Aaron Green

Building on the non-Markovian gambler’s ruin model with endogenous bias [1], this work integrates ergodicity economics (EE) to shift focus from ensemble averages (e.g., expected utilities) to time-average growth rates, which better capture individual experiences in non-ergodic systems [2; 3; 4]. The extended model computes ergodic growth rates g = (1/τ ) ln(Bτ /B0) for surviving paths, where τ is the stopping time. Simulations (now with 10,000 paths and fixed seed for reproducibility) show that endogenous bias β boosts g (from 0.041 to 0.062 for β = 0 to 0.2), reducing ruin rates but masking fragility through homogenized paths (reduced variance) and persistent tail risks, with bottom 5% growth rates saturating at ∼ −0.012 and conditional value at risk (CVaR at 5%) worsening from -0.019 to -0.025. Gini coefficients rise 18% with β, quantifying increased inequality. In multi-player settings, coarser resolution yields marginally lower g, suggesting an “ergodicity premium” where finer observation enables growth-optimal stopping. Continuous SDE approximations align closely (e.g., g ≈ 0.059 for β = 0.2). This bridges stochastic fragility with EE, highlighting how feedback amplifies inequality and crashes in markets. Results underscore EE’s value: optimism drives growth but fails to restore ergodicity, fattening tails [5; 6].

We extend the gambler's ruin problem by allowing the player's decision to continue to bias the transition probabilities of a symmetric random walk. The model incorporates memory-dependent persistence and optional stopping under concave utility. We derive ruin probabilities, expected utilities, and information premia via recursive transitions and Monte Carlo simulations. Results show heightened fragility: endogenous bias boosts gain-seeking and reduces ruin rates but amplifies tail risks and negative premia by up to 30% for moderate bias. Information premia, derived for multi-player settings with asymmetric observation, yield values of -1 to -4 units, reflecting resolution's impact in biased environments. This connects stochastic processes and behavioral economics, highlighting feedback-driven risks.