Eggenberger & Pólya, 1923 — the smallest possible machine for path dependence and increasing returns
Most random processes have a reassuring property: the more you draw, the steadier the ratio. Flip a coin ten thousand times and the fraction of heads sits practically on one half. That is the entire basis for "a big enough sample gets it right".
But a whole class of phenomena plainly refuses to behave that way: the queue that gets longer because it is long, the paper that gets cited because it is cited, the software that works better because more people use it. Here every outcome changes the odds of the next one.
The Pólya urn asks: if a random outcome raises the probability of its own recurrence, what happens in the long run? It is the minimum apparatus needed to answer that — balls and one rule, nothing else.
Only step 3 is new. Remove "add one of the same colour" and this is ordinary sampling with replacement, whose ratio dutifully converges to 1/2. Add it, and everything changes.
First thing: the ratio really does settle. Every run approaches some fixed share, and it does so quickly — after a few hundred draws it barely moves.
The second thing is the point: where it settles is anyone's guess. Histogram the endpoints of a few thousand runs and you get a flat line — every share between 0 and 1 is equally likely. Mathematically that is the uniform distribution, the special case of the Beta distribution when both parameters equal 1.
Put differently: every run converges, but what they converge to is itself random. Convergence supplies no certainty here — it merely freezes an early accident in place.
Change two numbers and you get a different world:
In general: start with a red and b blue, add c of the drawn colour each time, and the limiting share follows a Beta distribution with parameters a/c and b/c. That one formula carries both lessons — the harder the reinforcement (large c), the more extreme the outcome; the deeper the initial stock (large a, b), the less early accidents matter.
One more property deserves its own paragraph, because it explains why we can always construct a reason for the winner: the process is exchangeable — the probability of a sequence of draws depends only on how many reds and blues it contains, not on their order. From that one can prove something strange: the whole process is equivalent to "first draw a biased coin at random, then flip that coin forever". The bias p of that coin is the final share.
So looking back from the middle of any single run, you will see an "underlying tendency that was there all along" — and that tendency was picked out at random by the first few draws. That hindsight always finds a reason is a theorem of this model, not a defect of human cognition.
In 1989 Brian Arthur transplanted the urn directly into a model of competing technologies: two options, adopters arriving one at a time, each adoption making an option slightly more attractive. The conclusions carry over unchanged: lock-in is certain, which option wins is not predictable, and the better one is not guaranteed to win.
The same skeleton grows several other things: preferential attachment in network science (the "rich get richer" degree distribution), the Dirichlet process and the Chinese restaurant process in Bayesian statistics (a new guest is more likely to join a crowded table), and Pólya's own original application — contagion and "accident proneness", where having had one event raises the chance of the next.
It makes exactly one core prediction, and data can refute it: in a domain with increasing returns, the distribution of outcomes should be wide, possibly bimodal, rather than clustered around some reasonable value. If you look at the final shares across a batch of comparable contests and they huddle together, the reinforcement there is weaker than you assumed.