Game theory: Choosing the size of a present

BACKGROUND

Game theory is one of the more recent branches of mathematics, having been developed during the Second World War with the aim of maximising the advantage resulting from certain tactical moves. The pioneering work was done by Von Neumann and Morgenstern.

DIALOGUE

(Charlotte 8, Owen 12)

Children! I've decided to give you some money for Christmas this year.

Owen: Ripper.

Charlotte: How much?

I'll leave that to you. But if you're too greedy, you will receive nothing.

Owen: How do you mean?

Well, I'll just write an amount of money on this piece of paper, and cover it up. Then you tell me how much you want. If you ask for the amount I've written down, or less, then I shall give you what you ask for. But if you ask for more, you will be punished for your greed, and will receive nothing.

Would you like to practise? I've written down the amount. How much do you want?

Charlotte: five dollars.

Owen: twenty dollars.

(Uncovering paper) As you can see, I wrote down thirty dollars, so you would each have received the amount you wanted. Now we'll do the real thing.

Charlotte: twenty dollars.

Owen: five dollars.

Well, as you can see, I wrote down ten dollars. So Owen will receive the amount he asked for, but greedy Charlotte will receive nothing.

COMMENT

The children have no way of knowing what their "best" strategy would be. To work out the size of present which should be nominated to ensure (on the average) the greatest return, they need to know some information about your behaviour: Will you just pick at random any amount between $5 and $50, each being just as likely as any other? Or will you adopt the strategy which will minimise the child's present?

For further information on game theory, see Von Neumann and Morgenstern's original work or a modern introduction such as that by G Owen.

Bibliography

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