Consider a conditional prediction market, e.g. "if my cool policy is implemented, then widget production will increase by at least 15%". To my understanding, markets like this are intended as a tool for finding and the market just gets unwound or undone or refunded if doesn't occur.
I can work through the math and see that refunding the market indeed makes the price reflect , but this exacerbates one of the biggest issues with prediction markets: no one wants to lock up of capital to extract of profit in a year, so no one will lock up of capital to extract of profit in a year and only if some extra event happens.
My question is: are there any interesting or viable alternative ways to run a counterfactual or conditional prediction market? Off the top of my head, I could imagine using markets for and to derive , which would still pay out something if didn't occur.
P(A|B) is defined as P(A & B) / P(B), and both P(A & B) and P(B) are straightforward things to bet on in a prediction market.
The problem is that you get some estimates P*(A&B) and P*(B), and P*(A&B)/P*(B) is not necessarily a good estimate for P(A&B)/P(B) even when each of the component estimates were good. It gets much worse when the estimates aren't very good.
It gets worse still if what you really want is something more structured than a simple conditional probability, such as a credence for a causal relation. I suspect that there are many complications here that may be beyond the scope of any plausible prediction market structure.