Spent the evening on the couch with my wife watching TV--a game show called "Deal or No Deal," which, probably on account of the writer's strike, was tonight a two-hour investment in time. In the game the contestant wins money by choosing one of 26 boxes, each of which has hidden inside a money amount printed on a slip of paper. The amounts range from one cent to one million dollars. The twist is that, having chosen a box, the contestant names other numbered boxes, which a buxom model then opens, revealing a prize that the contestant did not win. A "banker" then offers the contestant a certain amount for the box she has chosen. The contestant may accept the offer--deal!--or choose to have the contents of more boxes revealed before receiving another offer.

Since the 26 different amounts are known quantities, it is a simple matter, if you have time, to calculate the "expected prize" at every point during the game. At the start of the game, for example, the contestant may expect to win the sum of the 26 different prizes divided by 26--which turns out to be about $131,478. In the first round, however, she must choose to have six boxes uncovered. If these six include two or three of the largest prizes, the banker's offer will be somewhat less than $131,478. You can see how the suspense can build as the contestant goes deeper into the game and reveals the prize hidden in more and more boxes. Do you take a sure thing or go for a really big prize?

The interest of the game, at least for me, rests in the tug-of-war between emotion and rationality. The rules of probability reveal, at every step of the game, the optimal strategy. But 26 is a somewhat awkward number with which to calculate, and there is of course the allure, mainly psychological, of the "million dollar prize." And it isn't always apparent that the best strategy for an individual is the one that maximizes the expected pay-out. Suppose the contents of all but two boxes are known, and the remaining two hide a ten dollar prize and a million dollar prize. The contestant's expected winning is now $500,005, so if the banker offers $300,000 to quit, the strategy maximizing expected value is to turn down the deal. But suppose you owe Tony Soprano a quarter million, and don't have it? For this contestant taking the $300,000 makes a lot of sense.

I think the game reveals that, regarding money, Aristotle was right: the object should not be to have as much as possible but, rather, to have enough.

## Comments