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October 3, 2013

How to wager in Final Jeopardy!: Part Three

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In Part Three, we’re back to where we began: with three players. But I’m hoping that it’s a little less ominous now that we’ve worked through some of the basic strategy.

If you want some practice with this, I suggest looking through the categories daily analysis and wagering practice. Try the situations on your own and see if you match what I suggest!

The key to three-player wagering is simple: break the situation down into three two-player games. Start with first vs. second; then second vs. third; then first vs. third.

Let’s return to the example I showed you in Part One.

Tim Liz Jason 1

We calculated optimal wagers for Tim and Liz in Part One, but we’ll do it again here for a refresher.

First vs. second

Rule #1 says that Tim wants to cover an all-in wager by Liz. If Liz wagers everything and gets it right, she’ll have 22,000. To match this total, Tim will need to wager at least 7,800. So 7,800 is his minimum wager.

We now look at Liz, who should focus on what will happen should Tim answer incorrectly. A wrong response will put Tim at no more than 6,400. To stay at or above this total, Liz can wager no more than 4,600.

For Rule #3, at most one player can ever safely wager to be above a zero wager by the other. When it’s close, look at the trailer. If Tim wagers zero, he’ll have 14,200. To match this, Liz will need to wager at least 3,200.

Second vs. third

Liz wants to cover a double-up by Jason, who will have 15,200 in that event. To match this, Liz will need to wager at least 4,200. This supersedes her previous minimum of 3,200 – after all, if she’s wagering at least 4,200, she’s wagering at least 3,200.

Now for Jason. An incorrect response from Liz will leave her with 6,800. To stay above this, Jason can wager no more than 800.

Finally, for Rule #3. 3,400 separates our two players. Jason cannot wager 3,400, as that violates his maximum wager of 800; Liz already must wager between 4,200 and 4,600, so we ignore Rule #3 for her, too.

First vs. third

We know that Tim must wager 7,800 to cover Liz. If he gets it wrong, he’ll have at most 6,400. Jason can wager no more than 1,200 to stay above this total, but he is already limited to at most 800, so we’ll keep his maximum at 800.

We’re done! Tim should wager 7,800; Liz should wager between 4,200 and 4,600; and Jason should wager no more than 800.

Tim Liz Jason 2

How are you feeling? That’s a lot of math. But remember that you get as much time as you need to do the calculations, and even math wizards will want to check their work. It’s a lot of money to blow on a bad bit of addition or subtraction.

Let’s look at another example to keep the momentum going.

John Helen Barbara 1

As with last time, we’ll start with our top two players, John and Helen.

First vs. second

John wants to cover an all-in wager by Helen. Should Helen pull that off, she’ll have 27,600. To be at or above this total, John needs to wager at least 11,200.

Now for Helen, who wants to stay above John should he answer incorrectly. In that case, John will have 5,200. To be safe, Helen can wager no more than 8,600.

For Rule #3, since the scores are close, we’ll start with Helen. To match John’s total on a zero wager, Helen needs to wager at least 2,600.

Second vs. third

Helen wants to control her own destiny against Barbara. If Barbara doubles her total, she’ll have 18,000. To match this, Helen must wager at least 4,200. This becomes her new minimum wager.

We turn our attention to Barbara. If Helen wagers properly and gets it wrong, she’ll have 9,600. This is more than what Barbara has right now, so she’ll need to get it right and wager at least 600 to match this.

Where do we start with Rule #3? Let’s try Helen. The difference between the two scores is 4,800, so Helen can wager no more than this to stay above a zero wager by Barbara. This becomes her new maximum wager.

First vs. third

John is going to wager at least 11,200. If he gets it wrong, he will have 5,200. To stay at or above 5,200, Barbara can wager no more than 3,800. This becomes her maximum wager.

We’re all done! Here are the acceptable ranges.

John Helen Barbara 2

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4 Comments
  1. Ken H permalink

    In the second game of this tutorial I think Helen should reconsider her max to be 4200, equal to he minimum. This would allow her to tie Barbara at 9600 if she gets it wrong and Barbara gets it right and bets the minimum of 600. And she would go on to play the next day if John also gets it wrong. This gives her an additional chance to go on without even changing her expected payout for the day assuming she has a 50% chance of getting it right.

