Musings on Sports Economics

Further Thoughts on Jeopardy! wagering (and comments from one of the actual contestants)

Yesterday, I dashed off a post regarding proper Jeopardy! betting tactics based on the unusual circumstance of a tie for first place on last night’s episode. I encourage you to go read it (How to bet, and not to bet, in Jeopardy!), but the conclusion was:

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.  … 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. … In this case, the correct bet for the player in third was no more than $2,800 and in reality, should have been zero.

To my delight and surprise, Natalie Hudson, the woman who came in third place on last night’s show, actually came here to Sportsgeekonomics and shared some of her insight.  First let me share that with you (it’s in the comments of the previous post, but it bears repeating):

… I will say that I felt time pressure when we were making our wagers.

I first calculated what Parker would end up with if he got it wrong, and I saw that I shouldn’t bet more than $2800.

I didn’t fully think through what Kristin might bet. Like you, I see players in second place bet all kinds of irrational things, so I think I mentally threw up my hands and said “who knows what she’ll bet!” So I didn’t notice the potential tie (and neither did Kristin, incidentally). I did think that I should try to beat her zero bet if I could, so that if I got it and she missed it, I would win. I saw that I would need to bet more than $3400 to do that. Darn.

And this is when the panic set in. Parker and Kristin had both finished wagering , and one of the contestant coordinators was hovering nearby, waiting to take up my scratch paper. They didn’t actually pressure me, and I think at some point they said we could take as long as we needed, but I felt like everyone was waiting on me (and, in a high-pressure situation, I was not thinking especially clearly).

I knew I had to bet either less than $2800 or more than $3400. The smaller bet was the smart one, but I took a look at the category, thought about all the people who would give me crap if I lost by not betting enough, and wrote down $3401. Oh well.

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:

"Voir dire" popped into my head immediately, and I never doubted it, though of course I should have, since if I’d thought for two more seconds, I would have realized it only had the "speak" part of the clue in it. I think putting "French" in the clue made it kinda neg bait for "voir dire," but maybe that’s just sour grapes on my part. I don’t know what the writers intended. But if they had said "Latin"

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.

… . No need to apologize for picking on me. You were fair, and you didn’t say anything I haven’t thought a million times since the taping in October.

Posted by
Andy Schwarz

Recent comments

Blog comments powered by Disqus