Indexing the archive…
Your Universe of Digital Possibilities
A sound bank, a rumour that it is not, and a queue that makes the rumour true. Run the same crowd of depositors twice from one seed: in one hall the rating is sealed, in the other it is printed on the wall. Find the point where a published rating starts a run on its own.
ℓ is the share of deposits withdrawn early; the bank pays them from its liquid share m and then by selling loans worth θ on the unit. Under common knowledge every θ with 0 < ψ(θ) < 1 has two equilibria — nobody runs and it survives, everybody runs and it fails.
Each depositor sees θ through private noise and withdraws when the chance the bank fails beats the cost ratio c. At the marginal depositor’s signal the share of others withdrawing is, to her, uniformly uncertain — so at the threshold the run is exactly 1 − c, whatever the noise. With ψ(θ) = θ this is Morris and Shin’s θ* = 1 − c.
α is the precision of a public signal, β of each private one. While the inequality holds the threshold is unique; beyond it the public signal becomes common enough to coordinate, and a precise enough headline can start a run on a bank the private evidence calls sound. With the bank’s resilience slope ψ′ the bound is α ≤ ψ′√(2πβ).
The third prophecy is Merton’s, the one that named the phenomenon: a definition of the situation that makes itself true. The Cobweb had one crossing and a forecast that could find it; this bank has two, and they are both prophecies that fulfil themselves — total safety and certain failure — with a jump between them where no forecast is right. Like The Threshold’s spin flip, the failure arrives all at once when a shared field crosses a line; like The Contagion’s outbreak, each withdrawal makes the next likelier. Insurance breaks the loop, and The Prophecy will say why a curve with a jump cannot promise a forecast that survives being read.