Final Jeopardy: A Bayesian Approach

Jeopardy

After years of submitting and resubmitting your application, you’re finally invited to try out for the TV quiz show Jeopardy. You complete a surprisingly easy written exam administered at one of the show’s testing centers in your city, and within just a few weeks of the initial invitation, you find yourself on an all-expenses-paid voyage to L.A. to be a contestant. On the day of the taping, just before entering the set, you get to meet and chat with your two opponents in the green room. Your stomach is in knots at the realization that you are about to have your intellect tested in front of an audience of millions and to make things worse, you’ve developed a rather inopportune case of the hiccups from nervously gulping the complimentary bottled water given to you by one of the production assistants.

Your opponents, on the other hand, couldn’t be any more calm. You soon find out that this is due, at least in part, to the fact that both are returning champions, having played to a draw on the previous episode. One is a professor of computer science who seems to be in her late forties and has been on a 5 episode tear over the past week. Her warmth and genuinely sunny disposition only serve to heighten your insecurities as they are blatant reminders of how utterly unfazed she is by the whole situation, thus reinforcing your sneaking suspicion that you’re about to be trounced on national TV. The other champion is a thirty-something journalist from a small town newspaper. His handshake strikes you as being cold and detached. Unlike his fellow champion, his experience spans only one episode and he has yet to secure a unitary victory. Perhaps consequently, unlike his co-champion, he’s not interested in making small talk and instead he retreats to the solitude to the make-up room, where he can focus his thoughts and get in the zone.

When the game gets underway, the professor gets off to a lightning fast start, opening the proceedings by correctly and consecutively answering all five questions in the category of her choice. It soon becomes clear that the small-town journalist is a one-hit wonder, likely having lucked out with the categories in his first outing. Visibly frustrated, he can’t seem to get off the ground and he ends the first round twelve hundred dollars in the red. After answering several questions correctly, you realize that you actually know most of the answers but you’re no match for the champion’s impeccable timing with the buzzer, a skill she honed with the benefit of a week’s worth of hands-on experience. When the host, Alex Trebek, announces a pause for the final commercial break, the journalist, having managed to amass a measly $800 compared to the professor’s $16 000, turns to the his co-champion and congratulates her on what is sure to be a sixth victory. You struggle to conceal your anger at having been written off by an opponent who hasn’t even managed to earn a fifth of your winnings so far and you vow to fight to the bitter end.

The professor begins the final segment much the same way she opened the match, by racking up a string of five correct responses. You can feel the game slowly slipping away from you, when all of a sudden the momentum shifts as you discover that you have the edge in before-and-after puzzles. Sensing that time is running out, you rush to get to as many of them as possible, choosing the highest-valued clues first to maximize your gains. Right after you request the $400 clue, your heart sinks as the bell sounds signifying the end of the double jeopardy round. Recalling that the professor has maintained a score significantly more than double your total for most of the game, you fear that she’ll run away with the victory and final jeopardy will end up being purely academic. You allow yourself an internal chuckle at your own pun and then realize that, in your haste, you hadn’t even noticed that the final clue happens to be the second daily double of the round. As Alex, asks for your wager amount, you quickly glance at the scoreboard and to your immense surprise you have exactly half of the frontrunner’s total. The score is $18 000 for the professor, $9000 for you, and the journalist’s score remains unchanged at $800.

Not only have you held your own by avoiding being doubled, but you could potentially go into final jeopardy tied with the champ if you chance it and risk everything. You estimate that, when the buzzer is not a factor, you have a 75% chance of responding correctly to any given clue, and you pessimistically peg the professor’s chances at 90%. Assuming that these estimates hold true irrespective of the category and that all players seek to maximize their chances of living to play another day, what should you wager in this last question before the final jeopardy round?

a) $0
b) $1
c) $1000
d) Everything

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Scroll down to see the answer.

 

 

 

 

 
The correct answer is a) $0.

The trick to solving this problem is to remember that, if there is a tie for first, both players will move on as co-champions and will face off against a third challenger in the next game as alluded to in the second paragraph. Also, notice the wording. Each player seeks only to maximize her chances of surviving into the next game. So by wagering $0, you go into final jeopardy with exactly half the professor’s total. Now, in final jeopardy, the professor can and will guarantee her chances of moving on by betting $0 because she knows that the best you can do is to tie her, in which case she remains the champion. On the other hand, if she bets even $1 she leaves open the possibility that she could end up with less money than you if you risk everything and answer correctly. Therefore by wagering $0 in the daily double, all you have to do is answer the final jeopardy question right, which you have a 75% chance of doing, and you will live to fight another day.

Wagers of $1 and $1000 both carry the same outcome. If you get the question wrong (25% probability) you lose automatically because the professor’s total will be out of reach in the final round. On the other hand, if you get it right, You’ll still trail the professor by a significant margin, and will have to answer correctly on Final Jeopardy to have a chance at staying alive. Essentially, this option requires you to answer two questions correctly instead of only needing one correct answer, as would be the case with option a.

The final option is tempting, because it puts the outcome entirely in your own hands. If you answer the daily double question correctly, then you go into double jeopardy tied with your opponent. But remember, you must first answer the daily double question correctly, which is only a 75% probability. But you know that your overall chance of survival is less than that because, in final jeopardy, you still require some additional luck. Assuming both players risk everything (a safe assumption according to game theory), you must either answer the question correctly, in which case you’ll win or tie, or both of you must get it wrong, in which case you’ll tie. All told, by choosing (d) and risking everything, you have only a 58.125% chance of survival.

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