IBM Watson's Jeopardy! Wagering Strategies HuffPost

I've seen some people throwing around some stats here on various things and I'm wondering if there's some good compiled source for that. I'm writing a paper for an intro game theory class on final jeopardy wagering strategy and could really use some stats on final jeopardy correct answer %, average winnings, that kind of stuff. I've taken some econometrics and statistics before but I'm not anywhere close to an expert (read: don't need each individual stat spoon-fed to me but can't do anything complicated either). Any help?

Let's meet today's contestants:

• Deanna, a communications specialist, did public relations for the San Francisco 49ers;
• Kevin, a screenwriter, almost ran into Stephen King's mailbox; and
• Andy, a writer, started a comedy festival at times headlined by Patton Oswalt. Andy is a two-day champ with winnings of $47,598.

Deanna took the lead on DD1 and extended it on DD3, leading into FJ with $17,800 vs. $12,600 for Andy and $7,900 for Kevin.
DD1 - $800 - AROUND THE WORLD - This largest Alaskan island is home to the largest type of grizzly bear (Deanna won $2,000 from her score of $4,000 vs. $5,200 for Andy.)
DD2 - $1,600 - STARTS WITH "E"- A Christian holiday, or any sudden insight (Kevin added $1,500 from his third-place total of $1,500.)
DD3 - $1,200 - ONE-SUBJECT NONFICTION - Vince Beiser's "The World in a Grain" is about this substance that concrete & glass, among other things, are made of (Deanna improved by $2,000 from her score of $13,800 vs. $11,400 for Andy.)
FJ - COUNTRY NAME ORIGINS - This island country was named for a 16th century Spanish king whose name comes from the Greek for "lover of horses"
Andy and Kevin were correct on FJ, with Andy adding $4,600 to win with $17,200 for a three-day total of $64,798.
Pop culture problems: No one knew the "E.B." who provided a voice in "The Lego Movie" is Elizabeth Banks, or could name "Sons of Anarchy" spinoff "Mayans M.C." (simply called "Mayans" by Alex).
That's before our time: The players didn't get the "this, this, this" you do after you "pack up your troubles in your old kit bag" in an old song is smile, smile, smile.
Pedantry corner: So it's Larry the Cable Guy, a character, that provides a voice in "Cars", not Dan Whitney, the actor who plays Larry the Cable Guy. Sure, why not.
Correct Qs: DD1 - What is Kodiak? DD2 - What is epiphany? DD3 - What is sand? FJ - What is The Philippines?

Please welcome today's contestants:

• Mandy, a project specialist, tried out for the show with her dad some time ago;
• Claire-Marie, a film advertising executive, watched Jeopardy! with her grandmother; and
• Garrett, a software engineer, is a dedicated mountain climber. Garrett is a one-day champ with winnings of $14,000.

Garrett extended his lead on DD3, but Claire-Marie finished nicely to stay well within range going into FJ with $13,800 vs. $18,200 for Garrett and $3,300 for Mandy.
DD1, $600 - WORDS AT THEIR FINAL RESTING PLACES - "Mother of the modern day civil rights movement" (Claire-Marie moved to the lead by adding $2,600 from her total of $2,000.)
DD2, $2,000 - 1990s BEST PICTURE BY TAGLINE - "It's a hell of a thing, killing a man" (Mandy lost $1,500 from her third-place score of $4,400.)
DD3, $1,200 - SPEED READING - It's the 4-letter title of Lisa Patton's novel about students caught up in the frenzy of matching students with sororities (Garrett won $4,000 from his leading total of $11,000 vs. $6,600 for Claire-Marie.)
FJ - GEOGRAPHY FUN - It's the largest country in area that begins & ends with the same letter
​​Garrett and Claire-Marie were correct on FJ, with Garrett adding $9,500 to win with $27,700 for a two-day total of $41,700.
Wagering strategy: By betting $13,000, Claire-Marie forced herself to be correct to have a chance to win. The percentage play would be to bet $5,000 or less, shutting out Mandy from second place and allowing Claire-Marie to win if Garrett missed even she was incorrect herself.
This day in Trebekistan: Alex helped fuel speculation about the next host by saying he's "looking over (his) shoulder" at Ken, who presented a category today.
Judging the writers: I'm not sure what was especially "fun" about the FJ clue as was promised by the category, but I guess that's just me.
​Correct Qs: DD1 - Who was Rosa Parks? DD2 - What is "Unforgiven"? DD3 - What is "Rush"? FJ - What is Australia?

Introducing today's contestants:

• Laurel, a graduate student from Florida, will have her novel out in 5-10 years;
• Steve, a psychology professor from Florida, raps about anything, including Canada; and
• Gabe, a researcher from Washington, D.C., has hundreds of thousands of baseball cards and might buy Ted Williams from Alex. Gabe is a one-day champ with winnings of $22,400.

