Quick note about a thing I didn't properly realize until recently. I don't know how important it is in practice.

tl;dr: Conditional prediction markets tell you "in worlds where thing happens, does other-thing happen?" They don't tell you "if I make thing happen, will other-thing happen?"

Suppose you have a conditional prediction market like: "if Biden passes the DRESS-WELL act, will at least 100,000 Americans buy a pair of Crocs in 2025?" Let's say it's at 10%, and assume it's well calibrated (ignoring problems of liquidity and time value of money and so on).

Let's even say we have a pair of them: "if Biden doesn't pass the DRESS-WELL act, will at least 100,000 Americans buy a pair of Crocs in 2025?" This is at 5%.

This means that worlds where Biden passes the DRESS-WELL act have a 5pp higher probability of the many-Crocs event than worlds where he doesn't. (That's 5 percentage points, which in this case is a 100% higher probability. I wish we had a symbol for percentage points.)

It does not mean that Biden passing the DRESS-WELL act will increase the probability of the many-Crocs event by 5pp.

I think that the usual notation is: prediction markets tell us 

but they don't tell us

One possibility is that "Biden passing the DRESS-WELL act" might be correlated with the event, but not causally upstream of it. Maybe the act has no impact at all; but he'll only pass it if we get early signs that Crocs sales are booming. That suggests a causal model

with

(I don't know if I'm using causal diagrams right. Also, those two "early-sales"es are meant to be the same thing but I don't know how to draw that.)

But here's the thing that triggered me to write this post. We can still get the same problem if the intervention is upstream of the event. Perhaps Biden will pass the DRESS-WELL act if he thinks it will have a large effect, and not otherwise. Let's say the act has a 50% chance of increasing the probability by 3pp and a 50% chance of increasing it by 5pp. Biden can commission a study to find out which it is, and he'll only pass the act if it's 5pp. Then we have

I expect that sometimes you want to know the thing that prediction markets tell you, and sometimes you want to know the other thing. Good to know what they're telling you, whether or not it's what you want to know.

Some other more-or-less fictional examples:

  • If Disney sues Apple for copyright infringement, will they win? A high probability might mean that Disney has a strong case, or it might mean that Disney will only sue if they decide they have a strong case.
  • If the Federal Reserve raises interest rates, will inflation stay below 4%? A high probability might mean that raising interest rates reliably decreases inflation; or it might mean that the Fed won't raise them except in the unusual case that they'll decrease inflation.
  • If I go on a first date with this person, will I go on a second? A high probability might mean we're likely to be compatible; or it might mean she's very selective about who she goes on first dates with.

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10 comments, sorted by Click to highlight new comments since: Today at 12:47 PM

Yeah this came up in a number of times during covid forecasting in 2020. Eg, you might expect the correlational effect of having a lockdown during times of expected high mortality load to outweigh any causal advantages on mortality of lockdowns. 

Do I understand this correctly as: "Countries with more deaths are more likely to do lockdowns, which may result in: 'the countries that did lockdowns actually had more covid deaths than the countries that did not'."?

Yep. 

Maybe a pithier title would be "Conditional prediction markets are evidential, not causal"?

Thanks! Yeah, I think that's making the same basic point with a different focus.

And that makes me more confident in changing the title, so doing that now. (Original title: "Conditional prediction markets are, specifically, conditioned on the thing happening".)

This ends up being pretty important in practise for decision markets ("if I choose to do X, will Y?"), where by default you might e.g. only make a decision if it's a good idea (as evaluated by the market), and therefore all traders will condition on the market having a high probability which is obviously quite distortionary. 

Hm, do you want to go into more depth? Intuitively I agree this is obviously distortionary, but I'm finding it awkward to think through the details of the distortion.

One thing that comes to mind is "if the market is at 10% but you think 5% is "correct" according to what seems like the spirit of the question, you're going to expect that the market just doesn't get resolved, so why bother betting it down". But I feel like there's probably more than that. (E: oh, the dynomight essay linked above mentions this one as well.)

Yeah this came up in a number of times during covid forecasting in 2020. Eg, you might expect the correalational effect of having a lockdown during times of expected high mortality load to outweigh any causal advantages on mortality of lockdowns.