    • Yep, there’s a stronger case for 4,200, because it’s unlikely that extra 600 (to wager 4,800) will make a difference against John (he’s probably not going to wager, say, 2,000). You can see this an alternative case of not wagering a dollar more than you “need” to.

  2. Ken H permalink

    Keith:

    In the second game of this tutorial (John, Helen and Barbara) I’m coming to a different recommended bet for Barbara, who enters FJ! in 3rd place.

    You state: “We turn our attention to Barbara. If Helen wagers properly and gets it wrong, she’ll have 9,600. This is more than what Barbara has right now, so she’ll need to get it right and wager at least 600 to match this.” Normally your advice in this situation is: “if a contestant has to get FJ! right to have a chance, she might as well go all-in.”

    It seems to me that an all-in wager by Barbara is the best option in this situation, but you suggest Barbara should cap her bet at $3800 to stay at or above John if he wagers 11200 or more and gets it wrong. However, If Barbara bets 600-3200 and John and Helen bet according to your recommendations Barbara has no chance at a win or a tie if she gets it wrong because Helen would have 9000 – 9600. So, even though Barbara COULD WIN with a 3200 bet and a solo-get wouldn’t 9000 + 9000 = 18000 be a better bet than 9000 + 3200 = 12200? (I say COULD WIN because there is a possibility that John and/or Helen might bet small and exceed 12200 if they get
    it wrong.) Note that betting 9000 also eliminates these possibilities of NOT winning with a 3200 bet and a solo-get.

    Here’s how I would analyze the situation (borrowing liberally from the logic I’ve learned from you:)

    We turn our attention to Barbara. If Helen wagers properly and gets it wrong, she’ll have 9,600. This is more than what Barbara has right now, so she’ll need to get it right and wager at least 600 to match this. Since she has to get it right to have a chance of winning she should consider going all-in with a 9000 bet. So, for now, let put Barbara’s minimum at 600 and her maximum at 9000.

    Now lets look at 1st vs. 3rd or Barbara vs. John with rule #2. If John wagers 11,200 and gets it wrong, he will have 5,200. To stay at or above 5,200, Barbara can wager no more than 3,800. So, should Barbara reduce her maximum from 9000 to 3800? I don’t think so. Note that if Helen bets within her recommended range of 4200 – 4800 and gets it wrong she’ll have 9000 – 9600 remaining. So if Barbara bets ANYTHING and gets it wrong she’ll fall below Helen. So, betting 3200 to stay at-or-above John is fruitless since she’ll lose to Helen anyway. The only rationale for Barbara to cap her bet at 3200 is to win-or-tie on a triple stumper IF Helen bets 8600 or more…far beyond her rational range.

    Now, sometimes the 3rd place contestant is in a situation where she SHOULD develop her strategy based on the possibility of an irrational wager by either or both 1st and 2nd. But, since Barbara has more than 50% of both John and Helen, this isn’t one of those situations. She can win with a solo get regardless of how John and Helen wager…so she should focus on the solo-get.

    Barbara will need to wager at least 7400 to cover a zero bet by John. If she gets it wrong, she’ll be left with only 1600…which is far less than John’s 5200 if he gets it wrong with a shut-out bet. So even if Helen (irrationally) goes all-in and gets it wrong, Barbara would not have enough to win on a triple-stumper. So, again, we come to the conclusion that Barbara should go all-in with a 9000 bet.

    Keith…the lengthy explanation above is not really needed to suggest what Barbara should do in this situation. It is just a rather long-winded explanation of why going all-in makes sense. It would suffice to just say:

    We turn our attention to Barbara. If Helen wagers properly and gets it wrong, she’ll have 9,600. This is more than what Barbara has right now, so she’ll need to get it right and wager at least 600 to match this. Since she has to get it right to have a chance of winning she should go all-in with a 9000 bet.

    • Thanks, Ken – you are exactly right. I’ve gradually refined my wagering techniques over the course of the season, and perhaps it’s time I update the wagering tutorials. Perhaps a project for the summer break.

What do you think?