Tuesday, April 25, 2017

Let's handwave some proofs for price compression

We need to prove some converging properties for price compression. In particular, when we have a Huffman tree, organized by significance, and it is balanced, then each value decodes to the same path length. But they are unbalanced.  That means the collection of values are bunched up for a lot of short paths.  These are values which are insignificant relative to a few large, infrequent events.

How can we balance a skewed Huffman tree? If the values incoming have constant returns to scale, then  successive input value rounding will lead to a balanced tree. In other words, consider what happens in S&L tech.  Suddenly large loans be inserted into the loan tree.  We need to restore balance by rounding (imprecision) so the tree again matches the deposit tree, and interest charges are a compute ratio across queue sizes, paid according to the percent of rounding, accumulated up the tree. We are measuring the imprecision in supply compared to the imprecision of demand, as reflected in the incoming deposit and loan events.  Rounding toward tree balance is like diagonalizing a matrix.

Does it work?

Well, in the extreme we now there exists a balance, one digit price.  But, any incremental rounding tends to shorten the long paths, and leave the short paths the same.  If scaling is constant, proof enough for me.  Do deposits and loan events scale? I think, unproven, that values scale sufficiently when queues are balanced.  The bit error, and currency risk in the model is the imbalance between loan payouts and deposit earnings.  It is the deviation of -iLog(i) as they differ from one. It is a denomination error because the real accounts never actually scale in queue size perfectly.

We want balanced trees becaue they have returns to scale. Rounding keeps all queues thew same size. In the real deployment, the rounding bounds on savings and loans are the variance bounds preannounced. Rounding is easy for the traders to understand.  Loans in 8 bins, deposits in 32 bins, bins sizes determined by market conditions.

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