Multivariate calculus for experts

As previously announced, Eulora now logs trades of players. So problem solved, right, we'll soon see prices of all items and that's that.

Wrong.

For one thing : supposing the players are P₁, P₂, ... P_i... P_n whereas the items are I₁, I₂, ... I_i... I_n, the logs look in practice like a whole lot of

1. P_q gave P_u : C_v of item I_s and C_n of item I_h and C_m of item I_j,
   whereas P_u gave P_q the sum of 17`230`715 ECu.
2. P_y gave P_i : C_z of item I_s and C_x of item I_h,
   whereas P_i gave P_y : C_b of item I_g and C_m of item I_j.

This doesn't readily reduce to anything, so Joe traded five matches and three ducks to Moe for fifty dollars and half a bag of tobacco. Is a duck worth two matches or isn't it ?

Well... if you aggregate enough of these trades to provide sufficient constraint you MIGHT be able to resolve the matter, in the sense of allocating exact values. If you can't allocate exact values, you should however always be able to provide the boundries for all items involved, so that Value(I_x) is within an [a,b] set so that b <= ∞ (conveniently enough a is not only positive, but there exist known lower bounds for the a of any given P_x, known as "base values").

So the first level problem here is, "provide an algorithm that eats trade logs and spits out [a,b] boundries for each encountered P". And, if you know statistics and are willing to get fancy, provide the confidence function for that interval, also, computed as that function which maximizes overall confidence for the collected set. And prove you're not in a local maximum. And indicate the confidence gain resulting from the deleting of each log line.ⁱ

By the way, the foregoing wasn't the "for experts" part of our little excursion in highest maths. Oh, no. The expert part is only now beginning : items I_n have qualities associated, Q_n, which obey the trivial liniarity that a for I_x where Q_Ix = 1 is twice the a for I_x where Q'_Ix = 0.5, and half the a for I_x where Q''_Ix = 2.

So now : considering for every item traded we capture a quantity C that was traded and a quality Q of the item traded, fit a price function of qualityⁱⁱ so as to maximize the confidence of the intervals discussed afore. You may, as a first approximation, split the Is into a number of classes and fit by class, but you don't have to.

That's it. Scare me.

PS. For as long as this matter stands open, there's nothing more wrong than LessWrong, nor may any Bayesian approach to science aspire to any other title than "Pure Voodoo".

1. This would be roughly equivalent to the "outlier" concept in undergrad statistics, except we're not working with simple datasets here, nor is the whole "and then exclude the two furthest out" rule of thumb much value in the way of science. [↩]
2. A function which takes quality and item as parameter and spits out the actual market value, as a ratio to the linearly computed base value for that quality (the item known as qabv in Euloran parlance). So your function takes two parameters and spits out a scalar. [↩]