Laurel extended her lead to nearly $13K on DD2, then Gabe doubled up on DD3 to close the gap. Going into FJ it was Laurel with $18,400, Gabe at $15,600 and Steve with $12,200.
DD1, $400 - MUSEUM-PODGE - The Vasa, a warship that spent 333 years under the sea, was salvaged and turned into a museum in this capital city (Gabe won the table limit of $1,000.)
DD2 (video), $1,200 - WATER - A painting depicts this river that, in Switzerland, plummets over the Schaffhausen waterfall (Laurel won $3,000 from her leading score of $14,600.)
DD3, $2,000 - SOME STERN WORDS - Unwilling to change your opinion? You're this, from the practice of coloring fibers before they were woven into cloth (Gabe won $6,400 on a true DD.)
FJ - EPITAPHS - From his own 1898 poem, the epitaph of this Irish-born man ends, "His mourners will be outcast men, & outcasts always mourn"
Only Laurel was correct on FJ, adding $12,801 to win with $31,201.
Language liabilities: No one knew an adjective for a stern person beginning with a rock that makes a spark is "flinty", or the word for soap foam that also means a state of being upset is "lather".
This day in Trebekistan: Alex is still pronouncing the name "Cartman" from South Park as if he's some kind of wagon-based superhero.
Correct Qs: DD1 - What is Stockholm? DD2 - What is the Rhine? DD3 - What is "dyed in the wool"? FJ - Who was Wilde?

So I've been thinking about this a bunch lately. If first place and second place are reasonably close, 2nd place should never bet. Why? Let's look at a scenario.
Bob has 10,000. Mary has 9,000. Bob bets 8,001, Mary bets 9,000.
Both wrong. Bob - 1,999; Mary - 0
Both right. Bob - 18,001; Mary - 18,000
Bob's right. Bob - 18,001; Mary - 0
Mary's right. Bob - 1,999; Mary - 18,000
The only way Mary wins is if she gets it right and Bob gets it wrong. Any other scenario leads to a loss. But what if she doesn't bet?
Both wrong. Bob - 1,999; Mary - 9,000
Both right. Bob - 18,001; Mary - 9,000
Bob's right. Bob - 18,001; Mary - 9,000
Mary's right. Bob - 1,999; Mary - 9,000
Now, Mary has two outs: if they're both wrong, or if Mary's right, then she wins. But what if the scores aren't so close?
Bob has 10,000. Mary has 6,000. Bob bets 2,001, Mary bets 0.
Both wrong. Bob - 7,999; Mary - 6,000
Both right. Bob - 12,001; Mary - 6,000
Bob's right. Bob - 12,001; Mary - 6,000
Mary's right. Bob - 7,999; Mary - 6,000
No paths to victory for Mary here. She needs to bet aggressively to even have a chance. But then where's the cutoff? It's the point at which Bob's score minus Bob's necessary bet to beat Mary's highest potential bet is less than Mary's score, which is to say, 2/3rds. Let's look at one more scenario:
Bob has 10,000. Mary has 6,667 . Bob bets 3,334, Mary bets 0.
Both wrong. Bob - 6,666; Mary - 6,667
Both right. Bob - 13,334; Mary - 6,667
Bob's right. Bob - 13,334 ; Mary - 6,667
Mary's right. Bob - 6,666; Mary - 6,667
Since the lead player can never count on the second player not to bet everything, they have to bet assuming they have a number to beat. By playing on that expectation, the second place player can double their potential to win by not betting, as long as their score is higher than 2/3rds of the leader's. If it's less then, but greater than a half, aggressive betting is necessary (or, at least, sets the odds back to one scenario in four). Obviously, if it's less than a half, they've already lost.

Jeopardy! recap for Tue., Jan. 26 - Excellent game in which champ Fred appeared to be in trouble after Sonam made $3,000 on DD2 to open a solid lead. But Fred came on very strongly with a stellar performance in a category about Africa to nail down first place going into FJ with $20,800 vs. $16,200 for Sonam and $8,600 for Gerry.
DD1 (video) - HOUSE PROUD - New Orleans has many of these narrow elongated dwellings named for a type of firearm (Sonam won $2,000)
DD2 - A LITTLE ALLITERATION - Helvetica is this kind of typeface that by definition lacks the fine cross-strokes at the top & bottom of letters (Sonam won $3,000)
DD3 - IRISH PLAYWRIGHTS - After World War II he began writing mostly in French instead of his native tongue (Gerry lost $2,000)
FJ - 20TH CENTURY AMERICA - The 1970 follow-up line “We’ve had a main B bus undervolt” was addressed to a listener in this city
Alex tipped us off that no one would be correct on this FJ with his comments about how tough the clue was and that the wagers would be important. Sonam would have come away with the victory if she had bet just enough to cover twice of Gerry's score, but instead she lost $12,201 and dropped to $3,999.
Fred made a strange wager of just $7,500, not enough to cover double of Sonam's score, so he would have lost if Sonam had been right, regardless of if he was correct. As it played out, Fred held on with $13,300 and a three-day total of $49,600.
One more thing: There was a bit of chuckling when the correct response to one of the clues was "daily double", in reference to the horse racing bet from which the Jeopardy! feature got its name.
Correct Qs: DD1: What is shotgun? DD2: What is sans sarif? DD3: Who was Beckett? FJ: What is Houston?

Simple and scary:

1. On a day-to-day performance basis, James is the new Ken Jennings. On just about every major statistical metric that Andy Saunders thoroughly tracks over at https://thejeopardyfan.com, James is even a bit superior, day to day, to 2004 streaking Ken Jennings. EXCEPT:
   
2. James has a far more aggressive and effective wagering strategy than Ken, increasing the likelihood that he goes into FJ with a prohibitive lead, as evidenced in:
   
3. In 75 games, Ken went into Final Jeopardy with a runaway/lock win 65 times. The other 10 times, he controlled the game, leading and therefore needing only to get FJ correct and bet to cover second place in order to win. James has an even better ratio, with only 2 of 22 games not being lockouts, and those two being in his control at FJ.
   

