Endogenous Bias in Gambler's Ruin: A Non-Markovian Model of Player-Driven Probability Feedback with Optional Stopping

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.