Pólya Urn: Reinforcement Dynamics

Pólya Urn Model

Start with a red and b blue balls. Draw one at random, then return it plus c new balls of the same color. This "rich get richer" rule leads to path-dependent outcomes.

P(red at step n) = rₙ/(rₙ+bₙ) After n draws: rₙ = a + (red draws)·c bₙ = b + (blue draws)·c Limit: X = lim rₙ/(rₙ+bₙ) X ~ Beta(a/c, b/c) E[X] = a/(a+b) Var[X] = ab/[(a+b)²(a+b+c)]

The limiting proportion is random (Beta distributed) — NOT concentrated at E[X]. Each urn run converges to a different fixed point: a manifestation of exchangeable sequences (de Finetti's theorem).

Initial red a: 5

Initial blue b: 5

Reinforcement c: 1

Trajectories: 20

Steps N: 200

Click Run Urns