Tomasz Kopczewski, University of Warsaw, Poland
Tomasz Potocki, University of Opole, Poland

Abstract: Two Worlds, Two Urns is a turnkey classroom experiment that empirically demonstrates and teaches ergodicity, while also serving as a compact research protocol for history-dependent dynamics in behavioral economics. By contrasting an ergodic i.i.d. process (Urn B) with a non-ergodic, self-reinforcing Pólya process (Urn A), students can experience instances where time and ensemble averages align, as well as instances where they diverge. The design highlights microeconomic methodological limitations that arise from 'freezing' time in static models, such as additive and symmetry assumptions, reliance on process averages, and neglecting path dependence. In the ergodic world, early shocks alter transition probabilities, resulting in stochastic symmetry breaking, cumulative advantage and increasing dispersion from identical initial conditions. We provide a replicable protocol comprising instructions, tracking sheets and analysis templates, and recommend reporting time-averaged growth, ensemble means, variance growth, rank mobility and inequality metrics, as well as process diagnostics (e.g. estimated reinforcement). The experiment supports theory-informed pedagogy and offers an entry point for integrating ergodic reasoning into behavioural research on risk, learning and inequality, where lived trajectories, rather than just cross-sectional snapshots, determine outcomes.

Keywords: non-ergodic dynamics, Pólya urn, path dependence, time vs ensemble average, stochastic reinforcement, inequality dynamics