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# Deal or No Deal – an expected value game

### Summary

This is a simple hand-run version of the worldwide TV game show, Deal or No Deal, with which all, or virtually all, of your students will be familiar. You can see an online version here. It is easy to set up with minimal equipment and is fun for the students to play. It can be used to demonstrate expected value and risk attitudes and students can use it to make calculations. It can also demonstrate the diminishing marginal utility of income.

### Equipment

• A data projector linked to a computer on which is uploaded an Excel file.
• One envelope for each of the students (up to 22 envelopes). Some students can be given more than one envelope if fewer than the expected number of students turn up. Each envelope is marked with a number from 1 to 22.
• 22 slips of paper with a sum of money written on each (see table below). These are shuffled and one is deposited at random in each envelope and the envelope sealed.

### Full description

This is a simple hand-run version of the TV game show Deal or No Deal. It can be played with up to 22 students (more than that and they will have to double up).

Virtually all students will have seen the game on television as it is shown around the world, with top prizes varying from country to country. In the UK the top prize is £250,000; in the USA it is \$1 million. The highest sum is in the Netherlands, where the top prize is €5 million.

Each student is given an envelope, inside which is written a sum of money. The sums range from 1p to £250,000. No-one knows which envelopes contain which sums of money, but the students are shown the full range on the screen. The 22 sums of money are:

 1p £1,000 10p £3,000 50p £5,000 £1 £10,000 £5 £15,000 £10 £20,000 £50 £35,000 £100 £50,000 £250 £75,000 £500 £100,000 £750 £250,000

These numbers are in the Excel file and are displayed on the screen. If there are fewer than 22 students in the group, then fewer envelopes need to be made up. The missing sums of money can easily be eliminated from the Excel file by deleting the relevant rows.

### Playing the game

One student is selected at random and comes out the front with his/her envelope. The student then selects one other student at a time. Each time the selected student opens his/her envelope and reveals the sum of money written on the slip inside. The lecturer then puts an asterisk before or after that sum in the Excel file. This shoots the number to the left side of the column as it is no longer recognised as a number. The students can thus easily see which sums of money have been eliminated and which remain.

Every third round, the tutor, playing the role of ‘banker’, makes the student at the front an offer for their envelope. The students aren’t told the basis of the offer, but the Excel file is set up to show both the mean and various fractions of the mean of the remaining numbers. The offer can be adjusted according to the perceived risk attitude of the student: the higher the degree of risk aversion, the lower the percentage of the mean should be offered. Also a lower percentage can be chosen, the greater the spread of remaining numbers.

When made an offer, the student had to decide whether to accept the offer or not: in other words, to say “Deal” or “No deal”. If the decision is “No deal”, the game continues for a further three rounds, when a further banker's offer is made. This processs continues as long as the student's decision is “No deal” until there are just two envelopes left: the student’s and one other. The student will be made one last offer and asked whether or not they want to deal. If the answer is still “No deal” the student should then open their own envelope to see what they have won.

A simple prize could be given to the student, such sweets or chocolates: e.g. one sweet per £10,000 or fraction thereof, or one per £50,000, depending on how generous you are!

If at any stage the student selects “Deal”, the prize can be awarded at that point, depending on the size of the banker’s offer. The student can continue selecting students to open their envelopes and banker’s dummy offers can be made until just the student’s own envelope remains and it is opened. This can then be compared with the deal the student accepted to see whether or not it was a good move (ex post) to have dealt rather than continued.

### Using the Excel file

The Excel file shows all the sums of money in a column and the mean is calculated at the bottom of the column, as is 90% of the mean, 80%, 70%, 60%, 50%, 40%, 30%, 20% and 10% of the mean.

#### Version 1:

The figures should be displayed. Each time an envelope is chosen, insert an asterisk before of after the number in the formula bar. This will shoot it to the left of the column since Excel will no longer recognise it as a number. Because the column is wide, the students can easily see which numbers have been eliminated.

Each time an asterisk is typed in against a number, the mean and the various percentages of the mean are automatically adjusted. This makes it easy to make the banker’s offer every third round.

#### Version 2:

The numbers are simply written on the whiteboard and rubbed out or crossed out as they are eliminated. The tutor can still use the Excel file as described above, but this is not projected. This version allows the tutor to keep the calculation of the banker’s offer private and to ask the students to try to work out on what basis the offer was made.

### Discussion

Discussion can centre around the decision-making process of the student if this were real money and what would cause them to choose “Deal” or “No deal”. Assuming that the banker wants to minimise the payout, on what basis will he/she choose the offer? How should the basis of the offer be adjusted according to (a) the risk attitudes of the student; (b) the spread of the remaining numbers yet to be eliminated?

Students who have seen the game on television (i.e. most of them) can discuss how the game show host extracts information about the contestant’s risk attitudes and how the banker’s offers vary with the attitudes of the contestant.