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## BEATING REGIS |

**How To Win On "Who Wants To Be A Millionaire?"**

"Who Wants to Be a Millionaire" is driving our society crazy, and
I'm convinced the reason is that we love watching apparently normal people throw
away money.** ** Do you howl, as I do, when a contestant balks rather than
answering a question she's almost sure to get right? Do you scream at your television,
as I do, when yet another contestant refuses to guess, even after eliminating
two of four possible answers?

**The Rational Strategy For "Millionaire"**

The way "Millionaire" works is simple. Regis asks a contestant a series of up to fifteen questions, starting at $100 and moving up to $1 million. For each question, the contestant chooses either to answer or to walk away with any money she already has won. If she answers correctly, Regis asks another question for double the amount of the last one. Once a contestant correctly answers the $1,000 or $32,000 question, she is guaranteed that amount.

Contestants generally assume they have three options: (1) answer correctly
and win the question's dollar value, (2) answer incorrectly and win only the
guaranteed amount, or (3) decline to answer, and walk away with what they have
in hand. But contestants are wrong. This assumption is true *only* for
the final question, for $1 million. For the other questions, contestants are
underestimating their average gain from future questions. This means they would
often benefit from answering a question even if they had to guess.

For the final question, the three-part analysis is absolutely right. You either (1) answer correctly and win $1 million, (2) answer incorrectly and win only $32,000, or (3) refuse to answer and walk away with $500,000 from the previous question. Given these three options, the rational strategy is simple. If you are neutral to risk (more on risk later) and if you believe the probability you know the correct answer is greater than 50 percent, you should go for it. Otherwise, walk away. This is just common sense: if you are going to be right more probably than not, go ahead and answer the question (again, assuming you are neutral to risk).

But for earlier questions, this three-part analysis is wrong because it undervalues
option (1). If you answer correctly, you don't just win money, you also win
the right to answer more questions -- and that means you might win even *more*
money. (You also win future rights to walk away from questions without answering,
which also are valuable options.) This additional wrinkle makes the rational
strategy much more complex. And it means that *it can be rational for a contestant
to answer a question even if she believes that more probably than not she will
get it wrong!*

Suppose, for example, you are confronted with the penultimate, $500,000 question. After reading the question, you think you have only a 40 percent chance of answering correctly. What should you do? If you walk away, you keep $250,000. If you answer, then on average you would get only $219,200 ($500,000 for a correct answer times the 40 percent chance you're right, plus $32,000 for a wrong answer times the 60 percent chance you're wrong). Right? Absolutely wrong. This analysis is incomplete, and the $219,200 figure is much too low.

Here's why: the value of a correct answer to the $500,000 question is greater than only $500,000 times 40 percent, because it carries with it the chance to answer the final question, and potentially to win a million dollars. How much is that chance worth? Here the math gets trickier. Suppose there's a 40 percent chance you'll answer the $1 million question correctly, a 10 percent chance you'll answer the question incorrectly, and a 50 percent chance you'll be clueless and therefore will walk away. Then the chance to answer the $1 million question is worth, on average, $653,200 ($1 million times 40 percent plus $32,000 times 10 percent plus $500,000 times 50 percent).

Now go back to the $500,000 question. If you walk away, you get $250,000. If you answer the question incorrectly, you get $32,000. But if you answer correctly, you get something worth $653,200 (factoring in, as we have, the chance of winning $1 million). This additional value means you should guess the answer to the $500,000 question, even though (in my hypothetical situation) you know that there's only a 40 percent chance you will get it right.

The analysis for earlier questions is, as you would expect, even more complex,
and the probabilities vary for each person and question. I used a computer spreadsheet
to calculate the values under various assumptions and learned that many risk-neutral
contestants should answer certain questions (specifically, the $125,000 and
$250,000 questions) *even when they don't have much of a clue about the answer*,
because of the high value of the chance to answer subsequent questions. In short,
they should go for it more often. So why don't they?

Well, we all know people don't always behave rationally. First, consider the "endowment effect," which seems to play a large role on "Millionaire." Simply put, the endowment effect means that we often irrationally value items we possess more than equivalent items we do not possess. In one famous study, students who are given a coffee mug require more to sell the mug than students without mugs are willing to pay to get the exact same mug.

Some "Millionaire" contestants seem to be overwhelmed by the endowment effect. They answer the $64,000 question correctly. Regis shows them a signed check in that amount, and they start to drool. You can tell they're about to head home. Game over. Others seem to pre-program themselves to try to avoid the endowment effect, saying things like "I told myself I would go for it until I got to the $125,000 question." But the endowment effect is powerful and many succumb, especially less wealthy contestants who would benefit greatly from the money they'd walk away with, and would be hurt more by losing even a smaller amount.

There's another distorting factor: risk-aversion. A hypothetical "risk neutral" person is indifferent to risk and would, for example, pay exactly $1 for the chance to win $2 on the flip of a coin. But human beings are often risk-seeking (paying more than $1 on the coin flip) or risk-averse (refusing to pay more than, say, 90 cents on the coin flip). The same people who are risk-seeking while gobbling up lottery tickets are risk-averse on "Millionaire," partly because more money is at stake, partly because those probability calculations above are too hard.

Moreover, several important studies have found that people are risk-seeking when it comes to losses (doubling down in blackjack) but risk-averse when it comes to gains (locking in a gain in a stock). Since "Millionaire" asks people to take risks to get more money, it's a magnet for risk-aversion -- in the same way day trading and casino gambling are magnets for risk-seeking. And risk-aversion and the endowment effect together mean contestants are more likely to take the money and run, than to take a risk of losing.

That means you and I will be screaming "Go for it!" from our sofas for as long as "Millionaire" is on -- to no avail. We can scream all we like, but we should not expect human decision-making to become rational. It's our nature. So, as Regis says, let's play.