One model, two readings

The model in symbols

Everything above is one stochastic process, stated here once. The dictionary after it names each symbol in both fields and says why each pairing is exact rather than an analogy; model.test.mjs checks the claims.

1. Arrivals. Sites $j$ = 1…$N$ sit at points of a map ⇄ sheet. Arrivals come as a [[poisson|Poisson process]]. Each has a type $k$ and a starting point $s$, together its class $c$, which arrives at rate $λ_c$.
2. What an option costs. For an arrival of type $k$ from $s$, site $j$ costs $c_j = t_k·d(s, j) + p_j + m·(1 − fit_{kj}) + ν·κ·r_{j,−k} − ρ·log(1 + r_j/r_0) + H − s·κ·Σ_q W_{kq}·(B_{kq} ∗ r_q)_j$ distance, price, [[mismatch|mismatch]], the other types’ load at $j$, a bonus for busy sites, a [[headtax|tax per resident]] $H$, the same everywhere, and the gain from being near the types this one deals with (the [[agglomeration|interaction]] term, item 7). (The panel above writes the same thing as a utility, $U = −c$.)
3. Choice. The arrival takes the option with the lowest $c_j − τ·ε_j$, where each $ε_j$ is its own draw from a [[gumbel|Gumbel distribution]], unless the outside option’s value $V_0$ beats them all. So $P(j) = e^{−c_j/τ} ÷ (e^{V_0/τ} + Σ_i e^{−c_i/τ})$ the [[logit|logit]] ⇄ [[softmax|softmax]] [[@mcfadden1974]]. A network produces the same law if each neuron fires at rate $e^{−c_j/τ}$ ([[escapenoise|escape noise]]) and the first to fire takes the input: of several independent exponential clocks, clock $j$ rings first with probability $rate_j ÷ Σ rates$. The two are one fact, because minus the log of an exponential waiting time is a Gumbel draw [[@yellott1977;nessler2013]]. The input is lost if a clock of rate $e^{V_0/τ}$ rings first.
4. Stays and load. Each arrival stays an [[exponential|exponential]] time with mean $T$, then leaves. $n_j$ counts those present, and the [[meanload|load]] is $r_j = n_j ÷ (sT)$, with $s$ agents per unit of demand. By [[little|Little’s law]] its average is the rate of arrivals at $j$.
5. Prices. $p_j = a_j + b_j$, where $a_j$ follows one of three rules:
   • Elastic supply ⇄ adaptation: $a_j = κ·r_j$, the slope of a cost $κ·r_j^2/2$. Each occupant ⇄ spike still in effect adds $κ ÷ (sT)$. With a memory, the price falls no faster than $e^{−time/memory}$ after demand drops ([[sticky|sticky]]).
   • Fixed supply ⇄ set point: $a_j$ rises by $τ·log(r_j ÷ c)$ per step until $r_j = c$.
   • Relative fees ⇄ balancing: the same with $c$ the average load, stepping by $τ·log(r_j ÷ r_{avg})$ ([[sinkhorn|Sinkhorn]]) or by a fixed amount $u$ toward the average ([[balancing|DeepSeek’s rule]]).
   A cap $n_j ≤ n_{max}$ either reroutes, so the arrival looks elsewhere and $b_j ≥ 0$ rises while a waiting list forms (the [[shadowprice|shadow price]]), or drops, so the arrival is turned away and $b_j$ = 0.
6. Tuning. $fit_{kj} = Σ_q w_{jq}·S_{kq}$, where $w_j$ is the share of site $j$ fitted for each type ⇄ the share of neuron $j$’s synapses tuned to each input, and $S$ says how well something fitted for one type serves another. $w_j$ moves toward the mix of types that $j$ and its neighbours serve, weighting neighbours by a Gaussian of width $σ$.
7. Interaction. $(B_{kq} ∗ r_q)_j = Σ_i B_{kq}(d_{ij})·r_{iq}$ is how much of type $q$ is near $j$, through a kernel $B_{kq}(d) = e^{−d/ℓ_{kq}} ÷ Z_{kq}$ scaled so that a uniform load $r$ gives $r$. $W_{kq}$ says how much type $k$ deals with type $q$: how many trips ⇄ connections run between them. With a starting point $s$ given (a centre ⇄ an input neuron) and $s = 0$, this is the [[downtown|land market]]; with no starting point (sources “anywhere”) the interaction term alone places arrivals, and the centre is an outcome. A second kernel with the opposite sign and a longer reach (competition for the same customers ⇄ [[lateralinhibition|lateral inhibition]]) can be added. The field $B ∗ r$ follows the load with a memory, since arrivals learn where the others are. Optionally a type’s gain saturates, $x_0·log(1 + field/x_0)$: one job, one input, is what a unit needs.
8. Geography and technology. A layer $g_m(j)$ over the sites (a coast ⇄ where a sensory input arrives) is worth $κ·w_{km}$ per unit to type $k$. Technology $T_j$ is a layer the arrivals make: it grows with the load using a site, spreads to neighbours through a kernel of reach $ℓ_T$ (knowledge propagation ⇄ myelin along used tracts) and decays, $dT_j/dt = α·r_j + D·((B_T ∗ T)_j − T_j) − δ·T_j$. It is worth $κ·v·T_j$ to everyone, it fades the given layers, $w_{km}·e^{−γ·T_j}$, and it lengthens the reach of dealings between two sites by their mean technology, $ℓ_{ij} = ℓ·(1 + β·(T_i + T_j)/2)$, with the kernel renormalized to its own reach so the same pull is spread over more partners. So an arrival’s cost gains $−κ·(Σ_m w_{km}·e^{−γT_j}·g_m(j) + v·T_j)$, and for a given $T$ every term is still the slope of one function: $F$ gains $−κ·Σ_j r_j·(Σ_m w·e^{−γT_j}·g_m(j) + v·T_j)$.
9. Taxes. A tax on buildings at rate $θ$ turns $κ$ into $(1 + θ)·κ$ in the price. A tax per resident adds the same $H$ to every $c_j$. A Pigouvian charge adds, a second time, the nuisance an arrival causes the others, $ν·κ·r_{j,−k}$. A tax on land value takes a share $L$ of the landowner’s residual $p_j·r_j − (1 + θ)·κ·r_j^2/2$, which is in no $c_j$. The table of taxes pairs each with its network counterpart.