In other words, James could be around for a very long time. He is the perfect storm Jeopardy contestant, A combination of high intelligence, broad knowledge base, quick recall, and precise buzzer timing together with optimal, aggressive wagering and board play. It will therefore take the perfect storm game for him to be defeated. What he is doing is stupendous.
It will be a nice change of pace to have some time off from his singularly intense and effective approach to the game with the two weeks of the Teachers Tournament starting Monday. As viewers, we will be thoroughly recharged and champing at the bit for more James come May 20.
The questions which remain are:

1. How long does James keep it up, and what ultimately trips him up?
   
2. How much does he earn before he is done, and how many records does he break?
   
3. How soon after he is done can we get him, Ken and Brad together for a little friendly competition? The Jeopardy universe commands that this take place.
   

Heading into Final Jeopardy, James Holzhauer was in a unknown territory. He was in second place with a score of $23,400. His opponents had $11,000 and $26,600 respectively. Down by $3,200, his answer is revealed and, as usual, he is correct. His wager? A paltry $1,399. At home, you're screaming at him from the couch. "How could you only bet $1,399? Are you throwing the game? Don't you want to beat Ken?"
However, he made the best possible bet to maximize his chances of winning the game, and here's why:
Emma has the lead going into Final Jeopardy. Her betting strategy is simple: Wager enough to win by at least $1 even if her closest competitor wagers everything. This guarantees a victory regardless of what her opponents do as long as she can come up with the correct answer. With James sitting at $23,400 she needs to bet enough to get to at least $46,801 if she gets it correct. A wager of $20,201+ will do the trick.
James knows what Emma is going to do. His only chance of winning is for Emma to get it wrong. Even if James bets everything, he still loses if Emma gets the answer correct since she will wager at least $20,201. By betting $1,399, James ensures a victory if Emma gets it wrong regardless of whether he or Jay answer correctly. If James bets more than $1,399, Jay still has a chance to win in the rare case that James/Emma both get it wrong and Jay gets it correct and wagers everything.
Let's calculate James' odds of winning with a bet of $1,399 vs a bet of $1,401+.
​
Going into this game, James has been correct on 31/32 Final Jeopardy questions (96.88%). This season, Final Jeopardy has been answered correctly 54.58% of the time (298 correct responses out of 546 possible responses out of the 187 games this season per J!Archive). James skews those numbers a little bit, so removing his data results in 267/514 or 51.95% for the average player to answer Final Jeopardy correctly this season. We'll give Emma the benefit of the doubt since she played extremely well and say she's an above average player that will get Final Jeopardy correct 60% of the time (a reasonable approximation. Ken Jennings himself was correct on Final Jeopardy 68% of the time for comparison). We'll label Jay as an average player and give him a 51.95% Final Jeopardy accuracy rate for the purpose of this analysis.
Below are 3 scenarios of various betting ranges from James along with the corresponding winning percentage for each player assuming Emma bets $20,201+:
Scenario 1: James bets between $0-$1,399.
James: 40%Jay: 0%Emma: 60%
It's fairly straightforward. James' bet of $1,399 ensures he will stay ahead of Jay regardless of what he does. James' only chance to win is for Emma to answer incorrectly, which should happen 40% of the time using the estimated FJ accuracy numbers. James does not need to be correct to win, so his 96.88% accuracy doesn't factor into this scenario.
Scenario 2: James bets between $1,401-$17,000, Jay bets everything.
James: 39.35%Jay: 0.65%Emma: 60%
Emma still wins if she gets it correct. However, James has more chances to lose now. If James and Emma are both incorrect, and Jay is correct, then Jay will now win. The odds of this occurring is 0.40*0.03125*0.5195=0.00649375 or ~0.65%
Scenario 3: James bets between $17,002-$23,400, Jay bets everything.
James: 38.75%Jay: 0.65%Emma: 60.60%
Emma's winning percentage is slightly higher now because she can now win even if she gets FJ wrong, as long as James and Jay also get FJ wrong.
​
Note: I excluded bets of exactly $1,400 and $17,001 because those would result in a sudden death tiebreaker in some scenarios which are too difficult to gather data on due to how infrequently they occur.
​
Being a professional gambler is all about taking advantage of small edges. Even though the winning percentage for all 3 betting scenarios is pretty similar, his bet of $1,399 is the highest amount he can bet where his winning percentage is capped at 40%. A higher bet will result in a higher game total if he wins, but lowers his winning percentage very slightly. In a game like Jeopardy where winning is everything, James made the correct wager to give him the best odds of winning.
​
Some other things worth mentioning:
​
The best possible bet for Jay is actually $4,600. This allows him to finish higher than Emma if FJ is a triple stumper, though his odds of winning are still 0% given James' bet of $1,399. His bet of $6,000 forced him to be correct on FJ to have a chance at finishing higher than 3rd place.
Although the FJ accuracy percentages for Emma and Jay are approximated for the purpose of this analysis, Scenario 1 still results in a higher win percentage for James than Scenario 2 and Scenario 3 for all FJ accuracy values greater than 0% and less than 100% as long as Emma always bets $20,201+
Unbeknownst to Emma, a FJ bet of $0 would have given her a lock on the game if she were to ascertain that James would make a brilliant counterbet of $1,399. If this episode was recorded long after James' episodes had begun airing, it might actually be a viable/realistic strategy given James' professional gambler background and betting strategies, though at this point his strategy when betting from 2nd place in FJ is unknown since this is the only time in his run that this occurred.
​
Data Sources:
http://j-archive.com/finalstats.php?season=35
https://thejeopardyfan.com/statistics/ken-jennings-final-statistics

