Uncertainty & dynamics

Some worlds remember. Some forget. Some don't care.

QWERTY keyboards. Driving on the left. VHS beating Betamax. Silicon Valley being in Silicon Valley.
None were inevitable. What separates "history matters" from "history doesn't"?

An urn. Two starting balls, one red 🔴, one blue 🔵. At each step you reach in, draw one, and then something happens. That "something" — the update rule — is the entire story. Try each of the three below and see how radically it changes what the long run looks like.

Pick a rule · draw balls · watch history behave

🔴 Red in urn

🔵 Blue in urn

Red share

50%

Draws so far

The whole story in three clicks. On 🧊 Bernoulli the urn never changes — draws are just random samples from the same jar forever. On 🌊 Pólya every draw adds one more of the colour you drew, so early luck compounds into a runaway. On ⚖ Balancing every draw adds one of the opposite colour, quietly pulling the share back to 50/50 no matter where you started.

Now run all three worlds at once

Same starting urn (1 red, 1 blue). Same random-number budget. The only difference is the update rule. Watch the three panels below diverge.

Three urns racing · 40 trajectories each

Each blue line is one trajectory's red-share over time. Bernoulli: everyone stays near the starting fraction — that flat line is what "no memory" looks like. Pólya: trajectories fan out to fill [0, 1] — that spread is what "history locked in" looks like. Balancing: everyone gets pulled to ½ regardless — that convergence is what "history erased" looks like.

Pólya's uniform-limit surprise. Give a Pólya urn infinite steps and the final red share is a random variable, and — for the classic (1, 1) start — it's uniformly distributed on [0, 1]. That means ending at 12 % red is exactly as likely as 87 % red is exactly as likely as 50 %. There is no "most likely" outcome. Every possible world is equally represented in the parallel-universe multiverse of Pólya. (Pólya, 1930.)

The path-dependence fingerprint

Run a lot of trajectories, record the final red share of each, plot the histogram. The shape tells you everything about whether history mattered.

Distribution of final shares · after 800 steps × 1500 trajectories per urn

Bernoulli: sharp spike at the initial share (50 %) — every universe ends up in almost the same place. Pólya: nearly flat over [0, 1] — every possible outcome is roughly equally likely. Balancing: sharp spike at ½ — history erased.

Entropy is the sharpest diagnostic. Compute the Shannon entropy of each final-share distribution. Pólya's is close to the maximum (log₂ of the bin count); the other two are near zero. That single number tells you whether a process is path-dependent. It's how Page distinguishes path dependence from tipping points in the book: same distributional shift, different mechanism.

Real-world Pólya-like moments

Technology standards. QWERTY vs Dvorak, VHS vs Betamax, TCP/IP vs OSI, GSM vs CDMA — all cases where slight early leads compounded into permanent dominance despite comparable or better competitors.
Cities and industries. Silicon Valley wasn't destined to be Silicon Valley — early Stanford spin-offs seeded a hub that pulled in more talent, which pulled in more spin-offs. Detroit for cars. Hollywood for film. Nashville for country music.
Institutions and law. Early legal precedents shape later ones. First-mover constitutional choices cast a long shadow. Common law is a Pólya urn where the balls are cases.
Cultural markets. Salganik, Dodds & Watts (2006) showed empirically that when people can see other people's music-download counts, popularity locks in randomly — a Pólya urn in the wild. See our Power Laws story for the two-worlds version of that experiment.

Related models

🌐 Markov chains — the exact opposite of path dependence: the past doesn't matter, only the present state.
📈 Power laws — preferential attachment is a path-dependent process that generates heavy-tailed outcomes.
🚶 Random walks — closely related stochastic process; the Pólya urn is a biased random walk in disguise.

← Markov chains NK landscape →