All of SG's Comments + Replies

Kind of. The point is to estimate P(Y). However, the value is not supposed to come from automatically estimating P(Y), but rather from providing an objective way of scoring manual predictions about P(Y) when we don't have an objective definition of Y. I agree that if the goal was to automatically predict P(Y) from P(Xi), structures of conditional prediction markets seems better. See also the "Latent variable markets vs combinatorial markets" section of my post; there are some cases where e.g. P(Y|Xi) is more interesting. So basically, yes, the value of LVPMs comes from estimating P(Y), but the way we estimate P(Y) is by having people make predictions on P(Y). All the elaborate P(Xi|Y) has the primary purpose of providing a way to objectively score predictions on Y, and a secondary purpose of providing information/common knowledge about the domain of applicability of the Y concept. One thing I guess I should note is that the market maker only has to pay the first person. To clarify how this works, suppose the market maker creates a market with initial state θ0, and a first trader then updates this state to θ1, and a second trader then updates the state to θ2. Assuming for simplicity that the initial distribution P(→X|θ0) is uniform over →X, when the market resolves to a specific →x, we can then read the payoffs here: * The second trader gets a payout of logP(→x|θ2)−logP(→x|θ1). * The first traders gets a payout of logP(→x|θ1)−logP(→x|θ0). * The market maker paid −logP(→x|θ0) for making the market, which we can think of them as getting a payout of logP(→x|θ0) to make the signs of the expressions all aligned. If we add all of these payouts up, we get (logP(→x|θ2)−logP(→x|θ1))+(logP(→x|θ1)−logP(→x|θ0))+(logP(→x|θ0))=logP(→x|θ2), which in general is a nonpositive number[1], and so the market will never give a greater payout than what is put in. Essentially what happens is that each trader pays off the next trader, guaranteeing that it all adds up. (This won't req

I unfortunately have to go to sleep now but I will respond to this tomorrow.

Thanks for the reply! When you say "a simple set of conditional and unconditional markets", what do you have in mind? Edit: I mean for a specific case such as the Russia/Ukraine war, what would be some conditional and unconditional markets that could be informative? (I understand how conditional and unconditional markets work in principle, I'm more interested in how it would practically compete with LVPMs.) So I am not 100% sure what you are referring to here. When you are talking about "model-provided updates as a result of people betting on indicator variables", I can think of two things you might be referring to: 1. If someone bets directly on a market Xi, i.e. changing P(Xi) to a new value, then (using some math) we can treat this as evidence to update P(Y), using the fact that we know P(Xi|Y). 2. If someone bets on a conditional market P(Xi|Y), then the parameters of this conditional market may change, which in a sense constitutes an update as a result of people betting on indicator variables. 3. (Possibly something else?) Your language makes me think you are referring to 1, and I agree that this will plausibly be of limited value. I'm most excited about 2 because it allows people to better aggregate information about the plausible outcomes, while taking correlations into account. You can treat the addition or removal of an indicator as a trade performed by the market owner. The payout for a person for a trade t1 does not depend on any trades t2 that happen after the person's t1. (This follows from the fact that we describe the payoffs as a function of the probabilities just before and just after t1.) So the owner of the market can add and remove however many indicator variables they want without any trouble. I think the logP(→X|θ)−logP(→X|ϕ) part of my scoring counts as an AMM liquidity provision? I might be misunderstanding what AMM liquidity provision means. To scale up the liquidity, one can multiply this expression by the logP(→X|θ)−logP(→X|ϕ) e