What the equilibrium is

With more and smaller agents ($s → ∞$) the loads follow the average flows exactly: the [[fluidlimit|fluid limit]] [[@kurtz1970]]. For steady demand, fixed tuning and prices without memory, those flows settle where the [[freeenergy|free energy]] $F = Σ_{c,j} x_{cj}·(t·d + m·(1 − fit)) + Σ_j κ·r_j^2/2 + ν·κ·Σ_j r_{jA}·r_{jB} − (s·κ/2)·Σ_{k,q} W_{kq}·Σ_{i,j} r_{ik}·B_{kq}(d_{ij})·r_{jq} − κ·Σ_j r_j·(Σ_m w·e^{−γT_j}·g_m(j) + v·T_j) − ρ·Σ_j G(r_j) + τ·Σ_{c,j} x_{cj}·log(x_{cj} ÷ λ_c)$ is lowest. $x_{cj}$ is the flow of class $c$ to site $j$, and $G$ is the function whose slope is $log(1 + r/r_0)$. Under a fixed supply, the constraint $r_j = c$ takes the place of $κ·r_j^2/2$. Every price is the slope of $F$ at its site’s load, a [[lagrange|Lagrange multiplier]], which makes the model a [[potential|potential game]] [[@monderer1996;sandholm2001]].

• As an economy, $F$ is the real cost of commuting, building, misfits and nuisance, minus τ times the variety of choices: Beckmann’s potential for traffic plus Wilson’s entropy, the [[sue|stochastic user equilibrium]] [[@beckmann1956;wilson1967;fisk1980;anas1983]]. As $τ → 0$ it gives Alonso’s [[bidrent|bid-rent line]].
• As a network, $F$ is an energy at temperature τ. The softmax is exactly the distribution that minimizes expected energy minus τ times entropy, and under a fixed supply $F$ is [[ot|entropic optimal transport]], which [[sinkhorn|Sinkhorn]] solves [[@cuturi2013;galichon2016]].
• Checked: at the flows’ resting point every price equals $κ$ times its load, and moving any arrivals from one site to another raises $F$.

