scaled (continuous) events with real (scalar) number outcome

[martinBrown]

Binary event LMSR is pretty easy, but continuous events are another matter. I've been trying to understand the scaled contracts spreadsheet. Just the basic scaled one for now (multi-variate scaled next time).

I annotated it with some notes and questions:



Main question is, how is the event resolved, and payouts done for a scaled contract?

[psztorc]

Hmm.

That particular tab may require some attention. Thanks for this feedback.

Notice the use of 'min' and 'scale'. So people vote on a value, it is transformed to a range(0,1) variable, and run through SVD. Scaled claims are not binned, instead the weighted median is used.

Then, post-SVD, they are rescaled back up by x*scale + min.

[psztorc]

I should point out that, the way I imagined it, each row was a different trader. It shouldn't really matter, though.

[psztorc]

I gave the Excel sheet another afternoon.

Please open it up and give feedback! Your screenshot with reactions was very helpful to me.
 
To directly answer your original question: The payouts are calculated the same way (SVD vote). The only difference is that, instead of being one choice of { 0, .5, 1 }, the value is anything within range(0,1). That difference means that instead of a nearest-selection, the median is used to pull out the consensus value, but the incentives should be the same.

Check the sheet to see how this single input (for example D35 of the DJIA example) ends up supplying the multiple pieces of information required to close out the market correctly.

[martinBrown]

Thanks a lot, the new spreadsheet is much more clear.

I was going to ask for a citation on using those kind of scaled outcomes with LMSR. But I think I found it here:

A user who wants to express his opinion on the expected value of GDP change, how- ever, might prefer “linear” assets such as “Pays $xˆ,” and “Pays $(1 − xˆ ),” where we have defined a rescaled variable:

x_hat = max(0, min( x - x_max / x_max - x_min ))

which is zero up to x =x, is one above x =x ̄, and moves linearly with x in between.

That's equivalent to the scaled formula in the spreadsheet, correct?

But if using a scaled outcome is this simple, then what's with all the fuss over continuous variables in the literature?

Cost Function Market Makers for Measurable Spaces

This allows us to overcome the impossibility results of Gao and Chen [2010] and design the first automated market maker for betting on the realization of a continuous random variable taking values in [0, 1] that has bounded loss without resorting to discretization.

That paper proposes a convex optimization problem for the cost function (much more complicated than the simple LMSR formula). And with similarly dense math: Betting on the real line.

What's the advantage of these sophisticated methods over the simple scaled LMSR? It claims to be the "first" automated market for a continuous variable... isn't that what a scaled outcome LMSR is? Maybe it's over my head but I'd appreciate if anyone can help me understand the difference here.

[psztorc]

Thanks a lot, the new spreadsheet is much more clear.

Good! I had a good feeling about it when I finished it.

I was going to ask for a citation on using those kind of scaled outcomes with LMSR. But I think I found it here:

A user who wants to express his opinion on the expected value of GDP change, how- ever, might prefer “linear” assets such as “Pays $xˆ,” and “Pays $(1 − xˆ ),” where we have defined a rescaled variable:

x_hat = max(0, min( x - x_max / x_max - x_min ))

which is zero up to x =x, is one above x =x ̄, and moves linearly with x in between.

That's equivalent to the scaled formula in the spreadsheet, correct?

Those equations do make it seem more complicated than it is, but yes. There is a simple range, and within that range (the max / max_min part), prices vary and are paid off. If you crash into max or min you're stuck there.

But if using a scaled outcome is this simple, then what's with all the fuss over continuous variables in the literature?

Cost Function Market Makers for Measurable Spaces

This allows us to overcome the impossibility results of Gao and Chen [2010] and design the first automated market maker for betting on the realization of a continuous random variable taking values in [0, 1] that has bounded loss without resorting to discretization.

That paper proposes a convex optimization problem for the cost function (much more complicated than the simple LMSR formula). And with similarly dense math: Betting on the real line.

What's the advantage of these sophisticated methods over the simple scaled LMSR? It claims to be the "first" automated market for a continuous variable... isn't that what a scaled outcome LMSR is? Maybe it's over my head but I'd appreciate if anyone can help me understand the difference here.

The devil is in the details, and in wacky-academic-math-world that means the devil is in the definitions:

For LMSR, we define the same interval betting language as in call markets. A trader can create a security by specifying an interval (l, u) to bet on. If the realized value of X falls into the interval, the security pays off $1 per share.

Read carefully! This is NOT "something paying x as long as x is within (l, u)", as Hanson discussed above. It is something else. I won't spoil the surprise unless you ask again. If you think about what this security is you'll see why [1] the potential loss is unbounded, and [2] the realized/practical loss, if anyone was crazy enough to actually do this with a LMSR, would be unbounded (ie the guy running this operation would go bankrupt cashing-out the first PM) as long as there was at least one profit-maximizing trader (assuming that l-u is not zero, ie that the interval (l,u) has some length).