Deal or no deal?

Mike Fletcher explores the optimal strategy for winning the popular television game show

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Deal or no deal is a television game show first developed in Holland, but now shown in more than eighty countries. It is loved by many, but considered by others to be no more exciting than watching paint dry. The format of the game, and the best strategy to win, have prompted much discussion, with articles on the programme appearing in some very serious economics journals!

The rules

In the UK version of the game there are 22 sealed boxes, each containing a sum of money ranging from £0.01 up to £250,000 with an average amount of £25,712. The contents of the boxes are shown in the table below:

£0.01 £0.10 £0.50 £1.00
£1 £5 £10 £50
£100 £250 £500 £750
£1000 £3000 £5000 £10,000
£15,000 £20,000 £35,000 £50,000
£75,000 £100,000 £250,000

Each box is allocated to a contestant, one of whom is selected at random to be the main player. Let’s call the main player Charlotte. Charlotte chooses a number of other boxes (in the first round this is five boxes) and their contents are revealed. Those boxes are now out of the game. A banker, Bill, then offers to give Charlotte some money in exchange for her box, based on the value of the boxes that are left in the game. If she accepts the offer the game ends, if she rejects it, more boxes are opened, new offers are made, until eventually either Charlotte accepts an offer, or she is left with whatever is in her box. One might think that since the chance of Charlotte choosing the box with £250,000 at the outset is 1/22 then one game in every 22 will end with the jackpot being won. In fact in the first 2600 UK shows only seven players have won the top prize. A staggeringly low figure of 0.3% of contestants! So how do the programme makers ensure that so few contestants win the jackpot?

The initial rounds

The only decision the contestants have to make throughout the whole programme is whether to take the banker’s offer. Suppose the contestant has had a good first round, ie all of the five boxes opened have a small amount of money. One might think that the banker will offer an amount close to the average in the unopened boxes. Not so. The amount offered by the banker at the start of the programme is much less than the average (mean) of all the remaining boxes. The banker does not want the contestant to have his offer taken at the start of the programme. It does not make for good TV. In general the offers made by the banker in the first few rounds are much less than the mean of all the remaining boxes. For this reason, and because they get caught up in the hype, contestants do not accept offers in the first few rounds.

In the later rounds the offers made, although always less than the average (mean) of the unopened boxes, (well nearly always–there have been a couple of times when the banker has miscalculated!) move closer to the average. This is when the banker needs to be careful with his offers. He does not want the contestant to win a large amount of money. The evidence shows that when there are only five boxes remaining unopened, the banker is at his most generous. His offers are typically less than the mean and greater than the median.

The later rounds

Suppose the final five unopened boxes contained £0.01, £1000, £3,000, £35,000, and £250,000. From the banker’s point of view he does not want the contestant to carry on and win the jackpot. If he offers £36,000, say, which is greater than the median and less than the mean (the mean is £57,800) he places the contestant in a difficult situation. The contestant knows that the offer is greater than the contents of four of the boxes. If he takes the offer then 80% of the time he will be making the right decision.

When there are only two boxes remaining the banker needs to make his offer even closer to the mean. Otherwise the contestant will not take the offer and may win the jackpot.

The clip below shows the first person to win the top prize on the UK show. The two boxes left contained £250,000 and £3000. The banker offered £45,000. The contestant judged that that this was insufficient. The banker had miscalculated and the contestant opened the box and won the jackpot.

Designing the game

The sums of money in the boxes are particularly interesting. At the end of every round, the median amount of money in the remaining boxes is always less than the mean. The programme makers have put a lot of thought into choosing the contents of the twenty-two boxes! There are 26,334 different ways the amounts can be distributed in the final five boxes and in every single case the banker can make an offer that is less than the mean and greater than the median. In other words the contestant can always be put on the horns of a dilemma! When there are five boxes remaining, 60% of the time the contestant will be better off by taking the banker’s offer rather than carrying on.

There are, however, some scenarios where the banker has a difficult decision to make.

Example 1: The last five boxes are £0.01, £250, £500, £750 and £1000.
The mean is £500 + 0.2p and the median is £500. The mean is only just bigger than the median! The banker only has one realistic offer, £500, and most contestants would take it.

Example 2: The last five boxes are £0.01, £5000, £10000, £15000 and £20000.
The mean is £10000 + 0.2p and the median is £10000. Again the banker has only one realistic offer, £10000, and most contestants would take it.

The programme makers have made a small miscalculation in choosing the contents of these boxes. Having a difference of £250 between the contents of four boxes can give rise to a situation where the median and the mean are virtually the same. Similarly having a difference of £5000 between the contents of four of the boxes can give rise to the same scenario.

This final clip demonstrates that although, in general, one should not accept the banker’s offer, if the banker makes an offer that is greater than the mean of the remaining boxes, one should definitely take it!

Mike is a former chief examiner for statistics, a lecturer in probability and a teaser writer for the Sunday Times.
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