Jeopardy! recap for Wed., Feb. 10 - The second round of the College Championship wrapped up with Sarah (Pitt) opening a large lead with a correct response to DD2, after which Ziad (NC State) was able to cut the margin by being right on DD3. There was a critical $4,000 swing late when Ziad missed a $2,000 clue that Sarah got, allowing her to lead into FJ with $18,200 vs. $17,800 for Ziad and $7,000 for Kate (Northeastern).
DD1 - SCIENCE TERMS - This imaginary line used to indicate 0 degrees longitude terminates the North & South Poles (Sarah won $2,000)
DD2 - SMALL COUNTRIES - About 600 small islands make up this Pacific Ocean country whose names means “small islands” (Sarah won $3,800)
DD3 - SUBCULTURES - Poetry read the Coexistence Bagel Shop was part of this 1950s movement (Ziad won $4,000)
FJ - CHILDREN'S AUTHORS - At 24 he began a verse retelling of the Cupid & Psyche myth, including a character named Caspian
Everyone was right on FJ with Sarah adding $17,401 to win with $35,601. She'll advance to meet Niki and Sam in the two-day final.
Wagering strategy corner: Ziad had a golden opportunity to take control of the game when he found DD3 with $10,600 vs. $15,800 for Sarah. His $4,000 wasn't large enough, as it didn't even put him in the lead with a correct response, but would have landed him in a very tough spot if he missed. Under these circumstances, he should have just gone all-in and would have won the game had he done so.
Triple Stumpers of the day: In a category about Cabinet departments, no one took a guess that the Office of Tax Policy part of the Treasury Dept. Also, the players missed several clues in a category about fight songs, including the school associated with the Cougars from Provo, Brigham Young.
Correct Qs: DD1: What is the Prime Meridian? DD2: What is Micronesia? DD3: What is Beat? FJ: Who was C.S. Lewis?

Jeopardy! recap for Mon., Nov. 19 - The two-day final of the Teen Tournament began with these players:

• Maya, a senior from Georgia, won two writing awards without knowing she was in the running;
• Claire, a senior from Florida, is captain of the golf team; and
• Emma, a junior from Maryland, is a five-time mythology contest winner.

Nearly all of the clues were responded to correctly in this tight three-way battle. Thanks in part to correct responses to both DDs in DJ, Claire showed the way into FJ with $18,600 vs. $15,600 for Emma and $12,800 for Maya.
DD1 - ELEMENTAL - The first letter of the medication Klor-con tells you that it's helping people with low levels of this element (Maya won $600)
DD2 - ART SUBJECTS - 1-word term for the Virgin Mary; there are paintings of her "and Child with Saints" & "...with Angels" (Claire won $2,000)
DD3 (video) - FLAGS - The flag shows the sun setting over wavy blue and white stripes that represent the Pacific Ocean, a reminder that this is Canada's westernmost province (Claire won $3,000)
FJ - AMERICAN WRITERS - In a twist of irony, he accidentally set fire to some 300 acres of woods at Fair Haven Pond near the Concord River in 1844
Claire and Emma were correct on FJ and both made modest wagers, with Claire improving by $3,200 to $21,800, while Emma added $3,400 to finish with $19,000. Maya dropped $5,000 and will take $7,800 into tomorrow's deciding game.
That's before our time: The players didn't give the response the clue wanted about the Y2K scare, based on the fear that computers would read the year 2000 as 1900. However, after initially being ruled incorrect, the judges accepted Maya's response of "00".
Wagering strategy corner: Since no one left in this tournament is walking away with less than $25,000 but can earn more if they can exceed that amount, the structure strongly encourages players to bet big at every opportunity (especially given the overall ease of the clues).
Correct Qs: DD1 - What is potassium? DD2 - What is Madonna? DD3 - What is British Columbia? FJ - Who was Thoreau?

OK, so not actually poker, but I think the strategy will be appreciated on poker. The story is here, among other places.
Jeopardy is basically equivalent to a 3-person tournament in which only first place has a substantial payout, so ICM calculations would put tournament dollars approximately equivalent to real dollars, so cash game strategies apply.
What this guy has done is to purposely seek out the Daily Doubles, which allow you to wager up to all of your money (i.e., your whole stack) on a question. Players get these Daily Doubles correct about 70% of the time (see here and here), so when he gets one he goes all-in. (Even when he is bad at a particular category, he still seeks out the Daily Doubles and wagers a token amount, which denies other players the opportunity to play that Daily Double.)
He is breaking Jeopardy tradition in a couple of ways. First, people typically work top-down in each category. But the Daily Doubles are usually in the 3rd, 4th, or 5th rows, so he goes for those first. Second, the average wager on a Daily Double is about $1500 (and typically don't vary hugely from that amount), but he's been going all-in (when he's 70% or better) or betting almost nothing (when he knows he's < 50%). So some people are annoyed. I'm just surprised that no one has done this before.

