The Pólya urn — a model for non-ergodic economics
23 Sep, 2025 at 16:35 | Posted in Economics | Leave a comment
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The Pólya urn — a deceptively elementary probabilistic construct — offers profound insights into the structural dynamics of many economic processes. The mechanism is straightforward. Following Scott Page’s example, we start with an urn containing one red and one blue ball. Each time a ball is drawn, it is returned to the urn along with an additional ball of the same colour. This positive feedback process — often characterised as ‘the rich get richer’ — serves as an analogue for capital accumulation and competitive dynamics, illuminating why economic systems are inherently non-ergodic.
In economics, ergodicity implies that individual long-term trajectories can be inferred from cross-sectional distributions; that is, time averages converge with ensemble averages. Yet the Pólya urn demonstrates that each realisation follows a distinct, path-dependent trajectory, converging towards a stochastic asymptotic distribution. Analogously, economic agents experience divergent outcomes shaped not only by deliberate choices but by contingent early events. While mainstream models often invoke representative agents and equilibrium expectations, the urn model underscores how small, stochastic advantages can be amplified through reinforcing mechanisms, entrenching inequality and market concentration.
The non-ergodic character of economic reality has significant implications for theory and policy. Models predicated upon ensemble averages systematically underestimate tail risks for individuals. The urn’s compounding logic elucidates why initial endowments exert a disproportionate influence, why wealth accumulates in a highly uneven fashion, and why aggregate growth metrics obscure the heterogeneity of lived economic experiences. Outcomes, therefore, cannot be ascribed solely to differential abilities or preferences; rather, they emerge from historically contingent, path-dependent processes in which chance events are magnified into enduring disparities.
Recognising the non-ergodic nature of economic systems, as exemplified by the Pólya urn, necessitates a reconsideration of foundational economic concepts ranging from risk assessment to social mobility. Economic success cannot be reduced to rational optimisation or meritocratic reward, but must be understood as the cumulative outcome of stochastic processes in which early contingencies compound into persistent inequalities. This perspective offers a more realistic analytical framework for studying inequality and designing policies that account for the irreducible unpredictability of individual economic trajectories.
To understand real-world, non-routine decisions and unforeseeable changes in behaviour, ergodic probability distributions are of no avail. In a world characterised by genuine uncertainty — where real historical time prevails — the probabilities that governed the past will not necessarily govern the future.
Time, as the saying goes, is what prevents everything from happening at once. To assume that economic processes are ergodic and to focus solely on ensemble averages — thereby implying a timeless framework — is an unsound approach for analysing the fundamental uncertainty that permeates open systems like economies.
Recognising that socio-economic processes are non-ergodic underscores a critical point: uncertainty, not mere risk, is the dominant force. This was a fundamental insight shared by Keynes and Knight in their 1921 works. The subsequent framing of uncertainty through the lenses of ‘rational expectations’ and ‘ensemble averages’ has had profoundly negative repercussions for the financial system.
Whilst Knight’s concept of uncertainty is epistemologically founded (rooted in what we do not know), Keynes’s is ontologically founded (rooted in the inherent unknowability of the world). This distinction has direct repercussions for understanding ergodicity in a strict methodological sense. Of the two, Keynes’s ontological view is the more compelling.
The most significant difference between these views is that the epistemological — Knightian — perspective fosters the mistaken belief that better information and greater computing power can ultimately reduce model misspecification, allowing us to calculate probabilities and depict the world as an ergodic system. As Keynes argued convincingly, this is often ontologically impossible, as it presupposes we inhabit Savage’s constrained ‘small world,’ which is clearly not the case.
For Keynes, the source of uncertainty was intrinsic to the nature of a non-ergodic reality. The issue is not merely the epistemological limitation of our current ignorance, but the more profound ontological fact that there is often no stable foundation upon which to form quantifiable probabilities at all.
Sometimes, we do not know because we cannot know.
