 23 '11 at 7:14

From what I hear about game-shows in general, if your performance does not make it to air, then you don't get anything. Hence you cannot just accept the first amount offered (if it turns out to be a better choice) and expect to get it, since it won't make an interesting show.

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I would be very surprised if you were right. Can you provide any kind of reference? – TonyK Dec 8 '10 at 12:34
I am inclined to doubt that there are unaired "deals". (After the game show scandals of the 1950s, some rules governing conduct were enacted. Presumably reneging on paying their winnings is against these rules!) Note that mathematically they discourage this by making the early offers a small fraction of the actual expected value. Probably they screen contestants as well to ensure that they are not going to do this. – Pete L. Clark Dec 8 '10 at 15:29
I'm afraid this is just what I've heard (a friend of a friend went on deal-or-no-deal and was made to sign a contract that stipulated...). – Douglas S. Stones Dec 10 '10 at 21:09
[PS. Maybe it's different in Australia] – Douglas S. Stones Dec 10 '10 at 21:11
Douglas Stoen si correct. My son just went on the show and was told very explicitly that if he took any of the first 5 deals he would not be paid. – user51690 Dec 4 '12 at 6:39

In maths, the expected value is the average of how much you'd win if you used the same strategy from the same position a large (approaching infinity) number of times.

We should first note that expected value alone doesn't decide the best option. We also have to take into account risk. Most people (other than gamblers) would prefer a certain dollar rather than a fifty percent chance of two and fifty percent of nothing. To deal with this, we typically define utility instead. Utility varies between individuals and is determined by their risk profile.

Since the dollar amount are quite large for contestants, it would be logical (in theory) to find someone who can afford to take the risk to insure you. This would allow you to receive the gains from a risker strategy.

For deal or no deal, they encourage you to play by making their offers worse than the expected value at the start of the game. Later, (according to Wikipedia) the offers may even exceed the expected value. Without knowing how exactly the offers are calculated (or forming a model), we can't answer this question accurately, but only make general statements.

If we ignore risk and offers being greater than expected utility, you'd always want to go as far as possible in the game. But, as is, the game is extremely difficult to analyse mathematically.

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