D'Alembert System: Arithmetic Risk Progression Analysis

The D'Alembert system is founded on the principle of arithmetic progression in position sizing, where each subsequent step is adjusted by a fixed increment Δ depending on the outcome of the previous iteration. Formally, S(t+1) = S(t) + Δ when X(t) ≤ 0 and S(t+1) = max(S₀, S(t) − Δ) when X(t) > 0, where S₀ is the base position size and Δ is the progression step. The mathematical model of this system is described by a Random Walk on a semi-bounded lattice with a reflecting barrier at point S₀. The stationary distribution of position size S(t) depends on the success probability p and is determined through the balance equation π(s) · p = π(s + Δ) · (1 − p) for s > S₀. The solution of this equation demonstrates that when p < 0.5, the position size tends toward unbounded growth, creating systemic risk.

Comparison of linear (arithmetic) and geometric progressions reveals fundamental differences in capital dynamics. Under arithmetic progression (D'Alembert), the position size after k consecutive failures is S(k) = S₀ + k·Δ, whereas under geometric progression (Martingale) it is S(k) = S₀ · 2ᵏ. The ratio of geometric to arithmetic progression R(k) = S₀ · 2ᵏ / (S₀ + k·Δ) grows exponentially as k → ∞, demonstrating the substantially lower aggressiveness of the D'Alembert system. The maximum position size after a series of 10 failures with S₀ = 1 and Δ = 1 equals 11 for D'Alembert versus 1,024 for Martingale — a difference of 93×. This property renders arithmetic progression considerably more resilient to serial losses, albeit at the cost of slower loss recovery.

The variance profile of the D'Alembert system is characterized by nonlinear dependence on streak length and model parameters. The total variance of the result over N iterations is expressed as Var(Σᵢ S(i)·X(i)), where S(i) is a stochastic process of position size correlated with outcome history. Unlike the Flat Model, where Var = N·S₀²·σ², an additional term arises here due to the covariance between S(i) and X(i). Numerical simulation demonstrates that at p = 0.48 and Δ/S₀ = 0.1, the variance of the D'Alembert system exceeds that of the Flat Model by 35–40% at N = 1,000, yet remains an order of magnitude below the variance of geometric progressions. Analytical computation of the exact variance requires solving a system of recurrence equations, most efficiently accomplished through the generating functions method.

Convergence analysis of the D'Alembert system to a stationary regime is conducted using the theory of ergodic Markov chains. The position size S(t) forms a Markov chain on the state space {S₀, S₀ + Δ, S₀ + 2Δ, ...}, which is irreducible and aperiodic when 0 < p < 1. The ergodicity condition (existence of a stationary distribution) is satisfied if and only if p > 0.5, i.e., when the probability of position reduction exceeds the probability of increase. When p ≤ 0.5, the chain is transient, and the expected position size E[S(t)] → ∞ as t → ∞ at a rate of O(t·(1−2p)·Δ). The practical significance of this result is that the D'Alembert system requires a mandatory upper bound S_max to ensure stability under adverse outcome distributions.

Comparison of the D'Alembert system with the Kelly Criterion enables evaluation of arithmetic progression efficiency relative to the theoretically optimal strategy. The Kelly Criterion defines the optimal capital fraction f* = (p·b − q) / b, where b is the payout coefficient, p is the success probability, and q = 1 − p. The fundamental distinction is that Kelly employs proportional management (S(t) = f* · C(t), where C(t) is the current capital), whereas D'Alembert uses additive management. The Kelly growth rate is G = p·ln(1 + f*·b) + q·ln(1 − f*), which is maximal among all proportional management strategies. The D'Alembert system does not maximize logarithmic growth; however, it provides lower sensitivity to errors in the estimation of parameter p, rendering it more robust under conditions of parameter uncertainty.