Thermodynamic limits of fairness in ergodic Markov decision processes

We propose a statistical-mechanics-inspired variational framework for algorithmic fairness in stochastic decision systems with ergodic dynamics. Fairness interventions are represented as entropy-regularized control policies acting on long-run occupation measures across multiple groups. The resulting formulation makes it possible to study fairness regulation through a combination of ergodic control, occupation-measure optimization, and large-deviation ideas. For finite systems, we derive explicit threshold conditions for robust fairness enforcement under finite-memory observation and imperfect mixing, thereby quantifying how memory length, instability, and regularization constrain the regulator’s ability to maintain prescribed fairness tolerances. For large systems, we introduce a precise thermodynamic regime and prove the existence of a limiting free-energy density described by a macroscopic variational problem on empirical laws of group-wise invariant summaries. This yields a mathematically well-defined thermodynamic interpretation of fairness regulation. Numerical experiments complement the theory by exhibiting parameter regimes in which robust fairness regulation is feasible or infeasible, and by illustrating the dependence of optimized fairness gaps and regularized costs on memory, instability, and entropy regularization.