I had a little bit of an issue with this when I first read it but now I'm not sure. It looks like you're completely right. Golf says the leader has two valid wagers, 1201 or everything. You, Vermonter, conclude that the leader has two valid wagers, 0 or everything. Scores are 18,800/18,800/10,000 going into FJ.Vermonter wrote: ↑Sat Jan 23, 2021 4:02 pmI don't take this lightly, since I believe this is the first time this has ever happened, but I'm going to disagree with you here on both counts. For one, Maggie needs to wager 8,801-9,999 to cover a potential zero wager by both leaders.Golf wrote: ↑Sat Jan 23, 2021 10:49 amFrom Maggie's spot there are two viable wagers, 0 or 8801. From the leader's perspectives there are two viable wagers, 1201 or everything.Bamaman wrote: ↑Sat Jan 23, 2021 10:30 amIn Maggie’s spot I would have bet $9,998. I would assume the leaders would bet it all but one might hold back a dollar so that thwarts that strategy. If I hated the category I would probably bet nothing.
In the lead I would absolutely bet it all. Even if I hated the category.
After Claire McNear emailed me last night with some questions for today's article, it dawned on me that I'd never explored the game theory of this specific situation. If you read below, you can kinda see why — conceptually, it's a mess. (Fear not: I plan to make a video.)
WARNING: GAME THEORY
Although it seems like it might be, wagering 1,201 is not a viable option because it assumes two outcomes that are mutually exclusive. Let's look at this situation from Brian's perspective.
First, consider your options against Jack. In three of the four possible Right/Wrong combinations — RR, RW, WR — wagering everything either gives you the best possible outcome (when you get it right) or is irrelevant (you miss while Jack gets it right and wagers more than zero).
The only combinations when an all-in wager could hurt you is WW, along with WR where Jack wagers zero. Therefore, against Jack, the only time you should wager zero is if you feel there's a >50% chance that BOTH (1) you'll get it wrong AND (2) Jack will either (a) get it wrong or (b) wager zero.
Next, assume you have only two options against Maggie: zero or 1,201. Assuming she wagers properly — a big if, of course, but in game theory, we consider only rational behavior — the sole scenario under which it makes a difference is when you both respond correctly. So by wagering 1,201, you are saying you feel there's a >50% chance that you both get it right.
Finally, let's combine those two statements. By not wagering everything, you're saying there's a >50% chance you'll get it wrong; by wagering 1,201 instead of zero, you're saying there's a >50% chance you'll get it right. Both of those cannot be true.
Conclusion: you should wager everything — unless you feel there is a >50% chance that (1) you'll get it wrong, (2) Maggie will get it wrong, AND (3) Jack will either (a) get it wrong or (b) wager zero, in which case you should wager zero.
At a first glance it might look like there is a scenario where the 1201 wager wins the game where the 0 wager would lose, and there is no scenario where the 0 wager wins the game where the 1201 wager would lose. Therefore the 1201 wager should be better than 0.
Except that is not true, there is a scenario where the 0 wager wins the game and the 1201 wager would lose, when the other leader bets 0 and you miss.