Twyla's $0 bet was less than ideal. Dean had absolutely no reason to risk falling below Twyla's $8400 on a miss. (The fact that he did so anyway is not entirely beside the point, though. I'll come to that in a bit.) Twyla should have bet as though Dean would make the shutout bet against her. Here were the pre-FJ scores:
Point 1: Sara's most likely bet is the shutout against Dean--$11,001. (And that in fact is what she bet.) Dean can't beat her if she gets FJ right. If she misses, she goes down to $5,799. So Dean needs to stay above that number for any realistic chance of winning. Any bet that takes him below $5,799 is foolish.
Point 2: Dean can safely make the shutout bet of $2,901 against Twyla. A get puts him out of Twyla's reach while a miss still keeps him ahead of her current score at $10,999. Twyla CANNOT win on a $0 bet against a rational wager by Dean. She needs to get up to $11,000 for a win.
Point 3: Twyla can get up to $11,000 for the win by betting $2,600 and getting FJ right. With this bet, she will almost certainly win as long as both her opponents miss. She can win on a TS or a sole get. That's the best offer she's going to get in her third place position.
Subtlety A: If Twyla loses that $2,600, she drops down to $5,800. That won't beat any rational wager by Dean. But it WILL beat a foolish wager such as the one he made. And--here's the fun part--it will also beat Sara BY ONE DOLLAR if Sara makes the shutout bet and misses. For that reason, I would have advised Sara to bet a dollar less than she did. (For similar reasons, I'd have advised Dean to bet $2,900 rather than $2,901.) If she got it right and Dean doubled his score, she would have tied rather than won outright. But if she missed, and Twyla made the ideal bet and also missed, Sara would tie rather than lose.
Subtlety B: What if Sara hates the category and doesn't make the shutout bet? What will she bet instead? Nothing. Zip. Nada. Twyla has exactly half Sara's score. If Sara bets $0, the most Twyla can do is double her score and tie. If Sara bets $0, all she needs is for Dean to miss FJ and she's a returning (possibly co-) champion. That means it is NOT unreasonable for Twyla to go all in, but it would be unreasonable to go ALMOST all in. This is not a situation where she can leave herself a buck or two on a miss. Bet it all or bet $2,600. I think $2,600 is a better bet, but that's a matter of feel rather than math.
CONCLUSION: Here are the Opus Approved bets for these players along with the grades they get for their actual bets. Deviating from these by even a dollar will reduce their chances, if only a little. Deviating by more than a dollar seriously reduces their chances:
Sara: $11,000 or $0 -- Actual bet - $11,001, Grade - A-
Dean: $2,900 -- Actual bet - $12,500, Grade - F
Twyla: $2,600 or $8,400 -- Actual bet - $0, Grade - D
Opus, you're assuming everyone bets smartly.
No, but I am trying to hedge against that possibility. I think that's worth doing. If I understand things, that's where our difference of opinion lies.
Sherm wrote:About half of the people on the show don't, and in betting logic in the real world, that should be taken into account. Twyla's A bet is 2600. There is little doubt about that. If she gets it right, she beats Dean if he gets it wrong (assuming he bets it right), and it beats Sara if she gets it wrong and bets it right (lock out bet).
What you are missing is really simple from last place. Almost, the only way Twyla is going to win is for the other two players to get it wrong on FJ.
Right. We agree on that part. (Unless you're saying that's part of what I'm missing, but I don't think you are.)
Sherm wrote:If that happens, you have to strongly consider it is going to be a triple stumper, and the way you maximize your dollar from last place, you bet nothing, and hope you will have enough. Someone on here might even be willing to look it up, but I'd be than willing to bet that if first and second place miss on FJ, more often than not, so does third, and its probably at around a 70% rate (2/3rds to 3/4ths of the time thats probably the result).
I think that's a fair argument. I'd definitely be interested in the statistics. Perhaps my grade for Twyla was too harsh. If she wants to appeal her grade, she's welcome to submit her own reasoning and I'll take it under consideration. I think I could see my way clear to a B+ if she reasons as you did. But if I catch her copying your work, it's an automatic F.
Out of curiosity, how much do you feel it's worth risking to open up the possibility of winning on a sole get against a rational wager? $2,600 is too much from your perspective. I assume $100 wouldn't be? I'm not sure what my own answer is since it's more than $10,000. It's hard to imagine a similar scenario that would put more than that at risk, so I don't have a feel for how high I'd go.