Final round of trivia
February 5th 2012 00:28
It's the final round of trivia. There's 20 points in it. Scores are:
-- We're on 34
-- Second place on 33.5
-- Third place on 28
-- Fourth place on 26
-- Fifth place on 17
-- Sixth place on 15
The compere says, "Before this next round starts, you can bet points on a double-or-nothing basis. For instance, you can bet 10 points that you'll get question 4 right. If you do get question 4 right, you'll get 20 points for that question; if you lose, you lose 10 points."
Should we bet? And, if so, how much?
Fourth place might think to themselves about the scenario in which no one bets. "If things continue at the current pace, and everyone gets about the same scores they did in the last few rounds, we won't win anything. We'll just fall further behind. Coming fourth is as good as coming last. Therefore we might as well bet big. How big? Well, in for a penny, in for a pound -- let's bet everything." Third to sixth places might adopt similar reasoning.
Second place and first place have a big enough lead that, without betting, the order isn't likely to change. Their thinking falls into the same mistake of not taking into account what other teams might do, because this is so difficult to factor in. "If we bet, and we lose, we'll lose our lead. We worked so hard for it. Why risk it?" -- They're worried about what they have to lose, whereas the other teams decided they have nothing to lose and are focused on what they have to win. They are also irrationally factoring in their hard work or justified winnings -- it seems counter to common conservative sense to gamble on something you've sweated over or earned.
We're likely to get at least 50% of our questions right, because this has been the pattern from all previous trivia competitions, and we're better than everyone else. Since we're on 34, and there's 20 points up for grabs, if we don't bet anything, we'll probably end up on 44.
If third place are scoring about 7 points a round, bet everything they have, and get it right, they'll be on about 62.
What to do?
I've got a feeling that, if you didn't take other teams into consideration, decision theory would say it doesn't matter whether you bet or not. I'm probably misrepresenting decision theory here, but anyway... Over the long run, the person who bets the same each round on a 50% double-or-nothing basis, and the person who doesn't bet at all, should score the same. (So if you get 2, 0, 2, 0, 2, 0... the average is the same as me getting 1, 1, 1, 1, 1, 1...)
But when you do take the other teams into account, what happens?
Maybe think about it this way...
-- If third place, on 28, have a 30% track record, then, out of 20 points, they're likely to score about 7. If they bet everything they have, then they have a 30% chance that one of those 7 points will instead be 28 points -- a 30% chance that their score will be 62; a 70% chance it will be 7. In the long run, their average is going to be approaching (30% of 62) plus (70% of 7) = 23.5.
-- If third place bet 20 points instead of 28 points, their long-run average will approach 26.7.
-- If they bet 10 points, their long-run average will approach 33.7.
-- If they bet 0 points, their long-run average will approach 35.
-- If their track record were 60%, and they bet everything they had, their long-run average would approach 45.
So, if your likelihood of a bonus is less than 50%, it makes more sense in the long run not to bet. If your likelihood is greater than 50%, it makes sense to bet. If it's 50%, it doesn't matter.
Since I'm claiming that each team is scoring 50% or less, then it doesn't matter what other teams do. If they bet, it's worse off for them. If you're right on 50%, then it doesn't matter what you do either.
If the playing field were different, and teams were scoring greater than 50%, then you'd have to make a speculative guess as to how likely it is that other teams would bet before deciding whether yourself to bet.
The long and the short of it -- we were coming first; we didn't bet anything; at the end of the game, we were last!
Big upset.
A shame that this sort of iterative thinking is no use for one-off situations...
-- We're on 34
-- Second place on 33.5
-- Third place on 28
-- Fourth place on 26
-- Fifth place on 17
-- Sixth place on 15
The compere says, "Before this next round starts, you can bet points on a double-or-nothing basis. For instance, you can bet 10 points that you'll get question 4 right. If you do get question 4 right, you'll get 20 points for that question; if you lose, you lose 10 points."
Should we bet? And, if so, how much?
Fourth place might think to themselves about the scenario in which no one bets. "If things continue at the current pace, and everyone gets about the same scores they did in the last few rounds, we won't win anything. We'll just fall further behind. Coming fourth is as good as coming last. Therefore we might as well bet big. How big? Well, in for a penny, in for a pound -- let's bet everything." Third to sixth places might adopt similar reasoning.
Second place and first place have a big enough lead that, without betting, the order isn't likely to change. Their thinking falls into the same mistake of not taking into account what other teams might do, because this is so difficult to factor in. "If we bet, and we lose, we'll lose our lead. We worked so hard for it. Why risk it?" -- They're worried about what they have to lose, whereas the other teams decided they have nothing to lose and are focused on what they have to win. They are also irrationally factoring in their hard work or justified winnings -- it seems counter to common conservative sense to gamble on something you've sweated over or earned.
We're likely to get at least 50% of our questions right, because this has been the pattern from all previous trivia competitions, and we're better than everyone else. Since we're on 34, and there's 20 points up for grabs, if we don't bet anything, we'll probably end up on 44.
If third place are scoring about 7 points a round, bet everything they have, and get it right, they'll be on about 62.
What to do?
***
I've got a feeling that, if you didn't take other teams into consideration, decision theory would say it doesn't matter whether you bet or not. I'm probably misrepresenting decision theory here, but anyway... Over the long run, the person who bets the same each round on a 50% double-or-nothing basis, and the person who doesn't bet at all, should score the same. (So if you get 2, 0, 2, 0, 2, 0... the average is the same as me getting 1, 1, 1, 1, 1, 1...)
But when you do take the other teams into account, what happens?
Maybe think about it this way...
-- If third place, on 28, have a 30% track record, then, out of 20 points, they're likely to score about 7. If they bet everything they have, then they have a 30% chance that one of those 7 points will instead be 28 points -- a 30% chance that their score will be 62; a 70% chance it will be 7. In the long run, their average is going to be approaching (30% of 62) plus (70% of 7) = 23.5.
-- If third place bet 20 points instead of 28 points, their long-run average will approach 26.7.
-- If they bet 10 points, their long-run average will approach 33.7.
-- If they bet 0 points, their long-run average will approach 35.
-- If their track record were 60%, and they bet everything they had, their long-run average would approach 45.
So, if your likelihood of a bonus is less than 50%, it makes more sense in the long run not to bet. If your likelihood is greater than 50%, it makes sense to bet. If it's 50%, it doesn't matter.
Since I'm claiming that each team is scoring 50% or less, then it doesn't matter what other teams do. If they bet, it's worse off for them. If you're right on 50%, then it doesn't matter what you do either.
If the playing field were different, and teams were scoring greater than 50%, then you'd have to make a speculative guess as to how likely it is that other teams would bet before deciding whether yourself to bet.
***
The long and the short of it -- we were coming first; we didn't bet anything; at the end of the game, we were last!
Big upset.
A shame that this sort of iterative thinking is no use for one-off situations...
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