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# PLAYING IT SMART

16 November 2009

You can learn something from prime-time television after all

Deal or No Deal is mindless but fascinating television. The genius is that it's an archetype of Occam's razor: the idea could hardly be simpler. Yet, opportunity is afforded to see the laws of probability, the principle of utility, and the concepts of risk-aversion and tolerance played out together. And, of course, big bucks are up for grabs in situations that quicken the pulse.

The American version of the game begins with 26 numbered valises on display. Each contains a sign designating a different amount of money, from a penny to a million dollars. Players select one, which stays closed. They then pick additional cases. These are opened in turn to show how much is inside. As cases are opened, the values are removed from the list of obtainable payoffs.

At various stages, contestants are offered "deals" -- payments on the spot -- in lieu of proceeding. Those who keep declining get to their original selection and one other case. This gives them a 50 percent shot at either of two remaining sums. They can take a final deal, swap cases, or have what's in their initial choice.

Offers during the game may seem to be somewhat arbitrary. They're not. They're based on using probability theory to compute average or "expected" values of the money known to be still at hand. No mathematical advantage would apply either way if deals were exactly equal to these values. That is, there'd be no statistical distinction between accepting the offer or going on. Over many such "fair" games, solid citizens as a whole would win roughly the same total whether or not they took deals, and when.

At the outset, with all payoffs active, expected value with the standard 26 levels is \$131,477.53. This rises or falls as low and high payoffs, respectively, are deleted. Say three cases plus the initial selection are unopened and the board shows \$500, \$5,000, \$50,000, and \$100,000 available. Each has 25 percent chance to be in any case. The average is \$155,500 divided by four or \$38,875.

Offers aren't precisely the expected values, however. Early on, they're rounded down, often sharply. This encourages players to reject premature deals and heightens tension as cases are eliminated. Later offers approach and occasionally even slightly exceed the average.

Expected value based on probability is an objective factor. But the game actually goes much deeper. Utility and risk-aversion versus risk-tolerance, issues for which economists have won Nobel prizes, enter the picture. These involve trading-off acceptance of assured intermediate gains rather than gambling on the hazards of losses or small returns versus hopes of major benefits.

Utility and risk can be illustrated by assuming two cases remain. Pretend the board shows \$100,000 and \$500,000. Expected value is \$600,000 divided by two or \$300,000. The deal might be \$250,000. Or, options might be \$100 and \$500,000. A fair offer would be half of \$500,100 or \$250,050. The deal might be \$200,000.

In the first example, the utility principle sets the offer as a choice between a certain \$250,000 and a 50-50 gamble between \$100,000 and \$500,000. The three amounts differ vastly in real terms, but all have high utility to many people. High risk aversion would favor accepting the offer; even moderate risk tolerance taking the gamble. In the second instance, \$100 would likely be perceived as having almost no utility compared with \$500,000, especially with 50-50 chances of either, so only the extremely high risk-tolerant would be apt to decline \$200,000.

Try it on yourself or some friends. Start with a choice based on two cases and varying differences. Then, make it more perplexing and interesting using three, four, or more possible payoffs. Your decisions will reveal a lot about the utility you assign to money and how you handle risk. They could also help you plan how you'll gamble when next you visit your friendly neighborhood casino. Why, they may even lead you to acute appreciation of this enigmatic epigram from the renown rhymer, Sumner A Ingmark: