Tim Urban has written a interesting analysis of James Holzhauer’s Final Jeopardy bet:
The post is terrific, including detailed game theoretic analysis with humorous twists and terms like “Mindfuck” and “Double-Mindfuck” strategies, like this one:
Readers should read Tim’s article first to get context on the decision and analysis, but here’s a summary excerpt of the situation:
On Monday, James, normally beginning Final Jeopardy in first place in the automatic Scenario A or only-need-to-get-it-right Scenario B situations, found himself in a Scenario C situation—in second place—behind an extremely strong player named Emma Boettcher. This was the score: [Emma: $26,600, James: $23,400, Jay: $11,000]
So James did what I described above. He figured Emma would bet at least enough ($20,201) to top twice his total ($46,800) in the case that they both got the answer right.
Following that logic, he figured that if Emma got it wrong, she’d end up losing at least $20,201, bringing her down to $6,399 at most. His only shot of winning would be if Emma got it wrong, so $6,399 became his “do not cross” floor. His maximum bet would then be $17,000, which would guarantee that he beat Emma if she got it wrong, regardless of whether he got it right or not.
But like in our case above, the third place player, Jay Sexton, was in the picture too, with $11,000. Twice Jay’s total ($22,000) became a second and higher “do not cross” line for James—so James made the perfect bet: $1,399.
With that bet, he’d win if Emma got it wrong, and Jay would have no chance of beating him—in both cases, regardless of whether he got the answer right or wrong. Being in second place isn’t a great situation to be in, but James gave himself the best shot he could.
Or did he?
https://waitbutwhy.com/2019/06/did-james-make-the-right-final-jeopardy-bet.html
This is classic Game Theory analysis — think: John Nash and the Nash Equilibrium from Beautiful Mind.
Overall, I think Tim is on target by questioning James’s betting strategy, though I’d add a few nuances. In the end, I have some agreement with Tim’s conclusion that James should have optimally bet Big instead of Small, but there are nuances and it depends on the parameters, like the probability of getting the question right and how correlated are right / wrong answers between top players.
His post is, naturally, funnier and more interesting than mine, but I thought readers might be interested in seeing my commentary…
Here’s the email I just sent to Tim (slightly edited):
Tim!
Great post! Loved it. I, too, was rooting for James. I tend to root for the favorite, instead of the underdog, in many situations. Excellence is beautiful.
I thought you’d be interested in a few thoughts. I have a PhD in economics, and I’m familiar with how one might think about this in terms of game theory.
The situation is very similar to the “penalty kick” analysis. Economists (including Steve Levitt of Freakonomics fame) have studied game theory in penalty kicks for soccer (see here). There is no pure strategy equilibrium — if the goalie goes left, I want to kick right, but then the goalie prefers to go right, and so I want to kick left, and so on. The only correct equilibrium is where goalies and kickers *both randomize* between guessing left and right. Similar analysis has been done in randomizing in serves in professional tennis.
[Update: Another good example is Rock-Paper-Scissors — pure strategies don’t work since you can always deviate to beat the other guy, but a mixed randomized strategy of 1/3 Rock, 1/3 Paper, 1/3 Scissors for both players is an equilibrium since neither player wants to deviate given the other player playing that strategy.]
What you’ve pointed out is that there is no *pure strategy* equilibrium between betting Big and betting Small. If James is known to bet Big, Emma should bet Big. If James is known to bet Small, Emma should bet Small (or zero). And vice versa. If Emma is known to bet Big, James should bet Small. If Emma is known to bet Small, James should bet Big. None of these is an equilibrium (called a Nash Equilibrium, after John Nash), because the best responses aren’t in sync with one another — everyone always wants to deviate if the other is playing pure strategy.
Here is the game in matrix form, with probabilities of victory in the boxes (based on 90% chance for each of Emma and James getting the question right). Emma wins with 90% probability if she bets Big. She wins with 10% chance if she bets Small and James goes Big, and she wins with 100% chance if they both go Small. James, on the other hand, wins with only 10% if (Big, Small), 9% if (Big, Big), 90% if (Small, Big), and 0% if (Small, Small). Here’s the payoff matrix, with best responses underlined:
Here, there is no pure strategy equilibrium, which would occur if both are underlined in the same box. We can also now think about a *randomized strategy*, where Emma chooses Big with probability x (and Small with probability 1-x) and James chooses Big with probability y (and Small with probability 1-y):
Then, we can solve for the indifference probabilities. When James goes Big with 1/9 probability, Emma is indifferent between Big and Small. When Emma goes Big with 99% probability, James is indifferent between Big and Small. So there’s a possible equilibrium where the strategies are randomized between Big / Small at these rates: James = {1/9 Big, 8/9 Small} = {11% Big, 89% Small}, Emma = {90/91 Big, 1/91 Small} = {99% Big, 1% Small}. Here’s the algebra:
So, what are the takeaways?
