Differential Sensitivity Analysis of Monte Carlo Poker Equity: Jacobian and Hessian Approaches

We establish a rigorous mathematical framework proving that heads-up play in poker tournaments yields a strictly higher advancement and win probability compared to any n-player configuration with n>2. Our analysis combines (1) combinatorial probability, (2) Monte Carlo convergence theory, (3) partial differential sensitivity of the equity estimator, and (4) regret-minimising dynamics derived from Counterfactual Regret Minimisation. Using blackboard notation and formal theorem–lemma–corollary structure, we derive explicit percentage advantages, asymptotic convergence rates, and structural dominance theorems. This constitutes an original contribution to incomplete-information game theory.