Everyone’s wondering how long it’s going to take for somebody to draw an inside straight and beat this great champion. And we have speculated about what it would take for him to lose. Much as he seems to be the perfect storm for Jeopardy in one champion, it will take something like a perfect storm to beat him.
So what do you think? Knowing what we know, if you lined up against him, what would be your approach?
Here’s my thinking: one potential “weakness“ in his approach is that, rather than pure daily double hunting in the first round, he goes straight to the bottom. So if I got control of the board in the early phases, I would immediately go to the $600 and $800 clues in the categories which seem more likely to yield a daily double. Strategic Goal: get DD1 before he can.
And monitoring my point total, to get the jump on James, I’d ring in through the rest of Round 1 before I had the answer in my head, particularly in a category where I feel comfortable. Go Ken Jennings, and hope you get it right.
Round 2, I would do everything I could, strategically ringing in preemptively if necessary, to get control of the board before James did, and pursue his single-minded hunt for the daily doubles, hitting middle value clues in categories more likely to yield a DD, hoping to grab at least one before him and convert. Strategic Goal: try to catch one of the DJ Daily Doubles, or both, before he can.
I would certainly Forrest Bounce him, given the chance, and call out partial names of categories in a way designed to minimize the hint to him and the other opponent. Strategic Goal: grab points before he can and try to get him off balance, if possible.
Realistically, to have a chance to beat him, unless I drew extraordinarily favorable categories that did not favor him, I would hope to come into Final Jeopardy with more than half his total. Then I would cross my fingers and hope to convert on final while he misses it. If really fortunate, at over 2/3 or 80% of his score, wagering strategy would come into play and I could hope for him missing or us both missing, provided the third-place contestant couldn’t snag me from behind on a miss.
And if ridiculously fortunate, and somehow ahead of him going into FJ, I would probably bet to cover in all events and hope I get it right.

Jeopardy! recap for Wed., May 9 - Introducing the contestants for game three of the Teachers Tournament:

• Judy, 8th grade English from Fortescue, NJ, whose student was baseball's Mike Trout;
• Scott, high school history and psychology from Portland, OR, who taught English in Mexico; and
• Mary Alice, 6th-12th choir from Dowagiac, MI, who is also an opera singer.

It was mostly a battle between Scott and Mary Alice, but after trailing badly most of the way, Judy got on a roll late, with a good chance to qualify if she could double up on FJ. The scores going into Final were Scott with $14,900, Mary Alice at $12,000 and Judy with $7,600.
DD1 - IN THE STATE MUSEUM - Andrew Jackson's leather wallet (Mary Alice lost $1,000)
DD2 - LITERATURE IN SPANISH - Mario Vargas Llosa based "The Time of the Hero" on his harsh treatment in military school in this South American capital (Scott won $500)
DD3 - POLITICAL TERMS - Sen Al Beveridge coined this term in 1912 for political support at the local level "grown from the soil" of people's needs (Judy won $2,000)
FJ - OPERA - A 12-minute piece of music from this opera depicts alpine dawn, a storm & the calm, & ends in a section called a galop
Even though the FJ category seemed to favor opera singer Mary Alice, only Judy was correct and made the proper all-in bet to advance with $15,200. Scott wisely made a small $500 wager to finish at $14,400 and in the wild card lead. Mary Alice went big as expected, losing $10,000 and was eliminated from the tournament with $2,000.
Wagering strategy corner: Scott had been doing very well in the Literature in Spanish category when he found DD2 with $7,200 to wager. Strangely, he only bet $500 despite his proven success, while an all-in bet would have virtually assured Scott of advancing, with plenty of time to recover if he missed.
That's before our time: No one knew the Disney live-action movie named after a Bratislavian sheepdog is "The Shaggy Dog".
Correct Qs:
DD1 - What is Tennessee? DD2 - What is Lima? DD3 - What is grassroots? FJ - What is "William Tell"?

Jeopardy! recap for Mon., May 13 - The opening game of second round of the 2019 Teachers Tournament featured these contestants:

• Sara, middle school instrumental music from Maryland, whose students have to study music;
• Trevor, high school Spanish from California, teaches Spanish because he likes Spanish; and
• Joe, middle school special ed from Washington state, says the foundation of teaching is respect.

Despite not finding either DD in DJ, Sara was able to maintain a lead most of the round and show the way into FJ with $14,400 vs. $10,600 for Trevor and $6,800 for Joe.
DD1, $600 - BIOGRAPHY SUBJECTS - Peter Guralnick's book about "The Rise of Elvis Presley" is aptly titled "Last Train to" this city (Sara won the window maximum of $1,000.)
DD2, $1,600 - COLORFUL SHAKESPEARE - At the end of a major battle in this play, the future Henry VII declares, "We will unite the white rose and the red" (Trevor lost $2,000 from his score of $9,800 while trailing Sara by $3,000.)
DD3, $1,200 - NOBEL PEACE PRIZES - The declaration of the rights of the child was a pivotal achievement for this organization (Joe won $3,400 on a true DD.)
FJ - AMERICANA - John and Priscilla Alden lie in the USA's oldest maintained cemetery, which like a poem about the couple, is named for this person
Everyone was incorrect on FJ, and despite a record of 1-for-4 on DDs and 0-for-2 on FJs, Sara advanced to the final, dropping $6,801 to win with $7,599.
Strategy corner: Trevor wagered nearly everything on FJ, but a better strategy would have been to bet $3,000, which would have given him a $1 victory when Sara missed. The only possible downside for that wager would have been forcing a tiebreaker with Joe if both were correct and Joe doubled up.
Triple Stumpers of the day: No one knew William Tell was a hero in Switzerland or could identify certain words in sentences as an adverb and a conjunction. Also, Alex was clearly disappointed the players couldn't name the Gulf of Saint Lawrence on a map.
Pavlov: Hull House = Jane Addams, referenced in at least 8 clues over the past decade.
This day in Trebekistan: Alex opened the show by saying that last week, "for some strange reason" the players didn't seem "comfortable with the categories". Translation: Sorry about the lousy gameplay last week, but don't worry, James will be back soon.
Correct Qs: DD1 - What is Memphis? DD2 - What is "Richard III" DD3 - What is UNICEF? FJ - Who was Miles Standish?

