Vacancy Chain Analysis

Tracking people scatters attention across many individual decisions, but tracking the vacancy reveals the system logic: an empty unit must be filled, and filling it usually creates the next empty unit, so the vacancy travels through the stock in a single connected chain. This inversion turns a tangle of moves into one orderly Markov process whose length and reach can be computed exactly, exposing how one new opportunity multiplies into many moves throughout the system.

The vacancy's movement among unit types is an absorbing Markov chain. Writing the transition matrix in transient and absorbing blocks, the transient block Q gives the fundamental matrix N = (I − Q)⁻¹, whose entry N_ij is the expected number of times a vacancy starting in type i visits type j before the chain ends. Summing a row of N yields the multiplier M — the expected total number of moves, and hence relocations, generated by one initial vacancy of that type.

A chain terminates whenever the vacancy is not handed on. The two canonical endings are entry from outside the system — a newly formed or in-migrating household, or an outside hire — who leaves no prior unit vacant, and removal of the vacated unit from the stock through demolition, conversion, or abolition of a post. These outcomes are the absorbing states of the Markov chain, and their relative frequencies determine how long chains run and how large the resulting multiplier is.