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January 14, 2014

How to wager in Final Jeopardy!: Part Five

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We conclude our tutorial on Final Jeopardy! wagering with a somewhat complicated topic: what I call “mind games”.The Final Wager Part Five

Our first example focuses on the subject of several of my Guide to Game Theory videos: a two-player situation in which neither player can safely execute a wager under Rule #3. (Math time: This happens when second place has more than 2/3, but less than 3/4, of first’s score.)

Final Jeopardy wagering October 17, 2013 Slide 01
(Click to see the game in the J! Archive)

As I show in those videos, if the trailer wagers “properly” (i.e., in accordance with Rule #2, to win if the leader gets it wrong) …

Final Jeopardy wagering October 17, 2013 Slide 02

… the leader can guarantee a victory by wagering small. To counteract this, the trailer might consider wagering “large”, like Katie did here.

Final Jeopardy wagering October 17, 2013 Slide 03

There are many “mind games” in motion here: how well do you know the category? How well do you think your opponent knows the category? How well does your opponent think you know the category? And how will each of you wager as a result?

The math dictates that when in doubt, both players should wager “large” – a lock-out bet for the leader and everything for the trailer. (You can see why in my entry on Expected Value.)


Our remaining two examples involve three players.

Here, first and second are fairly close, and third is trailing by a large margin.

Final Jeopardy wagering December 17, 2012 Slide 01(Click to see the game in the J! Archive)

For second place, there exists a range of wagers that will both (1) keep third place locked out of contention and (2) cover a zero wager by first place.

Final Jeopardy wagering December 17, 2012 Slide 03

Does first think second will make this “safe” wager? If so, he can make a play of his own: wager to cover the high end of that range. Second can respond in kind.

Final Jeopardy wagering December 17, 2012 Slide 04

Note, however, that this puts second at risk of falling below third if he gets it wrong. It might be worth a shot, though, if he thinks first is going to be sneaky.

This situation is called “Shore’s Conjecture” after Bob Shore, a champion who proposed it. It comes in three varieties depending on what happens if first bets to lock out second and gets it wrong:

Strong: first will fall below third’s pre-Final score
Intermediate: first will fall below third if third doubles up
Weak: first can’t fall below third (i.e., third has less than the difference between first and second)


Our third example is a very specific form of Shore’s Conjecture: where the amount second needs to wager to stay above third is equal to the amount he needs to wager to cover a zero wager by first.

Final Jeopardy wagering November 26, 2002 Slide 01(Click to see the game in the J! Archive)

In other words, the difference between first and second is equal to the difference between second and a double-up by third. In this situation, second has no reason to not wager that difference.

Final Jeopardy wagering November 26, 2002 Slide 02

If first notices this – as Faith Love was able to do in this game – she should strongly consider a wager of zero. Third should also wager everything in an attempt to tie second if the cards fall the right way.

Final Jeopardy wagering November 26, 2002 Slide 03

There are many other situations in which the wagering can get tricky. It’s just a matter of first figuring out the proper “normal” wagers, then using those as a springboard for mind games. In many situations, though – I’d say around 75% of the time – there are no good “alternative” wagers.

2 Comments
  1. mike m permalink

    i’d like to know more about the time clickers.

  2. Well, I only just saw this page (and while searching for Faith Love scenarios!) 12/17/12 was the second of my four FJ!s where I bet zero, and the only one where I was trailing going into FJ. I knew _some_ stuff about wagering strategy but I also knew that I hadn’t studied optimal strategies beyond two-thirds. I’m still comfortable with my zero bet, but Sue’s wagering in her four games turned out to be somewhat unpredictable, so advanced strategy didn’t actually work that well with her. Hats off though — she was a fantastic champion and an absolute beast on the buzzer.

What do you think?