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Another friend of mine is on Jeopardy! today, which seems like an appropriate occasion to post some other thoughts on J! strategy I've had recently.

Everyone knows that it's important to enter Final Jeopardy with the lead - if you have the lead, and you wager enough, and you get it right, you win. Almost everyone knows that it's very good to be in the lead with at least double the score of second place, what Alex calls a "runaway" and the rest of us call a "lock" - you can wager zero and win no matter what your opponents do.

Most people do not understand the relative importance of these two factors, and that there's a third threshold which is as important as either of them.

The third threshold is having a lead of at least 50% over your opponent, or conversely, trailing with a score at least two-thirds of first place.

Suppose you're leading with $20000, and second place has $12000. You can wager $4001 and guarantee a win with a correct response, and if you miss, you still have more than $12000. You only lose if you get it wrong and second-place gets it right. However, if you're leading with $15000 vs $12000, you have to wager $9001 to guarantee a win with a correct response. Second place can just wager zero, and you lose if you get it wrong, no matter what he does.

(Of course, if you know he's wagering zero, you could wager zero yourself, and guarantee a win. But if he knows you're wagering zero, he'd then wager large enough to win with a correct response. The optimal strategy is mixed, where you make different choices with various probabilities. However, in practice, this doesn't happen. Real-world second-place players routinely make irrationally large wagers, so real-world first-place players almost always wager large enough to cover a max bet and correct response.)

So if your opponent has $12000, and third place is out of the picture, the relevant ranges and winning scenarios are:

  • $24000+: You win.
  • $18000-$24000: You win unless you miss and he gets.
  • $12000-$18000: You win if you get.
  • $8000-$12000: You win if he misses.
  • $6000-$8000: You win if you get and he misses.
  • $0-$6000: You lose.
(Ignoring the endpoints, since they get weird... they're worthy of a post of their own.) Anyway, I knew all of this before I started looking at J! strategy more seriously recently. The implications are what surprised me.

What are the relative likelihoods of these scenarios? Fortunately, J-Archive has historical data. (My numbers do not match theirs because I've manually excluded certain irrelevant games, like college or celebrity tournaments.)

FJ is hard. The leader and second place respond correctly about 56% of the time. (Third place, usually a weaker player, gets at only 50%.) Even exceptional players don't get it much more often - the only one for whom we have a decent sample size is Ken Jennings, who got 68%. Because some FJs are unusually easy or difficult, the likelihood of leader missing and second-place getting is lower than it'd be if they were independent events, only 23%.

So, let's translate the above ranges into winning percentages:

  • $24000+: You win 100%.
  • $18000-$24000: You win 77%.
  • $12000-$18000: You win 56%.
  • $8000-$12000: You win 44%.
  • $6000-$8000: You win 23%.
  • $0-$6000: You win 0%.
Your improvement in winning percentage by extending a lead past the two-thirds threshold is 21%. Your improvement in winning percentage by extending a lead into a lock is 23%. Your improvement in winning percentage by getting a lead in the first place is only 12% - barely more than half the significance of the other two thresholds!

What this implies is that your strategy prior to FJ should not focus on taking a lead into FJ, but on taking a large lead into FJ, while ensuring that you don't fall below the two-thirds threshold. This often means making uncomfortably large Daily Double wagers.

Example: You're leading $17000-$12000, and hit a DD on the last clue before FJ. Almost all players in this situation wager somewhere in the $1000-$5000 range. This is a good bet. You're risking almost nothing, and if you get it right you're past the two-thirds threshold, adding 21% to your winning percentage. But is it the best bet?

No. You should wager $7500 instead, and it's not even close.

It's a substantial risk, because if you miss you give up the lead; but if you answer correctly, you don't just have a large lead, you have a lock. Locks are more important than leads. You're risking 12% to gain 44%.

If you're 50% likely to get the DD correct, then the small wager improves your winning percentage by 10.5%, while the large wager improves it by 16%.

But DDs are easier than FJs. You're probably more like 60% or 70% to respond correctly. If your chance of responding correctly is two-thirds, then the small wager improves your winning percentage by 14%, while the large wager improves it by over 25%. Making the right bet adds 11% to your winning percentage compared to the wrong bet.

This is huge. It's worth about $10000 in real money? It's worth about as much as taking the lead going into FJ? It's worth a half-dozen average non-DD clues? Take your pick.

Obviously most DD decisions aren't on the last clue of the second round, and aren't as easy to analyze as this. But the same principles apply, and usually point in the same direction: FJ is hard, so go big early.

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February 2013

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