## How to bet, and not to bet, in Jeopardy!

On tonight’s Jeopardy! a fairly rare thing happened — the game ended with a tie for first place. When Michael Felder (@InTheBleachers) announced this on twitter, my immediate reaction, without even knowing which of the contestants (in terms of their position prior to Final Jeopardy!) ended up in the tie, was that someone bet incorrectly. This post is designed to explore that outcome.

First, as an introductory matter, Jeopardy! runs in my family. My mother was on the show in the very old days, pre-Alex Trebek, when the dollar values started at $10, and although she lost at the Final Jeopardy! question, the host, Art Fleming, wished her “Good Luck with the Little One.” The little one in question was me, *in utero*.

Fast forward 44 years and her grandson, my son, was on Jeopardy! during kids week and was the Wednesday show’s champion. We talked about trivia a tiny bit before he was on the show, but really the only thing I coached him on was betting strategy. I told him, as I am telling you all now, that many very smart contestants on Jeopardy! seem to make very poor tactical choices when it comes to betting. And indeed, as I predicted this afternoon when I heard about the tie, tonight’s Jeopardy! is a great example of how poor betting strategy caused the third place player to miss her chance to win.

*(As I note below, you can also argue it caused the second place player to have to share the win rather than win it outright. But the specifics of the numbers meant she had to choose a scenario where she tied — there was no clear-cut winning bet, so I cannot call this an error ).*

So, let’s work through some Jeopardy! math.

**First Place**: This one is easy and in fact regular viewers of the show probably know how this goes. Unless you are absolutely convinced you know absolutely nothing about the category and so you’re certain you will get it wrong, the player in first needs to bet just enough to win one dollar more than double what player 2 bets. This is the lowest-risk way to ensure that if you get it right, you’re guaranteed to win and if you get it wrong, you will potentially have enough money to beat the other players nonetheless.

On tonight’s show, the guy who finished Double Jeopardy! in first place had $16,400 and the woman who was in second had $13,000. So, his optimal bet is to bet just enough to get $1 more than double $13,000. That number is $9,601. Not surprisingly, that is what he bet. So one of the contestants bet correctly.

**Second Place**: It is so rare that I see the person in second place bet correctly that I pretty much assume it will not happen. The thing to recognize is that if first place bets as described above and gets the answer right, there is NOTHING that second place can do to win. Betting every last dime only means that at best you will lost by $1. But since on Jeopardy! losing by $1 still means you lost (there is some minor difference b/w 2nd and 3rd, but in either case, it’s not the big money of winning). Instead, the only way the person in second place ends up winning is if first place gets the Final Question wrong. And you actually can calculate exactly what the person in first place will end up with if he/she gets it wrong, because you know what he/she is going to bet, i.e., just enough to beat you by $1 if you bet everything.

On tonight’s show, as we’re seen, that was $9,601. So the woman in second place should have known that her target was not to beat what the guy in first place had, but what he would end up with if he got the answer wrong, which was $6,799. Since she already had $13,000, she didn’t need to bet anything to beat that, but she could bet some if she wanted to. She just needed to make sure of two things. One, if both she and the guy in first got it wrong, she wants to make sure she loses only a small amount so she ends up ahead of him. Two, she needs to also make sure that if she and the woman in the woman in third place both get it right, that she will still emerge victorious.

As it turns out, the woman in third place had $9,600.

If you are quick at math you will see that this was actually an amazing number in this situation, because it put the woman in second place in a very delicate position. The maximum she could bet and still not fall behind the guy in first was $6,201. To see this, remember that the first place guy is going to bet $9,601 and thus, if he gets it wrong, he’ll end up with $6,799. That means, for the woman in second with $13,000, she can afford to lose $6,201. If they both get it wrong, they would both end up tied. Normally, I would say this means she ought to have bet a little less, say $2 less, but as you’ll see below, in this very rare case, betting exactly $6,201 is the right bet even if it means she would tie, rather than win, in the event that both first and second get it wrong.

This is because she also had to worry about the woman in third with her $9,600. If that woman she got the answer right and if she bet all her money (which she shouldn’t do but more on that later), she would end up with $19,200. Which means that the woman in second place needed to bet $6,201 (yes, the exact same number!) to make sure the woman in third place didn’t pass her if they both got it right. Which is actually pretty amazing. Her maximum possible bet to beat the first-place guy (if they both get it wrong) is identical to the minimum possible bet needed to ensure she beat the third-place player (if they both get it right).