$F$ leaves two things out, in both readings: [[sticky|sticky]] prices depend on history, and the tuning rule isn’t derived from $F$ (the dictionary says what it is instead). And it exposes one: the nuisance actually suffered is $2νκ·Σ_j r_{jA}·r_{jB}$, twice the term in $F$, because each arrival counts the nuisance it suffers but not the nuisance it causes, an [[externality|externality]]. Charging each arrival the nuisance it causes, a [[pigou|Pigouvian tax]], makes the true cost the potential, so equilibria sit where it is flat. With weak nuisance it has a single minimum, the charge finds it, and the tests confirm the true total cost falls. With strong nuisance several separated layouts are each stable, and the charge guarantees only one of them.

$F$ also needs nuisance to be mutual. If use A minds use B more than B minds A, no single function’s slopes give both their costs, and the model stops being a [[potential|potential game]]. Settling then needs another reason. One is enough: a [[stablegame|stable game]] settles without any potential [[@hofbauer2009]], and a chase, where B seeks A while A avoids B, is one, because its nuisance has no symmetric part. The tests find more. Two uses settle even under strong one-sided nuisance, which isn’t stable, because at a site the split between two uses can only move one way, with no room to rotate. Three uses that each mind the next never settle once $ν$ passes $2τ/(κ·x)$, with $x$ a use’s load at the busiest site, while a symmetric control with the same symmetric part separates there and stops. A Pigouvian charge adds the nuisance caused to the nuisance suffered, which makes any nuisance mutual, so it restores a potential and settles the three uses too.

The dictionary

Taxes

Only taxes whose network counterpart is exact are here. A network collects nothing, so in each case only what the tax does to choices carries over. Left out, because nothing in this model responds to them: a vacancy tax (the built stock affects no choice), a transfer tax (stays here don’t depend on the cost of moving), a self-assessed or Harberger tax (owners here set no prices), and a road congestion toll (commutes here don’t slow with traffic).

Where the correspondence breaks

• Floor space in use is a flow. Each household ⇄ spike trace stays a random time, so floor space in use is the arrival rate times the stay ([[little|Little’s law]]): a hotel more than a home. The [[built|built stock]] is bookkeeping in both readings: it follows use up over about 200 time units and down over about 1000, but no choice depends on it. In real cities a durable stock that outlasts its demand drags prices below the cost of building [[@glaeser2005durable]].
• Nobody collects the rent, or the taxes. In $F$ the rent is only a multiplier, and a tax only changes what choices cost. Payments to a landowner or a treasury have no counterpart inside one network with one objective, so the [[georgism|Georgist]] questions (who gets land rent, and should it be taxed?) [[@george1879]] need several agents with their own goals.
• Spikes don’t choose; the race chooses for them. The logit law is exact either way ([[tastenoise|taste noise]] ⇄ [[escapenoise|escape noise]]), but no spike experiences a utility.
• Agents and averages agree on averages. Individual agents reproduce the [[flows|average flows]]’ long-run pattern (the tests check it), and add what averages can’t show: [[erlang|Erlang loss]] at a capacity limit, [[drift|boundaries that wander]], and [[vacancychain|vacancy chains]]. Each arrival draws from prices refreshed every few arrivals rather than after every one.
• Prices that remember are outside $F$. A [[sticky|sticky]] rent depends on the past, not only on today’s load, so it is no longer the slope of $F$. Rents remember for years, far longer than a day, while adaptation recovers within a pause; the mixed-use setting has a knob for it. With average flows, prices that reset while a parcel is empty recover slowly, which hands a returning zoned use a spurious rush discount; individual agents mostly don’t get one.
• Tuning isn’t optimized. The fit follows the mix served. It is the same rule in both readings (refitting ⇄ [[competitive|competitive learning]]), but not the fit that would minimize $F$, which would fit every site entirely to its majority use. Neither landlords nor synapses are assumed to optimize here.
• Trained networks internalize [[externality|externalities]]. A network trained by gradient descent on one loss counts the interference each feature causes, as if a [[pigou|Pigouvian tax]] were charged, or as in [[coase|Coase]]’s world of free bargaining [[@coase1960]]. This model’s arrivals count only what they suffer, which is why zoning can matter here and in cities.
• Layers are out of scope. Backpropagation’s error signals are like the prices of intermediate goods in a production network; this model covers one layer’s allocation of capacity, not the chain.

References

Cited inline throughout the page: click a citation to see its entry here.