James should go Big only if Emma is >1% Small. If there’s a meaningful chance she deviates to Small, then he should go Big to capture that advantage. If she’s >99% chance of going Big, he should stay Small because the advantage to going Big is tiny. Similarly, Emma should go Big if she thinks James has greater than 1/9 chance of going Big. Emma should go Small if she thinks James has less than 1/9 chance of going Big.
Emma’s likelihood of going Big is probably close to 100%, but it’s a judgment call of how close. One consideration is that if she goes Small and James goes Big and then James wins after they both get the question right — she’d rather crawl in a hole and die rather than face all of her friends after losing that way. The payoff of that scenario is like negative infinity, because she could have won had she only bet Big. So it might actually be that she’s at or very close to 100% betting Big. Importantly, though, it depends on how bad this “negative infinity” payout really is — is it so bad to make her 98% Big, 99% Big, 99.99% Big? (Betting Big is also the standard strategy and also allows the leader to “control her own destiny,” which are two other factors indicating close to 100% betting Big for the leader.)
With that in mind, humans can be crazy and irrational, and can overthink things and make mistakes all the time. So if there’s even 1% chance of Emma switching to Small (maybe due to the Mindfuck analysis) — then James should go Big instead of Small.
The main intuition is that James only drops his likelihood of winning from 10% to 9% if he goes Big while Emma goes Big. But he raises his chance of winning from 0% to 90% if Emma goes Small. So, he’s better off gaining that advantage in the however unlikely chance that Emma goes to Small (unless she’s hell bent on Big at >99%).
Final consideration: Their likelihood of James and Emma getting Final Jeopardy right is probably pretty *highly correlated*, or at least modestly correlated. They’ll probably both get it right or both get it wrong. Because they’re so smart, it’s unlikely to stump one of them but not the other (i.e., the answer is either super obscure and neither gets it, or straightforward and they both get it). In that case, the analysis nudges towards James going Small because even if he goes Big he’s unlikely to get as high as 9% chance of winning (it’s likely lower than that). Still, James may also think he still has a good chance of getting it right if Emma gets it wrong, which maintains the attractiveness of going Big.
Also, the analysis is more interesting if chances of getting the question right are 75%, or 50%, or something lower than 90%. Then the randomized equilibrium is further away from 99% for Emma. Those probabilities really matter, and lower chances of being right should nudge James more towards Big, since his advantage of going Big is greater if Emma is more likely to miss the question (though, his advantage of going Small is also bigger, so I’m not 100% certain on the directional effect).
[Update: I calculate that if the chance of getting the question right is 75%, James should go Big if he believes Emma’s chance of going Small is >8% and stay Small if she’s going Big with >92% chance — here, it’s even more of a judgment call and makes the choice a bit more complicated than the 1% cutoff. So that probability really matters.]
Bottom line: James should go Small if he thinks there no reasonable chance (<1%) of Emma going Small. James should also go Small if there’s super low likelihood of getting a question right that Emma gets wrong. BUT, if James thinks there’s a reasonable chance of her going Small (>1%) and also a reasonable chance of him getting a question right that she gets wrong (which is totally possible since he knows so much), then he should go Big. On net, I agree that going Big is the better strategy for James. [Update: Actually, I think it depends quite a bit on the probabilities of correct answers, which explored in more detail below –> there some parameter ranges where James should go Small, which may have justified his decision.].
Anyway, hope you found this interesting — it’s a great game theory problem! It’s too bad we couldn’t see James go for another 50 episodes.
Take care-
DeForest
And here are some follow-up thoughts:
Still thinking… It may be better for James to bet BIG when probability of correctness is high – i.e., like 90% – and better for James to bet SMALL when the probability of correctness is low – i.e., more like 50-75%…
Assuming Emma always goes BIG (since she doesn’t want to look like a fool by going SMALL and losing)…
- At 50% correctness, James wins with 50% chance if betting SMALL and 25% chance if betting BIG (= (1-50%) x 50%).
- At 75% correctness, James wins with 25% chance if betting SMALL and 18.75% chance if betting BIG (= (1-75%) x 25%).
- At 90% correctness, James wins with 10% chance if betting SMALL and 9% chance if betting BIG (= (1-90%) x 90%)).
Overall, it seems James should bet SMALL if:
- 1. He thinks Emma has little or no chance of betting SMALL
- 2. He thinks the category is hard and/or has low expectations for correctness
And then he should bet BIG if:
- 1. He thinks Emma has a reasonable chance of Mindfuck strategy / betting SMALL
- 2. He thinks the category is easy and/or has high expectations for correctness
It would be interesting to see data on how frequently Final Jeopardy is answered correctly, by James and others…
Readers: I’d love to hear any thoughts on this, especially from economist readers…!