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.
So this was actually the first thing I wanted to write about upon reflection of my post from last night, which is time pressure. It’s one thing to be able to sit on my couch after dinner with the Tivo button on pause, pop open a spreadsheet, and figure out the optimal bets. Because my son was on the show, I’ve seen it being taped and doing math during the commercial break has got to be somewhat harrowing. As Natalie points out, she came armed with good strategy and knew her options. She had to make a call in the moment. That’s got to be hard. That’s why I told my son, who is good at math but was only 12 at the time he was on, that if he was in third and he thought his dollar total was more than the first place guy would end up with if he got the answer wrong, just bet zero (in our example, if you have more than $6,799). We all need simplifying heuristics at times and when one is under the Jeopardy! spotlight is probably a good time to keep it simple. So a bet of $2,799 was probably optimal (especially given the category, see below) but a bet of zero was adequate and easier to be armed with in advance. Economists might call zero the “satisficing” choice (i.e. it was good enough) but I might go further and say it’s the bounded-rationality-optimum because it lets you make a bet without doing too much math in a very stressful situation. (That doesn’t mean I think $3,401 was a correct bid, but I have deep sympathy for Natalie having to go through those gyrations in real-time in Culver City with Alex Trebek standing next to you for the photo ops they do during commercials, etc. it just means she, and all of us who entertain hopes of competing, should go in armed with knowing when to just say zero.)
The second thing I wanted to muse on was more of a Game Theory issue. Upon overnight reflection, I realized that as optimal as I thought the first two contestants bets were, they don’t seem to be a Nash equilibrium. Loosely, what I mean by this is that given what everyone else bet, each player would choose to change his/her bet in response if he/she could. A Nash equilibrium requires the bets to be stable even when the others are revealed (not whether the question is revealed, just the bets). So once the woman in second place bet her $6,201, the first place bet of $9,601 stopped being optimal. So if the first-place player had been allowed to see the other bets and then change his, he could have lowered his bet to $2,801 (which is actually close to the optimal 3rd place bet, so this is weirding me out how the numbers ) and been guaranteed a win whether both contestants got the question right or both got it wrong (though still not if he got it wrong and she got it right).
But then of course, if the second place player knew he was betting $2,801, she could bet more than $6,201 and then she could jump into the lead if they both got it right.
So without doing all of the math, my intuition tells me that the only Nash equilibrium is going to be what economists call a “mixed strategy,” meaning one where the contestants randomly choose among conditionally optimal choices (i.e., if she does this, my best bet is that). But there is economics and there is reality, and while they often coincide, in this case I think the reality is that players play to win when they know the answer, even if that ends up being suboptimal. By this what I mean is that if you are in first place going in to final jeopardy, and you get the answer right, there is no way that a competitive trivia player is going to want to lose when he/she has control of the game. So except for a person with very low confidence in his/her ability at Final Jeopardy!, the person in first place is going to bet to guarantee a win if he/she gets it right. it may not be a Nash equilibrium bet, but it is the only bet you can live with yourself over. How could anyone stand it to make it to Final jeopardy! with the lead, get the final question right, and lose. You just can’t, even if a mixed strategy is ideal and thus argues that you should sometimes mix things up by betting less.
So “optimal” as used in this entire analysis is always used with the recognition that we’re dealing with humans with human emotions and that one possibly optimal scenario, where the player in first place risks losing when he/she knows the answer to improve his/her chance of winning when he/she doesn’t know that answer, is pretty much off the table.
A few other thoughts. First — I just want to say again that it’s easy for me to criticize given I was not on the show. And the reason I was not on the show, despite the fact that my mother and my son have been on the show, is that in the one time I tried out (back in 1989), I screwed up my test and just missed the cut to go on to the more rigorous auditions. So this is a critique from a guy who woulda-coulda-shoulda, but didn’t.
Second, I was fairly generic in my assumptions about category-specific knowledge. I didn’t actually watch last night’s show (I Tivo-ed it and skipped to the end once I heard about the tie, because I was looking for something algebraic to think about) so I didn’t know, for example, that Natalie (a.k.a. “the woman in third place) was an attorney. With a category like legal terms, of course she’s likely to be inclined to take an even-money bet that she would still know the answer even if the two contestants ahead of her in score didn’t. It’d be like if the final Jeopardy! category were “White v. NCAA” (a case I helped to initiate and that I worked on) and I were sitting in third place. I’d probably say “math be damned, bet it all!”
As it turned out, the question was kind of tricky. I work in the law (as a consultant and expert witness) and while (on the comfort of my couch) I almost got to it before the 30 seconds were up, I didn’t get it. This is in part because there is a common, French-derived term that means something like “To see, to say” and it is used for the process of asking jurors about their biases, etc. It’s called “voir dire” and even though I knew that was the wrong answer, it was the only French I could think of. The real answer, “Verdict” is from Old French but retains a lot of its Latin origins and, well, let me let Natalie herself explain:
And finally, I felt kind of bad that I made an example out of Natalie, only to interact with her, but she was kind enough to forgive me for singling her out as an example of a phenomenon, even if it turns out that she was actually very clued in to proper betting strategy and just chose a suboptimal fork in the decision tree.