Who Wants To Be a $547,954-aire?

I recently finished reading The Know-It-All, A. J. Jacobs’s story of his quest to read the Encyclopaedia Britannica from A to Z. At one point in the book, the author is preparing to compete on Who Wants To Be a Millionaire, and he makes a striking discovery about the game show's rules (p. 331-332):

In the fine print of the ream of documents they sent me, it said that $250,000 is paid in one lump sum – but $500,000 and $1,000,000 are paid out over ten and twenty years, respectively. If you factored in inflation and lost investment opportunities, could $250,000 actually be a better deal? I hope so. I figure that would be a great moment in Millionaire history: I stop at $250,000 and explain to Meredith the intricacies of amortized payments. So I ask Eric – the former investment banker [and author’s brother-in-law] – to crunch the numbers.

He emails back that $1,000,000 over twenty years came out to $540,000 in today’s dollars. That’s before taxes, mind you – but it is still more cash than the other options. Damn. Now I really have to try for the million.

Jacobs is correct, of course, to discount a stream of payments to find its present value. But he reaches the wrong conclusion. While $540,000 is more than $250,000 or $500,000, that doesn’t mean his strategy shouldn’t change. The lower value resulting from payment over time should alter the degree of confidence he needs to guess on a question.

For simplicity, assume the contestant is risk-neutral (that is, he seeks the highest expected monetary value). Also ignore taxes. Finally, assume we’re operating under the new Millionaire rules, under which a wrong answer on the $500K or $1M question will knock you down to $25K (not $32K as under the old rules). Let p denote the contestant’s assessment of the probability he will get a question right, based on his knowledge of subject matter, how many answers he can rule out, etc. If the contestant has earned $250K and is pondering the $500K question, he will maximize his expected value by guessing when p > 0.47. Otherwise, he should quit with $250K. (The threshold p would be 0.5 if missing the question meant walking about with nothing.) If the contestant has earned $500K and is pondering the $1M question, he will maximize his expected value by guessing when p > 0.49 and quitting otherwise. So, as a simple rule of thumb, guess when you can rule out two of the four answers.

But those probabilities change if the money is parceled out over time. Assuming a 7.5% annual rate of interest (the rate that got me closest to Eric’s estimate above), the present value of $500K paid out over 10 years is $368,944, and the present value of $1M paid out over 20 years is $547,954. With these numbers, a contestant facing the $500K question would maximize expected value by guessing when p > 0.65 and quitting otherwise. A contestant facing the $1M question would maximize expected value by guessing when p > 0.66 and quitting otherwise. In other words, to take a stab at that million-dollar question, you need to think there's a 2 out of 3 chance you'll be correct. And for both the $500K and $1M questions, ruling out two of the four answers is not nearly good enough to make a guess worthwhile.

(Here’s another way to think about it. In present value, the potential gain from answering the million-dollar question is not an additional half million dollars, but merely an additional $547,954 - $368,944 = $179,010. So you have to risk $368944 - $25,000 = $343,944 for a chance at getting $179,010 more. You need to be pretty confident to do that.)

Of course, even this analysis is simplistic. It doesn’t account for taxes, which could make the yearly payout more desirable (you might be exposed to a lower marginal tax rate by spreading out your winnings). This will make it more worthwhile to guess. And it doesn’t take into account the diminishing marginal utility of money, which will make you risk-averse instead of risk-neutral. That will make you it less worthwhile to guess. And then, of course, there’s the pure ego value of winning a higher value, which is hard to put in monetary terms.

Still, I think it’s significant that the prize payout schedules, mostly unknown to Millionaire watchers and possibly ignored by many contestants, can make a big difference in optimal guessing strategies.

[NOTE: This post has been updated to correct an error pointed out by Blar in the comments. Also, for the $500K question, I'm ignoring the fact that a correct answer also gets you a chance at answering the $1M question.]