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more documentation tidying

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 \maketitle
 
 
-Each worker, $w$ indexed by $j$, has two fixed parameters: a knowledge value,
-$b^j$; 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.
+Each worker, $w$ indexed by $j$, has two fixed parameters: a knowledge value, $b^j$;
+and a culture vector $c^j$.
+Knowledge is presumed public,
+but the culture vector is visible to other agents only by its effects on productivity.
 
-$c^j$ is binary, of 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
-coin-toss from ${0,1}$.
+$c^j$ is binary, of length, say, $8$.
+$b^j$ is chosen uniformly from the interval [0,1],
+and the elements of $c^k$ are chosen by i.i.d. fair coin-toss from $\{0,1\}$.
 
 There is a fixed finite market whose per-turn returns are fixed at unity.
 
-Each firm, $F^k$ is associated with a set $W^k_i$ of workers at each timestep
-$i$, which partition the workforce, i.e. each worker has only one employer. It
-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
+Each firm, $F^k$ is associated with a set $W^k_i$ of workers at each timestep $i$,
+which partition the workforce, i.e. each worker has at most one employer.
+It 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 in series variables, when the timestep
 
 Each turn, each firm receives a pro-rata distribution of the market, $R_i^k$
 based on firm effectiveness $E_i^k$,
-
-\[R^k = \frac{E^k}{\sum_m E^m}\]
-
 \footnote{
 Alternately, a formal approach would be to use some game theory model, such as
 the Logit Quantal Response Equilibrium, in a simplified form
 at the moment since I have no reason to suppose that the the addition of a new free parameter is merited.
 }
 
+\[R^k = \frac{E^k}{\sum_m E^m}\]
+
 Firm profit, $P^k$ is given \[P^k=R^k-\sum_{j \in W^k} \omega_i^j\] - that is,
 the profitability is income minus expenses, where expenses are assumed to be
 comprised only of worker wages.
 agents employed by firm $F^k$.
 
 Then we calculate the performance of a firm $k$ by
-
-\[E^k=B^kC^kn^k\]
-
-or if you want to raise these factors to an arbitrary power because you have
-reason to suppose some kind of nonlinear scaling.
+\footnote{
+or if you wanted, you might raise these factors to an arbitrary power because you have
+reason to suppose as a kind of nonlinear scaling.
 
 \[R_k=(B^k)^g(C^k)^h n^k\]
 
 You could even enforce
 \[g+h=1\]
 and make this a weighted geometric mean, if you'd like. Since $B^k$ and $C^k$
-have both been specified to have range between 0 and 1, this allows a pleasing
+have both been specified to have mean between 0 and 1, this would allow a pleasing
 symmetry.
 
 For now, I presume, to the contrary, that $g = h = 1$, as it simplifies
 creating estimates of the marginal benefit of a given agent with a given
 knowledge factor.
+}
+\[E^k=B^kC^kn^k\]
 
-Now, we need to define the culture multiplier
-$C_i^k$. We associate a matrix $\kappa^k$ with each firm at each time step,
-comprised of the vertically stacked horizontal culture vectors of the agents
-employed at that firm. Then we calculate a culture factor from the column
+Now, we need to define the culture multiplier $C_i^k$.
+We associate a matrix $\kappa^k$ with each firm at each time step,
+comprised of the stacked (horizontal) culture vectors of the agents
+employed at that firm.
+Then we calculate a culture factor from the column
 means. 
 
 \begin{align*}
 \end{align*}
 
 for $y$ ranging over all the elements of the culture vector, and for a
-``matching'' reward function $d$ such as
-
-\[d(x) := 2 |x - 1/2|\]
-
+``matching'' reward function $d$ such as\footnote{
 Also fun might be:
-
 \[d(x) := \frac{1+\cos\pi x}{2}\]
+Later, asymmetric functions might be worth considering.
+}
+
+\[d(x) := 2 |x - 1/2|\]
 
-In general, asymmetric functions might be useful too.
 
 We give the knowledge multiplier more simply as the mean of the vector of all
 employee knowledge values.
 
-\[B_k=\frac{1}{|W_k|}\sum_j b_j\]
+\[B^k=\sum_{j=1}^{n^k} b_j\]
 
 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
+pro-rata according to $b^j$.
+This is a measure, discarding the effect of
 culture vector, and making strong linearisation assumptions, of the marginal
 return to a firm of the given worker.
 
 They are also assigned a``published'' 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.
-
+\footnote{
 This could also be attributed by Logit Quantal Response Equilibrium if we felt
 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 estimate the value of each worker.
+There are several schemes for this,
 but one is simply assuming the canonical valuation is accurate.
 
 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
-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 at timestep
+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 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
-have not done so.
+presumably on public information.
+This assumption can be relaxed.
 
 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, if the only information source is actual
+workers.
+But since we presume no knowledge of this underlying model we can
+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.
+vectors, but that instead 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.
 $f^k$ 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 over a timestep. That is, this valuation function imputes a mean change
+one over a timestep.
+That is, this valuation function imputes a mean change
 coefficient to all pairwise changes in employment.
 
 Symbolically we write
 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
+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
+well approximated by this to improve bidding performance.
+I'm thinking of
 taking the means in the log domain so that they decay naturally to 1 in the
 absence of new data points.