Numbers give the game away

By Burkard Polster and Marty Ross

The Age, October 20, 2008

We have a new idea for a game show. We call it Deal Or New Deal. We hope to sell it because we have a sneaky way of doing better than might be expected.

Here's how it works. The contestant, let's call him Barry Jones Junior, is presented with two suitcases, each containing a number (which may be positive, negative, a fraction, whatever). Barry chooses a suitcase and then the Game Host takes the other. If Barry's suitcase contains the higher number he wins a trip to Europe.

Ignoring ties, Barry clearly has a 50-50 chance of winning. But now we add a twist. We suppose that Barry is allowed to open his suitcase. Then, before the other suitcase is opened, Barry may choose to swap with the Host. Does this opportunity give Barry an extra edge?

It is hard to believe that it could, at least without some more information. For example, if Barry knows that the numbers are chosen randomly from 0 to 10, then his decision is straight-forward: he keeps his number if it is over 5 and otherwise he swaps.

But as it stands, how would Barry decide whether to swap? If his number is 1 000 000 that seems big. But maybe the numbers have been chosen as multiples of a million.

Amazingly, no matter how the numbers are chosen, Barry can give himself an edge. What he does is decide for himself which are the big numbers and which are small. Most simply, Barry can imagine choosing numbers according to a standard bell curve.

Barry uses his bell curve to randomly select a third number. If this random number is smaller than Barry's then Barry keeps his number, and the odds stay at 50-50. But if the random number is larger then Barry swaps with the Host.

When Barry swaps there are three scenarios to consider, as indicated in the diagram. If Barry's number happens to be smaller than the Host's (red zone) then Barry is actually winning, but then loses when he swaps. If Barry's number is larger than the Host's (green zone) a loss is turned into a win by swapping. These two scenarios are equally likely (because Barry's original choice of suitcase was random), and so here Barry's swapping cancels.

This leaves the blue zone, where the Host's number is largest. Here, swapping also turns a loss to a win. And, however the numbers in the suitcases were chosen, there is some positive chance of this third scenario occurring. This is Barry's extra edge!

How much extra edge? That depends upon how the numbers in the suitcase are chosen. If they are also chosen using Barry's bell curve then there are six equally likely orderings of the three numbers. Barry's strategy increases the winning orderings by one, and so his odds of winning go from 3/6 to 4/6. And, our plan is that Barry will be so grateful for the extra edge, he'll invite a Maths Master or two along on his trip.