House Edge: Mathematical Model of System Advantage

House Edge (HE) is a quantitative measure of the structural advantage of the system, formally defined as the difference between unity and the mathematical expectation of the normalized return: HE = 1 − E[R], where R is a random variable representing the ratio of the output value to the input value. In game theory terms, this parameter characterizes the non-equilibrium state of the payment matrix in favor of one of the participants. The HE value is strictly positive for any commercial analytical system with deterministic rules. This metric is an invariant of the algorithm and does not depend on the operator’s strategy.

The calculation of the House Edge is based on a full sweep of the outcome space, taking into account their probabilities. For a discrete model with k possible outcomes, the formula takes the form: HE = 1 − Σᵢ₌₁ᵏ pᵢ·mᵢ, where pᵢ is the probability of the i-th outcome, and mᵢ is the payout multiplier. In the continuous case, an integral form with a probability density function is applied. It is important to note that HE is a linear functional on the space of distributions, which ensures additivity during the composition of independent subsystems.

Long-term convergence of the empirical advantage to the theoretical value is guaranteed by the Strong Law of Large Numbers (SLLN). According to the SLLN, the sequence of averages S̄ₙ = (1/n)·Σᵢ₌₁ⁿ X_i converges almost surely to E[X] as n → ∞, provided the first moment is finite. The rate of convergence is estimated via the Central Limit Theorem: √n·(S̄ₙ − μ)/σ →ᵈ N(0,1). In practice, this means that to achieve an accuracy of δ, at least n ≥ (z_{α/2}·σ/δ)² observations are required, where z_{α/2} is the quantile of the standard normal distribution.

The statistical significance of the deviation of the observed HE from the theoretical one is verified using a z-test or Student’s t-test. The null hypothesis H₀: HE_emp = HE_theo is tested against the alternative H₁: HE_emp ≠ HE_theo at a given significance level α. The critical region is determined via the p-value: if p < α, the null hypothesis is rejected. Additionally, bootstrapping methods are applied to construct non-parametric confidence intervals, which is particularly relevant when the type of distribution of the initial data is unknown.