Jeopardy contestants are smart. But are they rational?

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“Describing anything very showy, in architecture it refers to a style using curves like tongues of fire”

Think you know the answer? Or rather I should say, do you know the question?
This challenge, the Final Jeopardy clue from an episode in February 2015, is a stumper. None of the three contestants on the show got it right. They each guessed “ornate”. The correct response is “flamboyant“. Dava-Leigh, the ultimate champion, won not because of her superior trivia knowledge, but because Ashley and Christina, her opponents, wagered suboptimally heading into the final round.

To succeed on Jeopardy you must be smart. You should know the world’s capital cities, all of the Presidents of the United States, and words that begin with the letter ‘Q’. If you made it onto the show then chances are you are very knowledgeable (or very lucky, a valuable skill in its own right). Candidates for Jeopardy must pass an online test as well as an in-person interview before qualifying to be a contestant. But brains will only take you so far. Keep in mind, that although you may do well to pass the qualifications exams, so do your opponents.

When watching at home I always shout out guesses at the TV screen. I blurt out responses before my friends can do the same. But of course, it doesn’t work like that on the actual show. To really succeed on Jeopardy, you must master the buzzer. And the key to the buzzer isn’t speed, it’s timing. Contestants must wait until Alex finishes reading the clue in order to ring in. Click the buzzer too soon and the system locks the player out for a fraction of a section. But waiting too long isn’t any better – an opponent will steal the chance to answer.

In talking about his secret to success, Jeopardy celebrity Ken Jennings explained, “Jeopardy! victory most often goes not to the biggest brain; it goes to the smoothest thumb. Timing on the tricky Jeopardy! buzzer is often what separates the winner from the, well, non-winners, and the Jeopardy! buzzer is a cruel mistress.” Jennings, as well as co-contestant Brad Rutter, also blamed the cruelty of the buzzer for their loss to IBM’s thoughtful computer Watson. “The key to Watson’s dominance lies in the famously tricky “Jeopardy!” buzzer, the signaling device that allows players to respond to the show’s clues . . . If it knows the answer, it makes the perfect buzz. Every single time.”

The one area of the game where the buzzer has no jurisdiction is Final Jeopardy. During the last round of play each contestant gets a chance to wager a dollar value and then answer the revealed clue. Speed is not the determining factor here, but instead math plays an integral role. Placing the correct bet could mean the difference between a $1,000 consolation prize and large cash jackpot.

I won’t get into the details of the game theory behind wagering strategy, but instead I’ll provide a very simplified view. The leader heading into Final Jeopardy wants to protect his lead and so should bet $1 more than double the sum of the second place contestant. The person in second place could assume that the leader will act rationally as just described and will want to bet just enough to beat out the leader in case he gets it wrong, while at the same time protecting against the usurp-ready trailer in third. The last place contestant is hoping the other two falter to take advantage of their mistakes.

Let’s look back to Dava-Leigh and walk through a real life example. Heading into Final Jeopardy, Christina had a small lead at $13,900 with Dava-Leigh behind her at $11,000, and Ashley trailing at $7,400.

Christina Dava-Leigh Ashley
$13,900 $11,000 $7,400
Scores heading into Final Jeopardy

How should they bet according to the rational, mathematical approach? Christina should try to cover a correct guess by Dava-Leigh by wagering $8,101 (2 X $11,000 – $13,900). Dava-Leigh should wager at least $3,801 to cover a doubled score by the trailing Ashley (2 x $7,400 – $11,000), but not wager more than $5,200 to beat Christina in case everyone gets the question wrong ($11,000 – [$13,900 – $8,100]). Ashley is praying her opponents both get the question wrong and she can rise from the ashes to steal the win. She should consider wagering $1,600 which would beat Christina in an all-wrong ending ($7,400 – [$13,900 – $8,100]).

Christina Dava-Leigh Ashley
$8,101 $3,801 – $5,200 $1,600
Suggested Final Jeopardy wagers

Phew! That’s a lot of algebra and game theory for a 30-second question. But it could mean a lot of cash in the pocket. These calculations assume each player acts rationally and does the same math. As you could imagine, that doesn’t always happen and we’ll look at some of the data in a bit. But let’s see how how our contestants wagered compared to suggestions we just figured out.

Christina Dava-Leigh Ashley
$13,900 $9,500 $7,400
Actual Final Jeopardy wagers

Christina and Ashley bet everything in an all-or-nothing gamble to win. This is actually fairly common behavior. On the positive side it maximizes winnings in a victory, but it can have negative consequences in case of an incorrect response. Dava-Leigh also bet high, although not everything. Perhaps she did not have a lot of confidence in the category: Word Origins.

In this particular game, everyone answered incorrectly, and so Dava-Leigh came out on top simply because she did not bet it all. The game would have played out much differently if the contestants used game theory. All players would end up with a score around $5,800 because they protected against just such a scenario.

So do Jeopardy contestants regularly bet rationally?
I looked at over 100 games from the current season (Season 31) to find some insight.

In about 25% games, one player clinched victory before the clue was even revealed. This can happen in two ways. The leading contestant can have more than twice the score of the next contestant, in which case betting zero would automatically win (see Cliff Clavin on how to ruin a sure thing). Or another way is if two out of the three contestants have a negative score and don’t even qualify for the final round. These are not very interesting examples for wagering analysis.

Let’s look at the other episodes and break it down contestant by contestant. The leader has the simplest strategy – bet to cover the person in second. Surprisingly only about 85% of leaders bet the recommended amount or a little more.The others bet under the amount needed to cover. Perhaps these were cases where the category frightened them?

The most interesting strategy potential lies with the candidate in second place entering Final Jeopardy. These players are stuck in the middle, but do they know what it is they should do? Actually yes. Based on the season 31 games, the middle contestant bets rationally about three-quarters of the time. That’s not to say they bet optimally though. Many times a reasonable bet will lie within a range of values, but the optimal bet will be to bet at the high end or the low end to maximize winnings or minimize loss. These situations are known as Stratton’s Dilemma, and although the contestant can bet within the proper range, she should probably max out if she’s going for the kill.

Most of the time when the second player did not bet rationally, she bet too high. This is likely because she was focused on beating out the leader without protecting against the third place contestant in case both she and the leader got the question wrong. Overall the middle player bet everything 25% of the time.

The last place contestant often times doesn’t even have much to think about. If his score is less than the difference between the scores of the first and second place players, as it was about 35% of the time, then there is not much strategy involved because the other contestants will always end with more if they bet rationally. In the remainder of the cases, the trailer is thinking about a situation where everyone gets it wrong, otherwise betting and praying he is the only one who answers correctly. I didn’t check how many times a last place contestant actually won, but I’m guessing not many.

The next time you watch Jeopardy on television, remember it’s not only trivial knowledge that’s important. Winners are quick-thumbed and gambling-saavy as well.

 

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