Why "Millionaire" Contestants Should Beware of Guessing

Like Frank Partnoy, I often find myself cringing in agony over the decisions "Millionaire" contestants make. Unlike Partnoy, however, I howl not when a contestant decides to walk away with, for example, $250,000 rather than hazarding a guess at the $500,000 question, but when she does guess at the answer rather than pocketing the cash. More often than not, it seems, such guesses turn out to be wrong. It's painful to watch a contestant slink offstage as she thinks ruefully about what she could have done with that extra 218 grand (the difference between the $250,000 she had in hand, and the $32,000 consolation prize she got after guessing incorrectly). Nevertheless, Partnoy suggests that contestants should take the risk and guess more often. I disagree.

When I first read Partnoy's analysis, I thought he might be right. As he insightfully points out, a correct answer to the $500,000 question on "Millionaire" is worth not just an extra $250,000, but also a chance to win the game's Holy Grail, a cool million. Thus, Partnoy's math shows that even when a contestant is less than 50 percent sure of the answer to a particular question, it may make sense for her to guess anyway, in order to preserve a chance at the bigger prizes ahead. I don't dispute Partnoy's calculations, but on reflection I do doubt their application to "Millionaire" contestants, even hypothetically rational ones -- for two important reasons.

Guessing Is Not A Science

First, Partnoy asks us to imagine a contestant who has won $250,000 and, after reading the $500,000 question, thinks she has a 40 percent chance of answering it correctly. Hold it right there! If the candidate is uncertain which is the correct answer, how is she supposed to know the precise magnitude of her uncertainty? Guesses are usually based on "gut feelings"; whatever alchemy goes into producing such feelings, it is hardly the stuff of precise calculations of likelihood.

not sure what Tokyo was called in pre-modern times. One of the four options she is given is "Edo," and since the sushi bar in her neighborhood goes by that name she thinks it might be the right answer. Such a guess is based on intuition -- the idea that if Edo were indeed the ancient name for Tokyo, it would also be a clever, hip name for a sushi bar. And, as it turns out, the intuition is correct. But at the moment the contestant is ready to guess, how is she to translate her intuition into a precise numerical assessment of how likely it is to be right?

Even if the contestant could determine somehow that she was 40 percent sure Edo was the right answer, wouldn't she also have to determine how sure she was that the 40 percent figure was right? And how sure she was about that level of certainty? And so on, and so on. Pretty soon the problem spirals out of sight and mind, and heading home with the $250,000 in hand starts to look a lot better -- and safer -- than doing dubious calculations until the sun burns out.

Only Repeat Players Should Care About Averages

Regardless of those problems, let's assume for the moment that the contestant facing this $500,000 question does somehow know with 100 percent certainty that she is 40 percent sure that pre-modern Tokyo was called Edo. Under Partnoy's approach, she must now calculate not only the average return for answering the $500,000 question but also the average return for answering the $1 million final question. To do so, she's got to calculate her probability of getting the $1 million question right without even seeing it -- a wild guess indeed.

And even if the contestant could make that wild guess, it still doesn't follow that she should. Assuming she has a 40 percent chance of getting the $500,000 question right and a 40 percent chance of getting the $1 million question right (and making a few other assumptions set out in the article), Partnoy calculates the contestant's average return, if she guesses, as $653,200 -- much more than the $250,000 she would win if she walked away without answering the $500,000. Not too shabby, and apparently a good reason to guess.

But only apparently. The problem here is that Partnoy's analysis describes not what the rational "Millionaire" player should do, but what the rational "Millionaire" repeat player should do. That is, his analysis deals in averages. His calculations are correct as far as they go: if a contestant participates in a statistically significant series of "Millionaire" games, then -- given the assumptions above and in his article -- on average the contestant should receive $653,200.

But "Millionaire" contestants don't play the game a statistically significant number of times. They play only once. Moreover, if the probabilities are as Partnoy posits, the most likely result of guessing is that the contestant will go home with only $32,000 -- and there's a reason the game isn't called "Who Wants To Be A Thousandaire?" That $32,000 result would be okay if the contestant could play again and again, because the steep increase in payoffs would mean that the times when she guesses correctly would more than compensate for the times when she's wrong. But it's not at all okay if the "Millionaire" appearance is a one-shot deal. Then, guessing when you're only 40 percent sure is most likely to result in a wrong guess and a loss of enough money to pay off the mortgage.

Partnoy would characterize the desire to pay the mortgage instead of going for the million as risk-aversion. But in a one-shot situation, risk-aversion is entirely rational for virtually everyone. With repeat play, of course, Partnoy's more risky (but, on average, more winning) strategy would pay off. Without repeat play, it too often will not.

For actual, real-life contestants on "Millionaire" -- who can't calculate uncertainties precisely, and who don't have the chance for repeat play -- I fear there is no system-cracking, one-size-fits-all strategy. But I do know this: if I ever appear on "Millionaire" and have to choose between keeping $250,000 and a 40 percent chance at $500,000 (with an additional, but unquantifiable, chance at $1 million), I'll take the $250,000 and run.

Trevor Morrison is an attorney for the federal government.

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