If I pay p1 for a contract in market 1, my expected payoff is:
(E[u|d1]−p1)P(d1)+0×P(d2) (since I get my money back if d2/market 2 is activated)
this is negative iff p1>E[u|d1] and positive iff p1<E[u|d1]
This is incorrect. There are two errors here:
The first expectation needs to be conditioned on the market activating. (That is not conditionally independent of u given d1 in general.)
Different people have different beliefs, so the expectations are different for different traders. You can’t write “E” without specifying for which trader.
I agree that if you assume u is conditionally independent of market activation given d1 and that all traders have the same beliefs then the result seems to hold. But those assumptions are basically always false.
The first expectation needs to be conditioned on the market activating. (That is not conditionally independent of u given d1 in general.)
If we commit to using futarchy to choose decision, then market 1 activating will have exactly the same truth conditions as executing d1, so “market activating and d1” would be the exact same thing as “d1“ itself (commiting to use futarchy to choose decision means we assign 0 probability to “first market activating & execute d2” or “Second market activating & execute d1”)
Different people have different beliefs, so the expectations are different for different traders. You can’t write “E” without specifying for which trader.
Yes, we can replace with E_i, and then argue that traders with accurate beliefs will accumulate more money over time, making market estimates more accurate in the limit
Yes, we can replace with E_i, and then argue that traders with accurate beliefs will accumulate more money over time, making market estimates more accurate in the limit
There’s a chicken-and-egg problem here. You’re assuming that markets are causal (meaning traders that are better at estimating causal probabilities) and then using that assumption to prove that markets are causal.
There’s a chicken-and-egg problem here[...] and then using that assumption to prove that markets are causal.
That argument was more about accomodating “different traders with different beliefs”, but here’s an independent argument for market being causal:
When I cause a particular effect/outcome, that means I mediate the influence between the cause of my action and the effect/outcome of my action, the cause of my action is conditionally independent of the effect of my action given me
Futarchy is a similar case: There may be many causes that influence market prices, which in turn determines the decision chosen, & market prices mediate the influence between the cause of market prices (e.g. different traders’ beliefs) and the decision chosen. Any information can only influence what decision will be chosen through influencing the market prices. This seems like what it means for market to be causal (In a bayesnet, the decision chosen will literally only have market prices as the parent, assuming we commit to using futarchy to choose decisions).
This is incorrect. There are two errors here:
The first expectation needs to be conditioned on the market activating. (That is not conditionally independent of u given d1 in general.)
Different people have different beliefs, so the expectations are different for different traders. You can’t write “E” without specifying for which trader.
I agree that if you assume u is conditionally independent of market activation given d1 and that all traders have the same beliefs then the result seems to hold. But those assumptions are basically always false.
If we commit to using futarchy to choose decision, then market 1 activating will have exactly the same truth conditions as executing d1, so “market activating and d1” would be the exact same thing as “d1“ itself (commiting to use futarchy to choose decision means we assign 0 probability to “first market activating & execute d2” or “Second market activating & execute d1”)
Yes, we can replace with E_i, and then argue that traders with accurate beliefs will accumulate more money over time, making market estimates more accurate in the limit
There’s a chicken-and-egg problem here. You’re assuming that markets are causal (meaning traders that are better at estimating causal probabilities) and then using that assumption to prove that markets are causal.
That argument was more about accomodating “different traders with different beliefs”, but here’s an independent argument for market being causal:
When I cause a particular effect/outcome, that means I mediate the influence between the cause of my action and the effect/outcome of my action, the cause of my action is conditionally independent of the effect of my action given me
Futarchy is a similar case: There may be many causes that influence market prices, which in turn determines the decision chosen, & market prices mediate the influence between the cause of market prices (e.g. different traders’ beliefs) and the decision chosen. Any information can only influence what decision will be chosen through influencing the market prices. This seems like what it means for market to be causal (In a bayesnet, the decision chosen will literally only have market prices as the parent, assuming we commit to using futarchy to choose decisions).