Jeopardy! recap for Fri., May 17 - The two-game 2019 Teachers Tournament final concluded with these players:

• Sara, middle school instrumental music from Maryland, won $4,800 in game one;
• Francois, 11th grade physics from New York, comes into game two well in front with $20,400; and
• Conor, high school world history from New York, is in second place entering this game with $5,600.

Francois, who was clearly the best player in the tournament, stayed in control and had it all wrapped up for the two days, entering FJ with $21,200. The battle was for second money between Sara ($14,800) and Conor ($13,400).
DD1, $1,000 - AND A BOOK - His 1913 novel "Sons and Lovers" was daring but not banned like a later work (Francois won $1,000 from his score of $6,000.)
DD2, $2,000 - V.I.P. - 3 North American bodies of water - a strait, a bay and a river - are named for this English captain last seen in 1611 (Sara won $3,000 from her total of $8,600.)
DD3, $2,000 - FIRE & ICE - The patron saint of Mediterranean sailors gave his name to this phenomenon seen on the masts of ships in stormy weather (Francois won $1,000 from his score of $13,000.)
FJ - PHOTO SHARING - Publishing its first photo in 1889, today it has more than 4 billion likes & 100 million followers on Instagram
Everyone was correct on FJ, with Francois betting $0 to win with a two-day total of $41,600, good the for title and $100,000. For some reason, Sara also wagered $0, allowing Conor to take second place, adding $13,000 for a two-day score of $32,000 vs. $19,600 for Sara. Conor took home $50,000 while Sara won $25,000.
Sara's $0 bet makes even less sense when you consider that if she finished third, she would keep anything over the $25,000 minimum that she won, so she declined what was essentially a freeroll.
Triple Stumper of the day: For a top-row clue, no one guessed the celebrity Amy who has a "Smart Girls" channel on YouTube is Amy Poehler.
Strategy corner: With one full category remaining in the first round and the DD still in play, both Sara and Conor declined to shop for the DD by picking top-row clues.
What are the odds? dept.: One game after Conor told his story about being brought to tears for having to settle for Derek Jeter's autograph, Jeter was the subject of the very first clue.
Coming up next: James returns to humble everyone in his path starting on Monday.
Correct Qs: DD1 - Who was D. H. Lawrence? DD2 - Who was Hudson? DD3 - What is St. Elmo's fire? FJ - What is National Geographic?

Jeopardy! recap for Fri., Oct. 27 - Champ Erica was correct on her DD opportunity in DJ while Anand missed his and late in the round, Erica was on the edge of a runaway.
But Anand did well in the last category and importantly pulled to within two-thirds of Erica's score on the final clue, so going into FJ it was Erica with $16,300, Anand at $11,300 and Ashley with $3,200.
DD1 - MODERN LINGO - This action signals a definitive end; it followed “Obama out” at the 2016 White House Correspondents Dinner (Erica won $1,000)
DD2 - COLLEGES & UNIVERSITIES - The Red Stick International Digital Festival is an annual event at this university (Anand lost $2,500)
DD3 (video) - BOTONY - Perfect for a drought-tolerant garden, the plants seen here are this type that store water in their fleshy leaves (Erica won $2,500)
FJ - THE WESTERN HEMISPHERE - The Mopan River flows about 20 miles from the capital of this country
After yesterday's embarrasingly easy FJ ("Clue"), the writers came up with a Triple Stumper here. Anand has clearly studied wagering strategy and it paid off, as he lost only $1,299 and won by $2 with a score of $10,001. Conversely, Erica's overly cautious wager on DD1 cost her the game.
Triple Stumpers of the day: No one guessed that the Mass. school museum with a slide rule collection is MIT, or knew the performer of the "Tea for the Tillerman" album with the hit "Wild World" is Cat Stevens.
Correct Qs: DD1 - What is a mic drop? DD2 - What is LSU? DD3 - What are succulants? FJ - What is Belize?

