Statistics

The Expected Time-Cost of Equipment Enhancement in Milky Way Idle: An Exact Green’s Function and Path-Integral Analysis

Authors: Lukas Saul
Comments: 13 Pages.

We develop an exact, closed-form statistical mechanics of the equipment-enhancement system of the idle RPG Milky Way Idle. An enhancement attempt advances an item by one level with probability p and, on failure, either resets it to level 0 (protection off) or decrements it by one level at the cost of a consumable protection item (protection on, from a chosen level R upward). We compute the expected total cost E[C] = T·E[τ] + L·E[M], where T is the per-attempt time cost, L the per-protection-item cost, τ the hitting time of a target level N, and M the number of protected failures incurred. The central object is the expected visit-count vector v_k, which we identify as the discrete Green's function of a piecewise birth-death operator and solve in closed form in terms of p, N, and R alone. We recover the unprotected limit C_n = (p^(−n) − 1)/(1 − p) as a special case, matching the result of Chae and Lee. We then present the full path-integral representation of the process as a sum over random-walk histories weighted by exp(−βC), give the transfer-operator resolvent G(x,y) = [(I − W(x,y))^(−1)]_{0,N}, and analyse the continuum limit: the WKB/instanton action S_inst = N·ln(1/σ) reproduces the exact exponent σ^(−N), while Gaussian fluctuations around the instanton yield an erfc-form budget tail P(C ≤ B) ≈ (1/2)·erfc((E[C] − B)/sqrt(2·Var[C])). All exact formulae are rational in p and are tabulated for the base success rates of the game.
Category: Statistics