In this case it would be best to use the language of counterfactuals (aka potential outcomes) instead of conditional expectations. In practice, the market would estimate E[Ya] and E[Yb] for the two random functions Ya and Yb, and you would choose the option with the highest estimated expected value. There is no need to put conditional probability into the mix at all, and it’s probably best not to, as there is no obvious probability to assign to the “events” a and b.
Phrasing it in terms of potential outcomes could definitely help the understanding of people who use that approach to talk about causal questions (which is a lot of people!). I’m not sure it helps anyone else, though. Under the standard account, the price of a prediction market is a probability estimate, modulo the assumption that utility = money (which is independent of the present concerns). So we’d need to offer an argument that conditional probability = marginal probability of potential outcomes.
Potential outcomes are IMO in the same boat as decision theories—their interpretation depends on a vague “I know it when I see it” type of notion. However we deal with that, I expect the story ends up sounding quite similar to my original comment—the critical step is that the choice does not depend on anything but the closing price.
a and b definitely are events, though! We could create a separate market on how the decision market resolves, and it will resolve unambiguously.
Potential outcomes are very clearly and rigorously defined as collections of separate random variables, there is no “I know it when I see it” involved. In this case you choose between two options, and there is no conditional probability involved unless you actually need it for estimation purposes.
Let’s put it a different way. You have the option of flipping two coins, either a blue coin or a red coin. You estimate the expected probability of heads as P(blue)=0.6 and P(red)=0.5. You base your choice of which coin to toss on which probability is the largest. There is actually no need to use scary-sounding terms like counterfactuals or potential outcomes at all, you’re just choosing between random outcomes.
We could create a separate market on how the decision market resolves, and it will resolve unambiguously.
That sounds like an unnecessarily convoluted solution to a question we do not need to solve!
However we deal with that, I expect the story ends up sounding quite similar to my original comment—the critical step is that the choice does not depend on anything but the closing price.
Yes, I agree. And that’s why I believe we shouldn’t use conditional probabilities at all, as it makes it confusion possible.
The definition of potential outcomes you refer to does not allow us to answer the question of whether they are estimated by the market in question.
The essence of all the decision theoretic paradoxes is that everyone agrees that we need some function options → distributions over consequences to make decisions, and no one knows how exactly to explain what that function is.
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))
In this case it would be best to use the language of counterfactuals (aka potential outcomes) instead of conditional expectations. In practice, the market would estimate E[Ya] and E[Yb] for the two random functions Ya and Yb, and you would choose the option with the highest estimated expected value. There is no need to put conditional probability into the mix at all, and it’s probably best not to, as there is no obvious probability to assign to the “events” a and b.
You can bet not on probabilities but on utility, see e.g. the futarchy specification by Hanson (Lizka’s summary and notes).
Phrasing it in terms of potential outcomes could definitely help the understanding of people who use that approach to talk about causal questions (which is a lot of people!). I’m not sure it helps anyone else, though. Under the standard account, the price of a prediction market is a probability estimate, modulo the assumption that utility = money (which is independent of the present concerns). So we’d need to offer an argument that conditional probability = marginal probability of potential outcomes.
Potential outcomes are IMO in the same boat as decision theories—their interpretation depends on a vague “I know it when I see it” type of notion. However we deal with that, I expect the story ends up sounding quite similar to my original comment—the critical step is that the choice does not depend on anything but the closing price.
a and b definitely are events, though! We could create a separate market on how the decision market resolves, and it will resolve unambiguously.
Potential outcomes are very clearly and rigorously defined as collections of separate random variables, there is no “I know it when I see it” involved. In this case you choose between two options, and there is no conditional probability involved unless you actually need it for estimation purposes.
Let’s put it a different way. You have the option of flipping two coins, either a blue coin or a red coin. You estimate the expected probability of heads as P(blue)=0.6 and P(red)=0.5. You base your choice of which coin to toss on which probability is the largest. There is actually no need to use scary-sounding terms like counterfactuals or potential outcomes at all, you’re just choosing between random outcomes.
That sounds like an unnecessarily convoluted solution to a question we do not need to solve!
Yes, I agree. And that’s why I believe we shouldn’t use conditional probabilities at all, as it makes it confusion possible.
The definition of potential outcomes you refer to does not allow us to answer the question of whether they are estimated by the market in question.
The essence of all the decision theoretic paradoxes is that everyone agrees that we need some function options → distributions over consequences to make decisions, and no one knows how exactly to explain what that function is.
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