If I understand your question correctly, then yes, interpretations that support correlation intrinsically get favored.
Strictly speaking, there is a sense in which we can entirely ignore Y when understanding what the latent variable markets are doing. A latent variable market over a set of variables X0,X1,…,Xn is really trying to predict their join distribution P(X0,X1,…,Xn). If these variables are all “getting at the same thing” from different angles, then they will likely “fall within a subspace”, and it is this subspace that the LVPMs are characterizing.
For instance consider the following plot of a probability distribution:
There is a ridge where x1 and x2 are similar in magnitude, and most of the probability mass is distributed along that ridge. It is unlikely for x1 to be 1 while x2 is −1, or vice versa.
The LVPMs characterize this ridge of high probability, so an LVPM on a probability distribution like the above would likely range from “x1 and x2 are both low” to “x1 and x2 are both high”.
If you are familiar with principal component analysis, it may be helpful to think of LVPMs as doing a similar thing.
So if I’m understanding correctly, you’re saying a LVPM market Y could be displayed without a title, and traders would still converge toward finding the joint probability distribution function? So instead of making a LVPM titled “Will Ukraine do well in defense against Russia?”, I could make a LVPM that says “Hey, here are a bunch of existing objective questions that may or may not be correlated, have at it”, and it would be just as functional?
If so that’s pretty neat. But what stops Trader1 from interpreting Y as “Will Ukraine do well in defense against Russia?” while Trader2 treats it as “Will Russia do well in the invasion of Ukraine?” IIUC, these two traders would make nearly opposite trades on the conditional probabilities between each Xi and Y.
So if I’m understanding correctly, you’re saying a LVPM market Y could be displayed without a title, and traders would still converge toward finding the joint probability distribution function? So instead of making a LVPM titled “Will Ukraine do well in defense against Russia?”, I could make a LVPM that says “Hey, here are a bunch of existing objective questions that may or may not be correlated, have at it”, and it would be just as functional?
Yep.
If so that’s pretty neat. But what stops Trader1 from interpreting Y as “Will Ukraine do well in defense against Russia?” while Trader2 treats it as “Will Russia do well in the invasion of Ukraine?” IIUC, these two traders would make nearly opposite trades on the conditional probabilities between each Xi and Y.
So this is in fact true: LVPMs will not be totally unique, but instead have symmetries where some options do equally well. The exact type of symmetry depends on the kind of market, but they mainly look like what you describe here: swapping around the results such that yes becomes no and no becomes yes.
The way I usually handle that problem is by picking one of the indicators as a “reference indicator” and fixing it to have a specific direction of relationship with the outcome. If the reference indicator cannot be flipped, then all the other indicators have to have a fixed direction too. (One could also do something more nuanced, e.g. pick many reference indicators.)
I don’t know whether this is necessary for humans. I mainly do this with computers because computers have no inherent idea which direction of the LV is most natural for humans.
I don’t know if it would be a problem for people. It’s not something that happens iteratively; while you can flip all the parameters at once and still get a good score, you cannot continuously change from one optimum to an opposite optimum without having a worse score in the middle. So if it is an issue, it will probably mainly be an issue with trolls.
So this is in fact true: LVPMs will not be totally unique, but instead have symmetries where some options do equally well. The exact type of symmetry depends on the kind of market, but they mainly look like what you describe here: swapping around the results such that yes becomes no and no becomes yes.
I’m not a savvy trader by any means, but this sets off warning bells for me that more savvy traders will find clever exploits in LVPMs. You can’t force anyone to abide by the spirit of the market’s question, they will seek the profit incentive wherever it lies.
(I feel somewhat satisfied that the “question reversal” symmetry is ruled out by the market maker restricting the possible PDF space (in half along one specified dimension, IIUC). I’m curious how that would be practically implemented in payouts.)
I worry about these other possible symmetries or rotations, which I can’t yet wrap my head around. I would love an illustrating example showing how they work and why we should or shouldn’t worry about them.
I’m not a savvy trader by any means, but this sets off warning bells for me that more savvy traders will find clever exploits in LVPMs. You can’t force anyone to abide by the spirit of the market’s question, they will seek the profit incentive wherever it lies.
