Here’s the context I’m thinking about. Say you have two options Ya and Yb. They have different true expected values E(Ya) and E(Yb). The market estimates their expectations as ^E(Ya) and ^E(Yb). And you (or the decider) choose the option with highest estimated expectation. (I was unclear about estimation vs. true values in my previous comment.)
Does this have something to do with your remarks here?
Also, there’s always a way to implement “the market decides”. Instead of asking P(Emissions|treaty), ask P(Emissions|market advises treaty), and make the market advice = the closing prices. This obviously won’t be very helpful if no-one is likely to listen to the market, but again the point is to think about markets that people are likely to listen to.
I believe we agree on the following: we evaluate the desirability of each available option by appealing to some map F:X→Δ(Y) from options X to distributions over consequences of interest Y.
We also both suggest that maybe F should be equal to the map x↦Q(x) where Q(x) is the closing price of the decision market conditional on x.
You say the price map is equal to the map x↦E(Yx), I say it is equal to x↦E(Y|x) where the expectation is with respect to some predictive subjective probability.
The reason why I make this claim is due to work like Chen 2009 that finds, under certain conditions, that prediction market prices reflect predicting subjective probabilities, and so I identify the prices with predictive subjective probabilities. I don’t think any similar work exists for potential outcomes.
The main question is: is the price map Q really the right function F? This is a famously controversial question, and causal decision theorists say: you shouldn’t always use subjective conditional probabilities to decide what to do (see Newcomb etc.) On the basis of results like Chen’s, I surmise that causal decision theorists at least don’t necessarily agree that the closing prices of the decision market defines the right kind of function, because it is a subjective conditional probability (but the devil might be in the details).
Now, let’s try to solve the problem with potential outcomes. Potential outcomes have two faces. On the one hand, Ya is a random variable equal to Y in the event X=a (this is called consistency). But there are many such variables—notably, Y itself. The other face of potential outcomes is that Ya should be interpreted as representing a counterfactual variable in the event X≠a. What potential outcomes don’t come with is a precise theory of counterfactual variable. This is the reason for my “I know it when I see it” comment.
Here’s how you could argue that E(Y|x)=E(Yx): first, suppose it’s a decision market with randomisation, so the choice X is jointly determined by the price and some physical random signal R. Assume YX⊥R - this is our “theory of counterfactual variables”. By determinism, we also have YX⊥X|R,Q where Q is the closing price of the pair of markets. By contraction YX⊥X|Q, and the result follows from consistency (apologies if this is overly brief). Then we also say F is the function x↦Yx and we conclude that indeed F(x)=E(Yx)=E(Y|x)=Q(x).
This is nicer than I expected, but I figure you could go through basically the same reasoning, but with F directly. AssumeF⊥R and P(F(a)=E(Y|a)|a)=1 (and similarly for b). Then by similar reasoning we get P(F(a)=E(Y|a)|Q)=1 (Noting that, by assumption, Q=E(Y|a))
Sorry, but I don’t understand what you mean.
Here’s the context I’m thinking about. Say you have two options Ya and Yb. They have different true expected values E(Ya) and E(Yb). The market estimates their expectations as ^E(Ya) and ^E(Yb). And you (or the decider) choose the option with highest estimated expectation. (I was unclear about estimation vs. true values in my previous comment.)
Does this have something to do with your remarks here?
I believe we agree on the following: we evaluate the desirability of each available option by appealing to some map F:X→Δ(Y) from options X to distributions over consequences of interest Y.
We also both suggest that maybe F should be equal to the map x↦Q(x) where Q(x) is the closing price of the decision market conditional on x.
You say the price map is equal to the map x↦E(Yx), I say it is equal to x↦E(Y|x) where the expectation is with respect to some predictive subjective probability.
The reason why I make this claim is due to work like Chen 2009 that finds, under certain conditions, that prediction market prices reflect predicting subjective probabilities, and so I identify the prices with predictive subjective probabilities. I don’t think any similar work exists for potential outcomes.
The main question is: is the price map Q really the right function F? This is a famously controversial question, and causal decision theorists say: you shouldn’t always use subjective conditional probabilities to decide what to do (see Newcomb etc.) On the basis of results like Chen’s, I surmise that causal decision theorists at least don’t necessarily agree that the closing prices of the decision market defines the right kind of function, because it is a subjective conditional probability (but the devil might be in the details).
Now, let’s try to solve the problem with potential outcomes. Potential outcomes have two faces. On the one hand, Ya is a random variable equal to Y in the event X=a (this is called consistency). But there are many such variables—notably, Y itself. The other face of potential outcomes is that Ya should be interpreted as representing a counterfactual variable in the event X≠a. What potential outcomes don’t come with is a precise theory of counterfactual variable. This is the reason for my “I know it when I see it” comment.
Here’s how you could argue that E(Y|x)=E(Yx): first, suppose it’s a decision market with randomisation, so the choice X is jointly determined by the price and some physical random signal R. Assume YX⊥R - this is our “theory of counterfactual variables”. By determinism, we also have YX⊥X|R,Q where Q is the closing price of the pair of markets. By contraction YX⊥X|Q, and the result follows from consistency (apologies if this is overly brief). Then we also say F is the function x↦Yx and we conclude that indeed F(x)=E(Yx)=E(Y|x)=Q(x).
This is nicer than I expected, but I figure you could go through basically the same reasoning, but with F directly. AssumeF⊥R and P(F(a)=E(Y|a)|a)=1 (and similarly for b). Then by similar reasoning we get P(F(a)=E(Y|a)|Q)=1 (Noting that, by assumption, Q=E(Y|a))
I’ll get back to you