Polya Process (Path-Dependent Probability)
A Pólya urn models reinforcement: after each draw, the selected color becomes more likely on the next draw. Early random events can therefore affect later proportions. Taleb uses this path dependence to question models that assume independent changes.
THE IDEA
MATHEMATICS INSIDE AND OUTSIDE THE REAL WORLD
A mathematical approach to the problem is in order. A Brownian-motion model uses independent increments: past changes do not alter the distribution of the next increment. Arthur suggests models with reinforcement, such as the Polya process, which can be explored with a Monte Carlo simulator.
Start with an urn containing equal quantities of black and red balls. Draw a ball, return it, and add another ball of the same color. Drawing red increases the chance of drawing red again; drawing black increases the chance of black. Repeating this process produces different final proportions across runs, because early draws change the probabilities of later ones. This is path dependence, not an improvement in a player’s guessing skill.
Compare this with an urn where each drawn ball is simply replaced and no balls are added. Its composition stays fixed, so successive draws are independent. Likewise, a win at an independent roulette spin does not improve the odds on the next spin. Dependence requires a model that accounts for the changing state; it does not put a process beyond probability mathematics.
What has gone wrong with the development of economics as a science? Answer: There was a bunch of intelligent people who felt compelled to use mathematics just to tell themselves that they were rigorous in their thinking, that theirs was a science. Someone in a great rush decided to introduce mathematical modeling techniques (culprits: Leon Walras, Gerard Debreu, Paul Samuelson) without considering the fact that either the class of mathematics they were using was too restrictive for the class of problems they were dealing with, or that perhaps they should be aware that the precision of the language of mathematics could lead people to believe that they had solutions when in fact they had none (recall Popper and the costs of taking science too seriously). Indeed the mathematics they dealt with did not work in the real world, possibly because we needed richer classes of processes—and they refused to accept the fact that no mathematics at all was probably better.
The so-called complexity theorists came to the rescue. Much excitement was generated by the works of scientists who specialized in nonlinear quantitative methods—the mecca of those being the Santa Fe Institute near Santa Fe, New Mexico. Clearly these scientists are trying hard, and providing us with wonderful solutions in the physical sciences and better models in the social siblings (though nothing satisfactory there yet). And if they ultimately do not succeed, it will simply be because mathematics may be of only secondary help in our real world. Note another advantage of Monte Carlo simulations is that we can get results where mathematics fails us and can be of no help. In freeing us from equations it frees us from the traps of inferior mathematics. As I said in [Chapter 3], mathematics is merely a way of thinking and meditating, little more, in our world of randomness.
When to use it
Investment strategies in markets with momentum effects (trend-following).
Technological competition where early adoption leads to increasing returns (e.g., VHS vs Betamax).
Scientific research funding: past success increases probability of future funding (Matthew effect).
Skill acquisition: practice improves performance, leading to more practice and further improvement.
FROM THEORY TO EVERYDAY LIFE
In practice
Polya Process Urn Game
Imagine an urn with 10 black and 10 red balls. After each draw, return the selected ball and add another of the same color. A run that happens to draw red early becomes more likely to draw red again, while another run may favor black. The mechanism reinforces colors drawn; it does not make a player better at guessing.
Contrast with Independent Roulette
With independent roulette spins, winning does not increase the chance of winning again. In a Pólya urn, drawing a color changes the urn’s composition and therefore the probability of drawing it again. The difference lies in reinforcement, not simply in whether a player has won.
A MOMENT TO REFLECT
Put it to work.
How might a Polya process apply to your personal learning or skill development? Consider a skill where success breeds further success.
Can you identify a real-world situation where outcomes are path-dependent, like a Polya process, rather than independent? What are the implications?
What are the dangers of using independent models (like Brownian motion) in finance when the real world might be more like a Polya process? How could this mislead decision-making?
CONTINUE THE CONVERSATION
Source texts
- Fooled by Randomness The Hidden Role of Chance in the Markets and in Life (Nassim Nicholas Taleb)