If you’ve been following The Jeopardy! Fan over the past few months, you’ll know that I use data dating back many years in my Coryat-based prediction model. Over the past few weeks, I’ve been expanding that spreadsheet in order to hopefully answer more and more questions about the show’s strategy!
I’ll be using this data, updated frequently, to provide more strategy-based articles that will hopefully help future contestants in their preparation!
One topic that has been alluded to in many strategic discussions, and which can be helpful in order to determine the best course of action, is being able to quantify the future value of winning a game. This is particularly important when it comes to talking about maximizing potential money won, as there are a few cases where a decision may sacrifice a couple of percentage points of winning percentage, but has a higher amount of average money won (and others where the extra money at stake is not worth the drop in winning chances, because the future value of a win doesn’t make it worth the risk).
To those preparing: It might be helpful to make a decision as to whether to try to maximize winning percentage, or money won, if you get The Call to be on the show. There is a difference, at least in a few cases.
The worth of a win on Jeopardy! is a function of a few stats:
- The average winning score;
- How often a defending champion wins;
- How often a defending champion finishes 2nd of 3rd;
- The expected value of a possible Tournament of Champions appearance.
The final point actually makes the worth of a win variable until someone reaches superchamp status!
All of the data used includes every regular-play game since October 4, 2004.
Why October 4, 2004? For the beginning of Season 21’s tapings, Jeopardy! realized that their challengers might not be getting enough rehearsal time on the buzzer; Ken Jennings’ streak made this very apparent. October 4, 2004 was the first episode of the “extra rehearsal time”. It’s made it slightly harder for a champion to win, and thus a good starting point for the start of the data.
So, the stats, from October 4, 2004, through to January 6, 2017:
- The average winning score: $20,137.
- A defending champion wins 46.26% of the time.
- If a champion does not win, they have finished second 57.14% of the time.
- The Tournament of Champions awards $490,000 in prize money to 15 players (not counting any alternates.) This gives a current expected value of a ToC guarantee of $32,667.
Using the formula 1 / (1 – r) to determine the sum of the infinite geometric series that begins 1, 0.4626, etc., we know that after a player’s first win, they are expected to win another 0.8608 games.
We also know that they are 21.40% to win 2 more games (46.26% squared), 9.90% (46.26% cubed) to win 3 more, and 4.58% to win 4 more (important because we need to add the chance that a player will qualify for the ToC to the worth of their first five wins.)
The fact that a champion finishes second 57.14% of the time means that we should apportion the $2,000 second-place prize slightly more often than the $1,000 third-place prize when adding that to a champion’s eventual winnings.
Putting this all together, we can determine the worth of a victory (not taking a ToC into account, we’ll do that later) with the formula:
(ExpectedFuture Wins * Average Win) + (% 2nd * $2,000) + (% 3rd * $1,000)
(0.8608 * $20,137) + (0.5714 * $2,000) + (0.4286 * $1,000) = $18,907.
The above $18,907 is actually the current expected value of a win once a player has guaranteed their spot in the Tournament of Champions.
If a player hasn’t qualified for the Tournament yet, we can use the percentages above to guess their chances of qualifying for the Tournament and add that percentage of $32,667 accordingly.
So, to a player with 0 wins, we can add 4.58% * $32,667 = $1,496 (as a 4.58% chance of winning 5 comes with winning one’s first, on average.)
Thus, we can say that:
A player’s first win is worth: 4.58% * $32,667 = $1,496 + $18,907 = $20,403.
A player’s second win is worth: 9.90% * $32,667 = $3,234 + $18,907 = $22,142.
A player’s third win is worth: 21.40% * $32,667 = $6,991 + $18,907 = $25,899.
A player’s fourth win is worth: 46.26% * $32,667 = $15,112 + $18,907 = $34,020.
A player’s fifth win is worth: 100% * $32,667 = $32,667 + $18,907 = $51,574.
A player’s sixth and subsequent win is worth $18,907 in future value. (as ToC status is generally seen as guaranteed at that point.)
I’ll refer to the above a great deal through my strategic posts, as the future value of wins #4 and #5 may make prioritizing maximizing winning percentage very important!
My next Strategy Talk article will talk about crush games, and whether it is worth it for a leader to bet more aggressively in Final Jeopardy! in those positions.