Hmm I’m not sure what exploits you have in mind. Do you mean something like, they flip the meaning of the market around, and then people who later look at the market get confused and bet in the opposite direction of what they intended? And then the flipper can make money by undoing their bets?
Or do you mean something else?
I feel somewhat satisfied that the “question reversal” symmetry is ruled out by the market maker restricting the possible PDF space (in half along one specified dimension, IIUC).
Yep, in half along one specified dimension.
I’m curious how that would be practically implemented in payouts.
There’s a few different ways.
If someone bets to set the market state to a θ that flips the fixed dimension, the market implementation could just automatically replace their bet with −θ so it doesn’t flip the dimension. The payouts are symmetric under flips, so they will get paid the same regardless of whether they bet θ or −θ.
Alternatively you could just forbid predictions in the UI that flip the fixed dimension.
I worry about these other possible symmetries or rotations, which I can’t yet wrap my head around. I would love an illustrating example showing how they work and why we should or shouldn’t worry about them.
Flipping is the only symmetry that exists for unidimensional cases.
If you have multiple dimensions, you have a whole continuum of symmetries, because you can continuously rotate the dimensions into each other.
Thanks for being patient with my questions. I’m definitely not solid enough on these concepts yet to point out an exploit or misalignment. It would be super helpful if you fill out your LVPM playground page in the near future with a functional AMM to let people probe at the system.
Flipping is the only symmetry that exists for unidimensional cases.
If you have multiple dimensions, you have a whole continuum of symmetries, because you can continuously rotate the dimensions into each other.
What do you mean by unidimensional cases? So like if the binary LVPM Y is made up of binary markets X1 to XN , I would have called that an N-dimensional case, since the PDF is N-dimensional. How many symmetries and “possible convergences” does this Y have?
No problem, answering questions is the point of this post. 😅
It would be super helpful if you fill out your LVPM playground page in the near future with a functional AMM to let people probe at the system.
The playground page already has this. If you scroll down to the bottom of the page, you will see this info:
To play with the payout and get an intuition for it, you can use the checkboxes below. Your bets have currently cost proportional to 0 Mana, and if the outcome below happens, you get a payout proportional to 0 Mana, for a total profit proportional to 0 Mana.
Japan hyperinflation by 2030?
US hyperinflation by 2030?
Ukraine hyperinflation by 2030?
If you make bets, then the numbers in this section get updated with the costs and the payouts.
But I guess my interface probably isn’t very friendly overall, even if it is there. 😅
What do you mean by unidimensional cases? So like if the binary LVPM Y is made up of binary markets X1 to XN , I would have called that an N-dimensional case, since the PDF is N-dimensional. How many symmetries and “possible convergences” does this Y have?
I am talking about the dimensionality of Y, not the dimensionality of →X.
In the markets I described in the post and implemented for my demo, Y is always one-dimensional. However, it is possible to make more advance implementations where Y can be multidimensional. For instance in the uncertainty about the outcome of the Ukraine war, there is probably not just a Ukraine wins <-> Russia wins axis, but also a war ends <-> war continues axis.
If I understand your question correctly, then yes, interpretations that support correlation intrinsically get favored.
Strictly speaking, there is a sense in which we can entirely ignore Y when understanding what the latent variable markets are doing. A latent variable market over a set of variables X0,X1,…,Xn is really trying to predict their join distribution P(X0,X1,…,Xn). If these variables are all “getting at the same thing” from different angles, then they will likely “fall within a subspace”, and it is this subspace that the LVPMs are characterizing.
For instance consider the following plot of a probability distribution:
There is a ridge where x1 and x2 are similar in magnitude, and most of the probability mass is distributed along that ridge. It is unlikely for x1 to be 1 while x2 is −1, or vice versa.
The LVPMs characterize this ridge of high probability, so an LVPM on a probability distribution like the above would likely range from “x1 and x2 are both low” to “x1 and x2 are both high”.
If you are familiar with principal component analysis, it may be helpful to think of LVPMs as doing a similar thing.
