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, Jay, and Emma placed the following wagers:
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?