The normalization is because we want to compare what happens conditional on Hillary being elected to what happens conditional on Jeb is elected. These probabilities will not be comparable unless we normalize.
Why would we want to do this? Your contracts aren’t structured in such a way that they encourage these sorts of conditional considerations. P(A|B) isn’t on the market. P(A and B) is. Maybe you meant for your contracts to be “If Hillary is elected, the U.S. will be nuked.” ?
You are right that P(A|B) isn’t on the market and that P(A and B) is. However, it is easy to calculate P(A|B) from P(A and B) /P(B) . The problem is that P(A|B) does not help inform you about the correct decision.
You are also right that what we want is something like “If Hillary is elected, the U.S. will be nuked”. However, the problem is that this natural language sentence is ambiguous: It can be interpreted as P(A|B), in which case it will lead to incorrect decisions. Alternatively, it can be interpreted as P[A| Do(B)], which is the information we need, but then it will be very challenging to write the rules of a prediction market such that the expiry conditions incentivize participants to bet their true beliefs.
I understand that we’re capable of calculating P(A|B), but if P(A|B) isn’t on the market, then the market won’t reflect the value of P(A|B). So I don’t understand your statement that the market will somehow get the answer wrong because of its estimate of P(A|B). The market makes no value estimate of that quantity.
Your market, as stated, is really strange in a lot of ways. By having the contracts include “Bush wins” or “Clinton wins” the market is essentially predicting itself. It’s going to have really strong attractors for a landslide victory. It seems like that isn’t what you intend, but it’s going to be the consequence of your current set up. Judging by the number of other people who have also replied that they are confused, you may want to rework this example.
You could have a market that estimates P(A|B) directly using the reversal mechanism (called-off bets). However, I maintain that this will give identical estimates as the markets I proposed. These things are probabilities, they follow the rules of probability logic.
The point I was trying to illustrate, was not that it is impossible to estimate P(A|B), but rather that P(A|B) is not the quantity that a rational decision maker needs in order to optimize for A.
I agree that using a market that directly estimates P(A|B) might have been a better example, because it avoids readers going in the wrong direction when they try to figure out what is going on. However, changing this will take some non-trivial rewriting of the text. I will try to do that when I have more time on my hands.
Your point about the markets predicting themselves is interesting. I was imagining a democracy informed by prediction markets rather than a pure futarchy. However, if the voters are influenced by the market, it does indeed predict itself to some extent. I don’t think this is a major problem, but I will keep thinking about it. I have relatively high confidence that my argument for prediction markets being confounded does not rely on this.
In the scenario I provided, the contracts will be traded at the following prices after the demon reveals his information:
Hillary elected and US nuked: $14.3 (1/7 of $100)
Hillary elected and US not nuked: $28.6 (2/7)
Jeb elected and US nuked: $14.3 (1/7)
Jeb elected and US not nuked: $42.9 (3/7)
Like you said, if people change their votes based on the market, the prices may be distorted by the market predicting itself.
Why would we want to do this? Your contracts aren’t structured in such a way that they encourage these sorts of conditional considerations. P(A|B) isn’t on the market. P(A and B) is. Maybe you meant for your contracts to be “If Hillary is elected, the U.S. will be nuked.” ?
You are right that P(A|B) isn’t on the market and that P(A and B) is. However, it is easy to calculate P(A|B) from P(A and B) /P(B) . The problem is that P(A|B) does not help inform you about the correct decision.
You are also right that what we want is something like “If Hillary is elected, the U.S. will be nuked”. However, the problem is that this natural language sentence is ambiguous: It can be interpreted as P(A|B), in which case it will lead to incorrect decisions. Alternatively, it can be interpreted as P[A| Do(B)], which is the information we need, but then it will be very challenging to write the rules of a prediction market such that the expiry conditions incentivize participants to bet their true beliefs.
I understand that we’re capable of calculating P(A|B), but if P(A|B) isn’t on the market, then the market won’t reflect the value of P(A|B). So I don’t understand your statement that the market will somehow get the answer wrong because of its estimate of P(A|B). The market makes no value estimate of that quantity.
Your market, as stated, is really strange in a lot of ways. By having the contracts include “Bush wins” or “Clinton wins” the market is essentially predicting itself. It’s going to have really strong attractors for a landslide victory. It seems like that isn’t what you intend, but it’s going to be the consequence of your current set up. Judging by the number of other people who have also replied that they are confused, you may want to rework this example.
You could have a market that estimates P(A|B) directly using the reversal mechanism (called-off bets). However, I maintain that this will give identical estimates as the markets I proposed. These things are probabilities, they follow the rules of probability logic.
The point I was trying to illustrate, was not that it is impossible to estimate P(A|B), but rather that P(A|B) is not the quantity that a rational decision maker needs in order to optimize for A.
I agree that using a market that directly estimates P(A|B) might have been a better example, because it avoids readers going in the wrong direction when they try to figure out what is going on. However, changing this will take some non-trivial rewriting of the text. I will try to do that when I have more time on my hands.
Your point about the markets predicting themselves is interesting. I was imagining a democracy informed by prediction markets rather than a pure futarchy. However, if the voters are influenced by the market, it does indeed predict itself to some extent. I don’t think this is a major problem, but I will keep thinking about it. I have relatively high confidence that my argument for prediction markets being confounded does not rely on this.
Given how you have set this problem up, what do you think will be the relative prices of the 4 contracts you specified?
In the scenario I provided, the contracts will be traded at the following prices after the demon reveals his information:
Hillary elected and US nuked: $14.3 (1/7 of $100) Hillary elected and US not nuked: $28.6 (2/7) Jeb elected and US nuked: $14.3 (1/7) Jeb elected and US not nuked: $42.9 (3/7)
Like you said, if people change their votes based on the market, the prices may be distorted by the market predicting itself.