Ok. That’s cool. Thanks.
So if I’m understanding correctly, you’re saying a LVPM market Y could be displayed without a title, and traders would still converge toward finding the joint probability distribution function? So instead of making a LVPM titled “Will Ukraine do well in defense against Russia?”, I could make a LVPM that says “Hey, here are a bunch of existing objective questions that may or may not be correlated, have at it”, and it would be just as functional?
If so that’s pretty neat. But what stops Trader1 from interpreting Y as “Will Ukraine do well in defense against Russia?” while Trader2 treats it as “Will Russia do well in the invasion of Ukraine?” IIUC, these two traders would make nearly opposite trades on the conditional probabilities between each Xi and Y.
Yep.
So this is in fact true: LVPMs will not be totally unique, but instead have symmetries where some options do equally well. The exact type of symmetry depends on the kind of market, but they mainly look like what you describe here: swapping around the results such that yes becomes no and no becomes yes.
The way I usually handle that problem is by picking one of the indicators as a “reference indicator” and fixing it to have a specific direction of relationship with the outcome. If the reference indicator cannot be flipped, then all the other indicators have to have a fixed direction too. (One could also do something more nuanced, e.g. pick many reference indicators.)
I don’t know whether this is necessary for humans. I mainly do this with computers because computers have no inherent idea which direction of the LV is most natural for humans.
I don’t know if it would be a problem for people. It’s not something that happens iteratively; while you can flip all the parameters at once and still get a good score, you cannot continuously change from one optimum to an opposite optimum without having a worse score in the middle. So if it is an issue, it will probably mainly be an issue with trolls.
I’m not a savvy trader by any means, but this sets off warning bells for me that more savvy traders will find clever exploits in LVPMs. You can’t force anyone to abide by the spirit of the market’s question, they will seek the profit incentive wherever it lies.
(I feel somewhat satisfied that the “question reversal” symmetry is ruled out by the market maker restricting the possible PDF space (in half along one specified dimension, IIUC). I’m curious how that would be practically implemented in payouts.)
I worry about these other possible symmetries or rotations, which I can’t yet wrap my head around. I would love an illustrating example showing how they work and why we should or shouldn’t worry about them.
Hmm I’m not sure what exploits you have in mind. Do you mean something like, they flip the meaning of the market around, and then people who later look at the market get confused and bet in the opposite direction of what they intended? And then the flipper can make money by undoing their bets?
Or do you mean something else?
Yep, in half along one specified dimension.
There’s a few different ways.
If someone bets to set the market state to a θ that flips the fixed dimension, the market implementation could just automatically replace their bet with −θ so it doesn’t flip the dimension. The payouts are symmetric under flips, so they will get paid the same regardless of whether they bet θ or −θ.
Alternatively you could just forbid predictions in the UI that flip the fixed dimension.
Flipping is the only symmetry that exists for unidimensional cases.
If you have multiple dimensions, you have a whole continuum of symmetries, because you can continuously rotate the dimensions into each other.
Thanks for being patient with my questions. I’m definitely not solid enough on these concepts yet to point out an exploit or misalignment. It would be super helpful if you fill out your LVPM playground page in the near future with a functional AMM to let people probe at the system.
What do you mean by unidimensional cases? So like if the binary LVPM Y is made up of binary markets X1 to XN , I would have called that an N-dimensional case, since the PDF is N-dimensional. How many symmetries and “possible convergences” does this Y have?
No problem, answering questions is the point of this post. 😅
The playground page already has this. If you scroll down to the bottom of the page, you will see this info:
If you make bets, then the numbers in this section get updated with the costs and the payouts.
But I guess my interface probably isn’t very friendly overall, even if it is there. 😅
I am talking about the dimensionality of Y, not the dimensionality of →X.
In the markets I described in the post and implemented for my demo, Y is always one-dimensional. However, it is possible to make more advance implementations where Y can be multidimensional. For instance in the uncertainty about the outcome of the Ukraine war, there is probably not just a Ukraine wins <-> Russia wins axis, but also a war ends <-> war continues axis.