So, suffice it to say, the woman in second place also bet correctly. This is fairly rare to see and I commend her for it. Indeed, in tonight’s show the woman in second was the returning champion, so she’s already shown she is smart and perhaps she’s had more time to think through these scenarios. In any case, she threaded the needle and found herself a way to eke into another day’s competition. Brava!

*Note: One can quibble over whether this means she should have bet $6,200 or $6,201. I’d argue the former bet of $6,200 was slightly smarter (and would have given her the outright win tonight), because it meant she would win outright on a really hard question that no one got right and still tie on one that only the first-place player got wrong. Whereas, by choosing $6,201 she guaranteed a win on a question that only the first-place player got wrong and was willing to accept a tie on one where everyone got it wrong. Since the person in third place is usually in third place for a reason — that is he/she is less good at the game — I’d prefer my tie come against third-place, who then advances with me, rather than having first-place advance with me. But that’s a minor quibble and a nuance we needn’t worry about.*

**Third Place**. Here is where I see the most bad betting and tonight’s show was no exception. As the player in third place, you need to know 2 things. One, you will only win if both of the players ahead of you going into Final Jeopardy! get the question wrong (unless they bet poorly). Two, you’ve already proven over the course of two rounds of Jeopardy! that they are at least a little bit better than you at the game. It may not mean that they will be better at Final Jeopardy! than you, but it means that the likelihood of both of them getting the final question wrong while you get it right is pretty low. You might all get it right. You might all get it wrong. But the scenario where both of them get it wrong and you get it right is pretty rare. You are best off betting as if the only way you can win is where everyone whiffs.

So, what do you do then? You know what the first place player will have if he/she gets it wrong. In tonight’s show again that was $6,799. So the woman in third place was in good shape — she just needed to beat this by a little — say $5 — if she lost (I say by a little because she also needs to beware of the person in 2nd playing an optimal bet that allows second place to edge out first if they both get it wrong). A bet of $2,801 means she would triple-tie for first, so betting just a little less, say, $2,795, should secure her a win in the event of everyone getting the final question wrong.

But again, she was in third-place. She is only going to win on a hard question that both of the other two contestants get wrong. Why risk any money on a question that you know is going to stump two other players, both of whom are have played better than her for the last 22 minutes or so? The correct bet here is probably $0. Yes, in the rare event that she got it right and no one else did, if she bets zero, she ends up leaving a little money on the table. But she still wins and gets to play another round the following day. And by betting zero, she gives herself a little more cushion in case someone else under-bets (and thus loses less). In this case, the correct bet for the player in third was no more than $2,800 and in reality, should have been zero.

Instead, tonight’s contestant bet $3,401. I know where she got that — it is what she would need to bet if player 2 bet $0 and she wanted to pass her, because $9,600 + $3,401 would be $1 more than player 2’s initial $13,000. But I cannot think of a scenario where player 2 would bet nothing. Indeed, in this case, it turns out player 2’s optimal bet is darn close to being locked in at $6,200 or $6,201. It’s usually not that easy to know player 2’s bet, but as a general rule it is safe for player 3 to assume that player 2 will bet enough to beat player 3 by at least $1 if they both get it right (i.e., player 3 can think about player 2 the same way player 2 thinks about player 1). So the bet where you just try to nudge out player 2 if she bets $0 is not a good one. You are better off beating where she will end up if she and player 1 both bet optimally. Which in this case is $2,800… except that I still think it’s $0 because if it’s a hard enough question to trip up the first- and second- place players, who would want to make an even-money bet that the third-place player will get it right? Not me.

So, I was right when I predicted that someone bet wrong. What’s surprising is that it was not the woman in second place, who did just about the only thing she could do to survive, and her smart betting tactics paid off with a tie and advancement. But, whether she knows it or not, the woman in third place had the game in her reach, and all she had to do was refuse to bet a dime.

Again, I am not just saying this because I saw how it worked out. Her best bet prior to even seeing the question was $0. (In economics, we use fancy terms like “this is an *ex ante*, not *ex post*, optimal strategy, but all we mean is that this is not just a Monday Morning QB sort of carping — it was the smart bet even before the question was revealed). But she bet too much and we ended up with two champions, neither of whom knew the correct answer to Final Jeopardy!, but both of whom bet optimally.

I live in full hope that some day third-place players on Jeopardy! will start betting more wisely. But I am not holding my breath.