A fascinating spectrum of both magical and strategic thinking about numbers has been on display this week on "Deal or No Deal," NBC's mind-bending nightly game show. As trembling contestants have made bird-in-the-hand gambles for high stakes, viewers have been treated to opinions about probability that range from the simple-minded to the grandiose. One man in the audience shouted, "Eight is my lucky number!" while another declaimed, "The race is not won to the swift, but he who endures to the end!"
"Deal or No Deal" is the invention of Endemol, the Dutch production company behind "Big Brother"; versions of the glitzy and ingenious show appear in some 38 countries, including, now, the United States, where the week's run was ably supervised by Howie Mandel as its host. After strong ratings, NBC has ordered more episodes.
Numbers fiends of every variety should take a look: the exact logic of the game is maddeningly elusive. Whether it might even be a perfect game, in fact, was a subject of real dispute when I watched; some at my house proposed that it might genuinely reward intelligence, being the product of a probability calculus that a sharp contestant might actually divine. Others believed that the game advanced with the goal only of being entertaining, that it had psychological - but not logical - coherence.
By now "Deal or No Deal" needs to be explained, which can only detract from the fun. Here we go: Twenty-six briefcases are distributed to 26 models before the show begins. Each case contains a paper listing a different dollar amount, from one penny to $1 million. At the start of the game, a contestant chooses one case, which becomes his; he is then allowed to see the sums in six of the remaining cases. After these have been disclosed, a mysterious figure known as the Banker calls the set, offering to buy the contestant's case for a sum derived, somehow, from the cash amounts that are still unrevealed.
The contestant can take the offer and cash out, or move on to the next round, during which he's allowed to open five more briefcases before the Banker's next offer. The second offer might exceed or fall short of the first offer, but it clearly reflects the newly adjusted odds about what the contestant is holding. If the contestant refuses it, he requests to see the contents of four, three, two, and then one more case, with offers from the Banker coming at the end of each round. Each time the contestant can accept and end the game, or proceed to the next round. If he doesn't accept any of the offers, he is left with the sum in his own case.
How the Banker arrives at his offers is the mathematical mystery of the game. As in many games, the offers seem based on the principle of mathematical expectation: the price of a given chance at winning a given amount. But mathematical expectation is a specific calculus: if your chances of winning $1 million are one in a million, for example, it is reasonable to pay $1 for a lottery ticket, but not $2. And the Banker does not appear to use that exact equation.
Is it wise to take a bank offer when it's below the mathematical expectation, as it always seems to be? As the game goes on, the offers asymptotically approach mathematical expectation; maybe contestants should wait. Or maybe there's another calculation involved, one that, if you could do it in your head, would allow you to outthink the Banker.
Alternatively, it's possible that the bank offers are not strictly mathematically derived, but made with a view to the contestant's psychology. After all, the Banker can be seen in silhouette, surveying the proceedings. Maybe he's just trying to jack up suspense to make the game more fun to watch.
This suspicion typically hovers around sloppier reality competitions: Did those aspiring models/designers/apprentices genuinely "fail" their challenges, or were they eliminated merely to make the show more engaging? On "Deal or No Deal," where the challenges appear more objective, the question is: Can this game be, if not beaten, at least played with the head and not solely with the heart?
So far, no game theorist from the Institute for Advanced Study has appeared to try his hand at "Deal or No Deal," and play as a cool-headed rationalist. Instead, the players on the American show are, like most game-show contestants, hysterics. They scream and jump around, tearfully consulting with loved ones who are ceremoniously called to the stage. In general, they claim to make their selections on good feelings or lucky numbers or all manner of superstition.
The show encourages this woolly numerology, personifying the numbers of the suitcases by identifying them with the scantily dressed models holding them. Contestants implore the models, by name, to be "good" to them by revealing the low numbers that lead the Banker to increase his buy-out offer. They also form attachments to the cases they select in the beginning, seeing "faith" in that case as something to be cultivated.
On the other hand, these people aren't stupid. And they're not immune to the lure of a sure thing. As one contestant's mother told her, looking at a decent six-figure offer: "The million dollars is a great American dream, but America was made by working hard, and that's four years of wages and salary!" (The girl proceeded to gamble anyway.)
Fittingly, the American contestants on this international show reveal themselves to be pragmatists, willing to change creeds based on what works. The man who had initially counseled endurance eventually begged his brother to cut and run - take the Banker's $67,000 offer. The brother refused, announcing that risk-taking was in his genes. The next time around, he was offered $99,000 to quit. Emotions ran high.
"I am my father's child!" the contestant shouted proudly. Then he hesitated. "But I'm no damn fool!" he cried.
He smiled broadly. "Deal," he said.
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