I think you've conflated different sorts of "do" here. Your argument about correlation vs causation is strong when the people making the predictions have the power to do or not do. It's weaker when the bettor has little to no control over the person(s) who might do. It holds virtually no water if the "do" in question is itself a statistic covering millions of people (e.g. a market conditional on the outcome of an election).
This post does not address the criticism in the mentioned article. Musk may be bad for Tesla, but firing him may indicate something even worse happened that hurts the company even more. For example, his archenemy manages to get him fired and is now out to destroy Tesla. Suppose: firing Musk is observable, but his archenemy prevailing is not (so, one cannot bet on the joint event: Musk gets fired and the archenemy prevails). The conditional prediction market suggests that keeping Musk is good for Tesla, not because Musk himself is good for the company (he is, in fact, bad) but because Musk keeping his job indicates that his archenemy failed to prevail in his attempt to hurt both Musk and Tesla.
I don't think the proposed solutions to the selection bias work (at least, not always). Say you set a prediction market at time T to evaluate the effect of keeping or replacing Must at time T+1 on Tesla’s stock price at time T+2. However, Musk will be automatically fired if his archenemy prevails at time T+1 (leading to the stock price collapsing at T+2). At time T the decision makers (say Tesla’s board) may have no idea whether Musk’s archenemy enemy ends up prevailing or not. One can interpret this situation as a hostile takeover where the archenemy is the corporate raider.
If Musk might be fired by some other process than the decision, then the decision is not actually deciding if he is fired. But they could have a market on whether to approve his firing, as that they have control over.
This assumes they retain control and can reverse the archenemy's decision. This is not always the case. Think of a corporate raider taking over the firm and replacing not only Musk but also the board. So, to summarize, your argument is premised on those using prediction markets on T to help decide on T+1, still being in charge on T+1. This may or may not be the case.
Sometimes decision markets show a different causal relationship than the one you'd like to know.
For example, consider a decision market with 2 stocks that track Tesla stock in 6 months, conditional on Elon being or not being CEO. A futarchist suggests that the board should decide whether to replace the CEO based on which price is higher.
But the board's CEO decision isn't the only causal factor affecting both Tesla stock and the CEOship. Suppose the market knows that there's a risk of a war with China that will cause Tesla stock to tank and Elon to leave to go build drones. That would reduce the expected price of Tesla stock conditional on him not being CEO and increase the price conditional on him being CEO, but that price differential doesn't reflect the cause that the board is in control of: firing him.
Individual market participants might be able to articulate which causes they're considering when pricing bets, but all the possible causes get combined together to drive the market price.
You are talking about the decision selection bias, which I discuss in the post. This post is mainly about which P(x if A) do markets use to estimate prices.
I think you've conflated different sorts of "do" here. Your argument about correlation vs causation is strong when the people making the predictions have the power to do or not do. It's weaker when the bettor has little to no control over the person(s) who might do. It holds virtually no water if the "do" in question is itself a statistic covering millions of people (e.g. a market conditional on the outcome of an election).
This post does not address the criticism in the mentioned article. Musk may be bad for Tesla, but firing him may indicate something even worse happened that hurts the company even more. For example, his archenemy manages to get him fired and is now out to destroy Tesla. Suppose: firing Musk is observable, but his archenemy prevailing is not (so, one cannot bet on the joint event: Musk gets fired and the archenemy prevails). The conditional prediction market suggests that keeping Musk is good for Tesla, not because Musk himself is good for the company (he is, in fact, bad) but because Musk keeping his job indicates that his archenemy failed to prevail in his attempt to hurt both Musk and Tesla.
That is the decision selection bias, which I discuss in the post.
I don't think the proposed solutions to the selection bias work (at least, not always). Say you set a prediction market at time T to evaluate the effect of keeping or replacing Must at time T+1 on Tesla’s stock price at time T+2. However, Musk will be automatically fired if his archenemy prevails at time T+1 (leading to the stock price collapsing at T+2). At time T the decision makers (say Tesla’s board) may have no idea whether Musk’s archenemy enemy ends up prevailing or not. One can interpret this situation as a hostile takeover where the archenemy is the corporate raider.
If Musk might be fired by some other process than the decision, then the decision is not actually deciding if he is fired. But they could have a market on whether to approve his firing, as that they have control over.
This assumes they retain control and can reverse the archenemy's decision. This is not always the case. Think of a corporate raider taking over the firm and replacing not only Musk but also the board. So, to summarize, your argument is premised on those using prediction markets on T to help decide on T+1, still being in charge on T+1. This may or may not be the case.
It would have been clearer to talk through the Elon Musk example, and how your resolution looks like exactly for that case.
"And that winning concept seems to be the correct decision theory concept, which decision makers are also well advised to use."
Also, this might or might not be purely a question of phrasing, but I disagree with the interpretation that you can pit decision theories against each other in a market, and the winner will be the "correct" decision theory. See here for more: https://casparoesterheld.com/2022/07/17/the-lack-of-performance-metrics-for-cdt-versus-edt-versus/
Sometimes decision markets show a different causal relationship than the one you'd like to know.
For example, consider a decision market with 2 stocks that track Tesla stock in 6 months, conditional on Elon being or not being CEO. A futarchist suggests that the board should decide whether to replace the CEO based on which price is higher.
But the board's CEO decision isn't the only causal factor affecting both Tesla stock and the CEOship. Suppose the market knows that there's a risk of a war with China that will cause Tesla stock to tank and Elon to leave to go build drones. That would reduce the expected price of Tesla stock conditional on him not being CEO and increase the price conditional on him being CEO, but that price differential doesn't reflect the cause that the board is in control of: firing him.
Individual market participants might be able to articulate which causes they're considering when pricing bets, but all the possible causes get combined together to drive the market price.
You are talking about the decision selection bias, which I discuss in the post. This post is mainly about which P(x if A) do markets use to estimate prices.
You make a compelling case. Great post.