Jeopardy! recap for Thur., Feb. 9 - Kate really knew her Alexander Hamilton-related history, running the category on the subject to take immediate command. Champ Kirstin kept up the chase most of the way, finally getting to the front on the final two clues to lead into FJ with $17,100 vs. $14,600 for Kate. Megan was unfortunate to run into two top-notch opponents and finished DJ at -$1,000.
DD1 - MAN VS. BEAST - In a Hawthorne story, a witch makes one of these to protect her corn, then brings him to life (Kirstin lost $1,000)
DD2 - LOVE - From the Greek for "carry", these scents are released by some insects to attract members of the opposite sex (Kate won $3,000)
DD3 - THE OED, FROM POISE TO QUELT - "One who practises the art of boxing" (Kirstin won $2,500 - she was told by Alex that she was trailing by $3,600, but with just two other clues remaining, she didn't bet enough to take the lead, a mistake she was able to overcome.)
FJ - THE OSCARS - Since "The Godfather Part II", this film has been the only sequel to win Best Picture
For the second time in recent weeks, we had an FJ with a rather lengthy response that a contestant knew, but ran out of time to complete writing it down. That proved very costly to Kate, who bet it all and dropped to $0, while Kirstin had no guess, losing $2,499 to win with $14,601 and a two-day total of $25,602.
Rather than covering double of Kate's score, Kirstin's FJ wagering strategy was to finish $1 ahead of Kate if Kirstin missed and Kate bet $0. That's a solid approach in a game where the top two players are close in score and the third player is no factor, as it gives the leader two ways to win: if the other contestant misses and/or bets nothing.
Triple Stumper of the day: No one knew that the West Coast city a book called "The Barbary Coast" is about is San Francisco.
That's Before Our Time dept.: The contestants couldn't guess that Candy-O is a cover band of The Cars, and even when spotted the word "little", had no idea that the Skipper's nickname for Gilligan is "little buddy".
This Day in Trebekistan: Alex seems to really love Pepé Le Pew. Why am I not surprised.
Correct Qs: DD1: What is a scarecrow? DD2: What are pheromones? DD3: What is a pugilist? FJ: What is "Lord of the Rings: Return of the King"?

Yesterday's episode (March 5, 2018) had the leader uncover the Daily Double as the very last clue in Double Jeopardy. This is an immensely strategic decision point: Some say to bet big, aim for a runaway, essentially turn the clue into a Final Jeopardy just for you. Is that truly statistically superior? This time I decided to crunch some numbers. In competition, you don't have time to do much more than eyeball the scores and apply much more than a hunch - or ideally, a rule of thumb. Let's see if we can figure one out.
You: $12000. Her: $9800. Him: $9600. Let's look at three strategies. I'll call them (S)mall, (M)edium, and (L)arge.

• S) Bet small to stay in the lead. Let's say $0.
  
• M) Bet medium to try and make it a "crush," i.e. put both opponents below 2/3 of you if you're right. You'll have to wager $2701 for that. I'm gonna go with $2800 to make the numbers cleaner. In real life our champion, as it turned out, wagered $3000, a roughly equivalent amount.
  
• L) Make the Hero Bet. Bet big to try and lock the game right there. You'll have to wager $7601 for that. I'm gonna go wtih $7700.
  

To estimate which strategy has the highest chance of winning, we need to consider the probabilities of Daily Doubles as well as Final Jeopardy rounds. J-Archive provides DD and FJ statistics by season. I'm too lazy to sum all 34+ seasons into one statistic, so I'll just go to Season 33, the last complete season, and see what it says.
For DDs: http://j-archive.com/ddstats.php?season=33
It says contestants in Season 33 answered DDs correctly 61% overall, but only 50% (30/60) when it was in the bottom row (as it was in our real-life example). that's a small sample size, though, so I don't want to treat 50% as gospel. Let's say 55%.
(So right off the bat we know Strategy L has at least a 55% chance of winning - plus some small chance of coming from behind from a distant third in FJ.)
So there are six possible outcomes of the DD:

• SR) You won $0. You have $12000. You have a narrow lead entering FJ.
  
• SW) You lost $0. You have $12000. SW is the same as SR.
  
• MR) You won $2800. You have $14800. You have a "below 2/3" or "crush" lead in FJ.
  
• MW) You lost $2800. You have $9200. You're in a close third entering FJ.
  
• LR) You won $7700. You have $19700. You've clinched it.
  
• LW) You lost $7700. You have $4300. You can still win in FJ if your opponents bet big and both lose.
  

What should you bet in FJ? This is a game-theory question especially when if you chose Strategy S to keep the scores close. This analysis is going to multiply in complexity if I start looking at all possible strategies, so let me just pick one that I think is the most realistic: Because both your opponents have similar scores and are practically in the same boat, I wouldn't bother getting tricky or underbetting. With two almost-tied opponents to consider instead of one, the chances of at least one of them betting big is huge. And since all three of you are using this same logic, chances are they're both betting big. You can't count on them both getting it wrong. You gotta bet to cover them if you can, is what I say. So let's say your FJ wager is:

• SR) Bet $7601 to cover Her.
  
• SW) Bet $7601 to cover Her. SW is the same as SR.
  
• MR) Bet $4801 to cover Her.
  
• MW) Toughie. Your opponents have slightly more and are probably gonna bet big. This is the case where you should probably cross fingers for a Triple Stumper. Bet $601 just to cover a $0 bet from them.
  
• LR) Bet $0 and write something funny.
  
• LW) You're only winning if your opponents both miss. Bet between $0 and $3899 and hope they're betting big.
  

Final Jeopardy is not three independent weighted-coin-flips. There is a high degree of correlation between two contestants being right or wrong in FJ (as you would expect - after all, they're getting the same question).
http://j-archive.com/finalstats.php?season=33

Distribution of Final responses by score position out of the 217 games in which all three players played Final:
WWW = 43
WRW = 35
RWW = 31
RWR = 28
RRR = 26
RRW = 19
WWR = 19
WRR = 16

This is a pretty small sample size so I don't want to read too much into the 1st-place vs 2nd-place stat breakdown. Instead, let me aggregate them:

• Three contestants are right 26 times
  
• Exactly two contestants are right 63 times
  
• Exactly one contestant is right 85 times
  
• No contestants are right 43 times
  

So from your perspective, there are six FJ outcomes which I'll label A through F.

