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March 22, 2016

Lock-tie situation in Final Jeopardy!:
should I wager $0 or $1?

In last night’s analysis, I suggested that Steve, the who led with twice second’s score, wager no more than $1 in case Melissa held back a dollar for some reason.

This is the second such situation – a “lock-tie” in technical parlance – since the new tiebreaker rules went into effect in November 2014. The first was January 14; the leader wagered zero (risking the tiebreaker) and second place went all-in.

The previous occurrence was July 8, 2014. It was a total disaster.

The trailer’s wager, historically

Looking at J! Archive data, I see only 29 occasions in regular play (excluding last night) where second place had exactly half the leader’s total heading into Final. In one of those games, there was a tie for second.

The trailer wagered:

  • Everything: 20 times (67%)
  • All but $1: 3 times (10%)
  • Something else: 7 times (23%)

(First place properly wagered zero in all but 3 of the 28 pre-tiebreaker games – something I’d consider a Cliff Clavin move – but we’re not really interested in that anymore.)

Final Jeopardy! correlations

At the beginning of this season, I ran some stats for a feature on the Jeopardy! website.

Data from seasons 27-31. Source: J! Archive

Data from seasons 27-31. Source: J! Archive

To figure out whether the leader will prefer to wager $0 or $1, we only need to check what happens when second responds correctly. (The issue is moot if second misses.)

Player(s) who respond correctly:

  • Only first: 20%
  • Neither right: 31%
  • Both right: 31%
  • Only second: 18%

That means in circumstances where second place gets it right, first place will also get it right around 63% of the time – better than a coin flip. If you think you have a better-than-63% shot of winning a tiebreaker, feel free to bet zero and hope your opponent misses (either the clue or the proper wager). Otherwise, I’d play to win.

Your comfort with the category might also come into play, as I discuss in the video I now put up every time there’s a wager-to-tie situation.

The possibility that second withholds a dollar lowers that 63% number by a slight amount, since you’ll win no matter what if you wager zero. I don’t think it’s worth considering in this tiebreaker era, since the correct play should be even more obvious. (Don’t quote me on that, of course…)

As for that tie for second, Elin Gaynor committed a cardinal sin, depriving us the possibility of three-way co-champs in 1997.

Doug Elin Michael
10,200 5,100 5,100
0 5,099 5,100
10,200 10,199 10,200
0 ! 5,100 ! 5,100 !
4 Comments
  1. So, isn’t another way to think of this to *include,* rather than ignore, what happens if second place misses? If I’m the leader (and if I assume that I only care about winning, not about earning max dollars) then I could consider it this way:

    a) If I get it wrong, my best bet is $0
    b) If the second-place person gets it wrong, my best bet is also $0.

    The question then becomes: what are the odds that either I or the second-place person (or both) get it wrong? If we guess (generously) that each of us will get Final Jeopardy right 2/3 of the time, that makes it a 44.44% chance that we both get it right. That means that 55.56% of the time, $0 would be the best bet. Doesn’t it?

    And if we use the 48.4%-get-it-right number you calculated for that (excellent) Jeopardy! story, that would make it only a 23.4% chance we both get it right, and a 76.6% chance that betting $0 is the best move. Doesn’t it?

    I’m no mathematician, so I’m wondering where the flaw is here.

    • Kelly permalink

      You’re forgetting that if BOTH of you miss then your choice is irrelevant. Here’s how it goes:

      You win with a $1 bet under the following circumstances:
      You get FJ! right (regardless of whether or not second does).
      Both you and second miss FJ!.
      Second makes a bad wager when you miss FJ! and your opponent gets it (if that wager is everything but a dollar it’s a tiebreaker).
      That’s at least 3 out of 4 possible outcomes (and probably more because of the correlation of outcomes and the possibility of a bad wager).

      You win with a $0 bet under the following circumstances (note in all cases your outcome is irrelevant):
      Second misses FJ!.
      Second wagers properly and gets FJ! right – and you win the tiebreaker.
      Second makes any bad wager.
      Once again that’s at least a 3 out of 4 chance – but here you gain the slight benefit in the case of an everything-but-a-dollar bad wager at the (probably greater) expense of losing the correlation benefit if you both get FJ! right.

      The moral of the story is that in this situation either $0 or $1 is perfectly acceptable (and perhaps any amount that would not let third back in, but going for more than one dollar needlessly adds risk if second does make a boneheaded wager).

    • If second place misses, your wager is irrelevant. Once you reach that point, that’s where you stop using brain power and focus on where the wager does matter. :)

What do you think?