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Anonymous committed 949a776

revise equations, fixing up ntation and attempting to construct decent estimator.

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 bounded above by the requirement of predicted profitability.  Workers may
 optionally presumed to have tenure, and only leave for a better wage rather
 than having their own contract re-evaluated. We denote the valuation function
-$v$ for the value to $F^k$ of worker $w^j$ for the next timestep $i+1$ by
-\[v(F^k, w^j, i+1)\].
+$v$ for the value to $F^k$ of worker $w^j$ for the next timestep at timestep $i$ by
+\[v(F^k, w^j, i)\].
 
 This notation implies that all firms share a valuation function, based
 presumably on public information. This assumption can be relaxed, but I
 
 I have partially implemented the latter.
 
-The model is that value of a worker to one firm $f(j,i+1)$ is the canonical
+The model in this case is that value of a worker $w^j$ to an arbitrary firm $f^k$ at time $i$ is the canonical
 value they have at their current firm $f(j,i)$ multiplied by a coefficient,
 reflecting the typical change in canonical values experienced by workers
-moving from that previous firm to this one. That is, this valuation function
-imputes a mean valuation coefficient to all pairwise changes in employment.
+moving from that previous firm to this one over a timestep. That is, this valuation function
+imputes a mean change coefficient to all pairwise changes in employment.
 
 Symbolically we write
-\[v(f^k, w^j, i+1) := V_{f(j,i),F^k}p_i^k\]
+\[v(f^k, w^j, i) := V_{f(j,i),F^k}p_i^k\]
 
 where $V_{x,y}$ is a matrix of these pairwise firm coefficients.
 
 The estimator for the matrix entries is simply a decaying mean of datapoints
-so far. I'm still nutting out the precise form for it. Basically, if the
+so far. We maintain a running estimate of $V_{x,y}$ based on earlier transitions. So, define
+
+\[S_ab^j:= \{i:f(j,i-1)=F^a \land f(j, i)=F^b\}\]
+Then we may construct the estimator, $\bar{V}_{x,y}$.
+But it leads to a nasty notation explosion. I might need to (re)define some terms to keep this under control. Will come back to that.
+
+In any case, I'm still nutting out the precise form for it. Basically, if the
 employment at firms over time remains similar, the difference in culture
 matrices and hence value of given workers, between them might be sufficiently
 well approximated by this to improve bidding performance. I'm thinking of