Monday, December 26, 2005

Deal or No Deal

A new US game show started last week, Deal or No Deal hosted by Howie Mandel (the same Howie from this post). A New York Times article describes the game as a exercise in probability.
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.

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.

If the contestant just wanted to maximize the expected value of their winnings they should always turn down the Banker, but many do accept the Banker's offer. Are they acting rationally?

When the amount of money involved becomes a significant fraction of the contestant's net worth, a contestant becomes risk averse and is often willing to accept a sure amount rather than an expected higher amount.

Economists model this phenomenon using utility theory. A person has a utility function of their net worth, usually with first derivative positive and second derivative negative (think square root or logarithm). They aim not to maximize expected net worth, but expected utility which leads to risk aversion. For example, if you had a square root utility then you would be indifferent to having a guaranteed net worth of 360,000 and playing a game that would give you a net worth 1,000,000 or 40,000 with equal chance.

Economists can't afford to run these experiments at their universities; they can't offer enough money for serious risk averse effects to kick in. But television game shows like this do give us a chance to see risk aversion in action.

8 comments:

  1. And an economic paper for the show has already been written.

    Here's another blog entry with a link to play it online.

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  2. I think the word you want here is "aversion" and not "adversion."

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  3. Thanks. I fixed the post. Spell checking doesn't catch all my mistakes.

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  4. There is a 2nd paper, too:
    Post, Thierry, Baltussen, Guido and Van den Assem, Martijn J., "Deal or No Deal? Decision Making Under Risk in a Large-Payoff Game Show" (January 2006). http://ssrn.com/abstract=636508

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  5. Looking at the scores from the U.K. version, (don't have the wp on me, think it's dond dot something) it's true that, overall, the Banker makes about a 47% profit on all the "deals" on the show.

    But you have to understand that a player of this game only gets to play it once, instead of many times, as most mathmatical models dictate. Risking $30,000 for a 50/50 shot at $100,000 or nothing seems like a good deal, but $30,000 is more than most people make in a year. Even on those kinds of odds, few would choose to go for that bet.

    And, from person to person, the banker's deal makes sense.

    While there are some big losers at the large end of the scale - mostly people who pick up the 250,000 pound box (the max in the UK version, who by definition cannot do better by a banker's deal) taking the banker's deal resulted in 22 individuals -losing- money, while 71 individuals gained money from the banker's deal.

    In short, most people, individually, are better off taking the banker's deal. The banker, on the other hand, is able to make profit by securing the few very large boxes for few sums.

    Psychology enters into the games as well, but most of those factors would likely affect the decision-making of the contestant, and not the "banker" who is likely a computer algorithm.

    In short, in each individual game, it is better for the individual to take the banker's deal. In the aggregate of all the games, however, it is better for the banker to make the deal.

    I would not say that this is risk aversion at all, but merely the reality of one-time-only play.

    :)

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  6. By the way, the "winners" won an average of 14,775 GBP. The "Losers" lost an average of 79,020 GBP

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  7. http://www.javajargon.com/deal.html
    This guy created an "Ultimate Decision Maker." It's somewhat accurate. Obviously the banker doesn't follow a set pattern in his offers with respect to the expected return. But this excel program gets you pretty close to the actual offers.

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  8. Can you please explain your comment: "For example, if you had a square root utility then you would be indifferent to having a guaranteed net worth of 360,000 and playing a game that would give you a net worth 1,000,000 or 40,000 with equal chance."? I'm not sure what you mean.

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