• A) 26/217 chance all three of you get it right (12.0%)
  
• B) 63/217 x 2/3 chance you and one opponent get it right (19.4%)
  
• C) 63/217 x 1/3 chance you get it wrong and both opponents get it right (9.7%)
  
• D) 85/217 x 1/3 chance only you get it right (13.1%)
  
• E) 85/217 x 2/3 chance you get it wrong and one opponent gets it right (26.1%)
  
• F) 43/217 chance nobody gets it right (19.8%)
  

At long last, let's put it all together. Again, She has $9800 and He has $9600.

• SR) You have $12000 and wager $7601 in FJ. You win in scenarios A, B, D. You lose in C, E. You win in F unless one of your opponents (probably Him in 3rd place) bet small. Let's make a wild guess and say you win/lose half the time in scenario F. Adding the likelihoods of A, B, D, and half of F gives us 12.0 + 19.4 + 13.1 + 19.8/2 = 54.4% success
  
• SW) Same as SR. 54% success.
  
• MR) You have $14800 and wager $4801 in FJ. You win in scenarios A, B, D, and this time, you're guaranteed to win in F. You lose in C and E. Adding A, B, D, and F gives us 64.3%.
  
• MW) You have $9200 and wager $601 in FJ. Assume your opponents are betting big. You win in D and F only. D + F = 32.9%.
  
• LR) You won in a runaway. 100.0%.
  
• LW) You have $4300 and wager smallish in FJ. Interestingly, this turns out to be not much worse than MW above. You win in D and F if your opponents bet big. 32.9%.
  

Going back to our DD statistics, we're using 55% as our chance of getting the DD right. So remember the question? Should you choose Strategy S, M, or L? Let's finally find out.

• (S)mall - bet to keep the lead: SR x 55% + SW x 45% = 54.4% overall chance of winning the game
  
• (M)edium - bet to put your opponents below 2/3: MR x 55% + MW x 45% = 50.17%
  
• (L)arge - bet to make it a runaway: LR x 55% + LW x 45% = 69.81%
  

Looks like betting big on the DD is the winner!... but hold on. Not only did I make several assumptions along the way, about your wagering and your opponents' wagering, and about your right/wrong chances. In addition, the analysis depended a lot on your opponents' scores - the results may not hold if their scores are just a little bit different. What is putting Strategy L over the top is the "second chance" value, your ability to still win even if you bet big on the DD and miss. If this real-life example were just a little bit different, then betting big and missing would give you no second chance at all. Like, if your lead were just a little smaller, or if She and He weren't almost tied. Without the "second chance" appeal, Strategy L success rate falls from almost 70% to just 55% which is essentially no better than Strategy I. So it's precarious.
Furthermore, contestants take note, this real-life example had the DD as the very last clue. Even one or two more clues left on the board would change things - because you can't 100% guarantee a runaway unless you take into account what happens if your 2nd-place opponent collects all the remaining clues. You might not even have enough money to wager enough to guarantee a lock, even if you're in the lead. If you can only give yourself, say, an 80% chance of a lock instead of a 100% chance of a lock, that also neuters a lot of this strategy's appeal. So, contestants, I would not march into the studio memorizing "always bet the farm on a late DD." The conditions have to be just right.
On top of all that, remember when I noted that DD success rates on the bottom row are only 50% in season 33. I said, yeah but it's a small sample size and the overall rate is 61%, so let's call it 55%. But what if it really is 50% on the bottom row and 61% on the fourth row? That might actually be enough to make Strategy L look much worse or much better, based on how important it is to get the DD clue right.
So, when betting big isn't ideal, what is?
What hurts Strategy M is that the benefits from pushing your opponents below 2/3 aren't actually that great. The benefits get outweighed by the risk of falling into 3rd place. That's why Strategy M has a lower overall winning percentage than Strategy S.
But again, this is based on a particular real-life scenario where your two opponents are very close to each other. If instead, one of them were in a distant third, then they'd be much more likely to have disparate strategies, and I'd have to rerun all this analysis again, and we might very well find Strategy M on top. If your crush over 2nd place is big enough to lock out 3rd place, for instance, then I would definitely expect Strategy M to improve.
Finally, remember all these empirical probabilities are based on just one season's worth of historical episodes. Small sample sizes! Who knows if FJs are still this hard/easy as they were in 2016.
So I'm afraid I haven't come up with a "rule of thumb" for this situation. Too many things could change the analysis. If anything I'm tempted to advise betting small - because that way minimizes your chances of making an error and wagering a wrong amount.
(I bet nobody is still following this.)
I think I learned the following through this:

• Making a DD Hero Bet for a lock CAN be vastly superior.
  
• The DD had better be VERY close to the end of the round if you're thinking of playing for the lock.
  
• The larger your lead, the safer it is to play for the lock. (Duh)
  
• The closer your opponents are to each other, the safer it is to play for the lock, because that keeps you "still alive" if you miss.
  
• If you're torn between going for it and playing it safe, why not choose based on whether the DD is in the bottom row (scary) or not (easier).
  
• I'm not sure whether betting "medium" to play for a 2/3 "crush" is ever better than betting "small" to preserve the lead. It doesn't seem to be in this real-life example.
  

Please let me know if I screwed up any of my probabilities or strategy assumptions!