Strategy Talk: Zerg’s Fallacy (Part 1) and Why Trailers Often Make The Wrong Wager

Often, you’ll see a trailing player in Final Jeopardy! bet to take the lead by $1, seemingly forcing the leader to get Final Jeopardy! correct and make a wager in order to win the game. Even though many trailers think it’s a great idea, it is, in many cases, the worst thing overall that you can do.

Here at The Jeopardy! Fan, I have decided to give the name of this specific wager “Zerg’s Fallacy”. It gets its name from the famous November 30, 2004 game which saw Nancy Zerg’s defeat of Ken Jennings. Down $14,400 to $10,000 going into Final Jeopardy, Nancy Zerg elected to bet $4,401 in Final Jeopardy!, getting lucky when she got Final Jeopardy! correct and Ken did not.

As it turns out, between $4,401 and $5,600 is both frequently used and quite harmful to one’s winning chances.

As I often do here in Strategy Talk, the point that I often make is “What has to happen for me to win the game?”

Let’s start with the specific situation in the Nancy Zerg game, namely that the trailer is between two-thirds and three-quarters of the leader’s score, and third place is out of contention (Specifically, in this case, Nancy has 69.44% of Ken’s score.)

Since October 4, 2004, Final Jeopardy! with two players in contention for the win has played as follows:
Both players get Final: 29.24%
Leader only gets Final: 21.91%
Trailer only gets Final: 19.05%
Both players miss Final: 29.81%

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Once again, the scores are:
Ken $14,400
Nancy $10,000

The leader’s standard cover bet here is $10,000 × 2 = $20,000 + $1 = $20,001 – $14,400 = $5,601. Since October 4, 2004, the leader in regular play has made this (or a similarly good) bet 91.86% of the time. Obviously, if Ken gets the question right here, he wins, regardless of what Nancy does. If Ken is incorrect, he falls to $8,799.

Nancy could choose to bet very small here, and considering the frequency of how often the leader makes the standard cover wager, this is overwhelmingly still the best play for her. Thus, Nancy needs to stay above $8,800, meaning her wager needs to be in the range of $0 to $1,200.

There are some second-order games at play, here. If Ken is confident that Nancy is going to bet small, then he can attempt to stay above $10,000 + $1,200 = $11,200 himself and limit his bet to $3,199. However, this has only happened 2.33% of the time, and the leader has not wagered $0 in this position in the entire data set (dating back to October 4, 2004). Of course, here, Nancy could protect against this herself with any bet $5,600 or greater.

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So, what has to happen for Nancy to win in each case:

With a bet of $1,200 or less: Nancy wins 0% of the time if Ken bets $3,199 or smaller, between 0-19.05% of the time if Ken bets between $3,200 and $5,601 (19.05% assumed here), and (19.05% + 29.81% = 48.86%) of the time if Ken bets $5,601 to $8,800 (trailer only gets, or both players miss).

As you can see in the chart below, Nancy wins with a small bet 45.99% of the time.

2nd Bets Small 1st Win 2nd Win Frequency
1st Small 100% 0% 2.33%
1st Cover 51.14% 48.86% 91.86%
1st Not-Cover 80.95% 19.05% 5.81%
Total 54.01% 45.99%

With a bet of $5,601 or greater: Nancy wins (19.05% + 29.24% = 48.29%; trailer only, or both players get Final) if Ken bets $3,199 or smaller, and 19.05% of the time if Ken bets between $5,601 and $8,800.

As you can see in this chart, Nancy wins with a big bet 21.43% of the time.

2nd Bets Big 1st Win 2nd Win Frequency
1st Not-Cover 51.71% 48.29% 8.14%
1st Cover 80.95% 19.05% 91.86%
Total: 78.57% 21.43%

However, if Nancy bets in the middle range (between $1,201 and $5,600), including the “Zerg’s Fallacy” range between $4,401 and $5,600, she only wins in the case that she gets Final and Ken does not. This happens 19.05% of the time. The leader has not wagered zero in this position since before Oct. 4, 2004, thus defending against it is not to Nancy’s advantage.

So, to summarize:
Bet small: 45.99% chance of winning
Zerg’s Fallacy: 19.05% chance of winning
Bet big: 21.43% chance of winning

When the future value of winning a game can run between $18,000 and $50,000, mistakes like this can be very costly!

Amazingly, since October 4, 2004:
The trailer has bet small only 20.93% of the time;
The trailer has bet in the middle range 31.40% of the time (the upper middle, between $4,401 and $5,600 in the case above, 22.09% of the time)
The trailer has bet big 47.67% of the time;

In fact, the trailer’s propensity to bet too much in this situation is precisely why the leader is 74.71% to win in this situation with the standard cover wager, and 71.00% to win with the small-range bet mentioned above (even though, assuming rational wagering, a leader could guarantee victory with a small bet).

In my next “Strategy Talk” post, I’ll outline another situation where “Zerg’s Fallacy” is used to a slightly less detrimental (but detrimental nonetheless) effect.

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