Were the wagers in the June 3, 2019 Final Jeopardy! round rationale?

The long-running U.S. television program Jeopardy! is a trivia question-and-answer (or answer-and-question) game show, involving three players competing to be the fastest to correctly answer questions, while accumulating their respective scores. Upon entering the third round ("Final Jeopardy!"), the players can wager up to their maximum score accumulated in the first two rounds.

• Each player is aware of the other players' scores going in to Final Jeopardy!
• Each player is also generally aware of the strengths of the other players.
• In Final Jeopardy! each player secretly commits to his/her wager, and secretly answers their question.
• If a player answers correctly his or her wager is added to his or her score.
• If a player answers incorrectly, his or her wager is deducted from his or her score.
• The player with the highest score after Final Jeopardy! wins the cash value of his or her score (the other players win consolation prizes).

The winner is invited to continue playing in another game, after winning the cash prize.

The June 3, 2019 game was noteworthy, as one player (James Holzhauer) had shattered a number of single-game records and was close to overcoming another record set 14 years earlier (by Ken Jennings).

However, the game of June 4, 2019 included two other very strong competitors, Emma Boettcher and Jay Sexton.

Going in to the Final Jeopardy! match, the scores were as follows:

James: $23,400 Jay:$11,000

Emma: $26,600 James, Jay, and Emma placed the following wagers: James:$1,399

Jay: $6,000 Emma:$20,201

All three answered the Final Jeopardy! clue correctly, and Emma was the winner, winning \$26,600+20,201=$46,801.

Emma's bet clearly covered James' all-in (if both she and James answered correctly). James' bet covered Jay's all-in (if Jay was correct and James was incorrect).

However, given James' history of betting large sums in Final Jeopardy, because James' bid was abnormally low some have wondered whether his wager was rationale.

Assuming that each player has the same probability $$p$$ of answering correctly, and that each player knows that the other players are equally strong, what techniques could be used to show whether each of James', Jay's, and Emma's scores were rationale?

Are these bets rationale in some mixed strategy? Or should James have gone all-in? If not, then should Emma have wagered only enough to cover James' covering of Jay's all-in?

• Your stated conditional "...given James' history of betting large sums..." is rather incomplete, not to say deceptive. Out of the 32 games that Holzhauer played, 29 were runaway games, in which his scores going into final jeopardy were more than two times the highest of the other two scores. In such a situation the strategy for determining the size of the bet is entirely different than the strategy in the situation of his final game. – Lee Mosher Jun 6 at 15:26
• The assumption that $p$ is the same for all players is a very simplifying (and bold) assumption. Also, we want to maximize our expected win, not merely the probability that we win. E.g., if $p$ is close enough to $1$, Emma should go all in - the extra profit she is very likely to make outweighs what happens when she loses and is not theonly to lose. However, we do not win only the amount but also the chance of playing again - so we'd need to take the expected win from continuing in account ... – Hagen von Eitzen Jun 6 at 15:53
• @LeeMosher I personally agree; however, this is the rationale that I have seen others recite when questioning whether James' bet was too low. – Mark S Jun 6 at 17:29
• @HagenvonEitzen The players see other players' capabilities on answering questions correctly for the questions that they are able to buzz in on; additionally, all players have to pass a "tough" test, so they know that the other players are pretty competitive. They know their own private $p$ - because they know the subject matter to which the question will be asked. Emma was an English major who wrote a thesis on Shakespeare, so her $p$ could be high on Shakespeare questions, but I don't think she should assume the others have much lower $p$? – Mark S Jun 6 at 17:36

• How did you determine that James deduced that Emma would bet \$4,600? Why \$4,600? What methods are used to analyze such $3$-player games? – Mark S Jun 6 at 17:38