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# File docs/dan.tex



 Each worker, $w$ indexed by $j$, has two fixed parameters: a knowledge value,
-$b^j$; and a culture vector $c^j$.
+$b^j$; and a culture vector $c^j$. Knowledge is presumed public, and the culture
+vector is visible to other agents only by its  effects on productivity.

 $c^j$ is binary, sparse, length, say, around 8. $b^j$ is chosen uniformly from
 the interval [0,1], and the elements of $c^k$ are chosen by i.i.d. fair
 pays each worker a wage $\omega_i^j$, determined by means that will be
 expanded momentarily. We denote by $f(n, i)$ the index of the firm whose
 worker set contains $w^n$ at time $i$. (We sometimes omit the $i$ parameter in
-this function, and the $i$ subscript when the timestep may be implied from
-context.)
+this function, and the $i$ subscript in series variables, when the timestep may
+be assumed from context.)

 Workers are initially assigned to firms at random with zero wages.

 Each turn, each firm receives a pro-rata distribution of the market, $R_i^k$
 based on firm effectiveness $E_i^k$,

-$R_i^k = \frac{E_i^k}{\sum_k E_i^k}$
+$R^k = \frac{E^k}{\sum_m E^m}$

 Alternately, a formal approach would be to use some game theory model, such as
 the Logit Quantal Response Equilibrium, in a simplified form
 Firm funds, $\Phi^k$, are given
 $\Phi_i^k = \sum_{m=1}^i P_m^k + \Phi_0$ for $\Phi_0$ the startup capital.

-When a firm has no funds it may not spend any money. It could optionally be replaced by a new firm with different hiring rules.
+When a firm has no funds it may not spend any money. It could optionally
+be replaced by a new firm with different hiring rules. This is not yet
+implemented.

 Let performance of a firm $k$ be given


 $d(x) := \frac{1+\cos\pi x}{2}$

+In general, asymmetric functions might be useful too.
+
 We give the knowledge multiplier more simply:

 $B_k=\sum_j b_j$

-Workers are assigned, each turn, public canonical'' performance measure
+Workers are assigned, each turn, a canonical'' performance measure
 $p_i^j$, based on a share of their firm's market distribution, attributed
 pro-rata according to $b^j$. This is a measure, discarding the effect of
 culture vector, and making strong linearisation assumptions, of the marginal

 $p_i^k = \frac{b_i^j}{\sum_{j \in W_{f(j)}} b_i^j}$.

+They are also assigned apublished'' valuation, which, in this simplest version
+of the sim, is assumed to be the same, but which could be made different to
+introduce information asymmetry.
+
 This could also be attributed by Logit Quantal Response Equilibrium if we felt
-like maintaining that symmetry.
+like maintaining that symmetry between worker value imputation and firm
+return allocation, though I see no particular justification for it here.

 Wages are set by single-side auction with firms as bidders. At each timestep,
 firms estimate the value of each worker. There are several schemes for this,

 Firms make bids for each worker each round randomly distributed in the
 interval between their valuation of the worker, and the worker's current wage,
-bounded above by the requirement of predicted profitability. Workers may
+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.
+than having their own contract re-evaluated. We denote the valuation function
+ for the valuate to $f^k$ of worker $w^j$ for the next timestep $i+1$ by $v(f^k, w^j, i+1)$. (Note
+ that this notation implies that all firms share a valuation function, based
+ presumably on public information. This assumption can be relaxed, but I
+ have not.)

 Now, this makes the problem of whom to hire and at what wage, in general, a
 complex game with the potential for nontrivial strategy from both firms and
 workers. But since we presume no knowledge of this underlying model we can
-hopefully ignore that. In general, we can calibrate the game so that there is
-not enough time to get enough information to play strategically.
+hopefully ignore that. In general, if the only information source is actual
+hiring then over plausible lifetimes for the experiment (real human workers
+not often having more than on the order of a dozen jobs in the same industry)
+then we can presume that there are not enough data points to gain true
+knowledge of the form of the culture vector plus the value of all worker's
+vectors, but that some kind of imperfect approximation is used.
+
+So, if we wish to revise the firm strategy when can do this by giving them,
+say, risk aversion, or a model of worker value whose parameters they estimate.
+
+I have partially implemented the latter.
+
+The model is that value of a worker to one firm $f(j,i+1)$ 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.
+
+Symbolically we write
+$v(f^k, w^j, i+1) := V_{f(j,i),f^k}p_i^k$
+
+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.
+
+

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