State-Dependent Asymmetry in Soft-Pity Gacha Waiting-Time Models: Exact Recurrences, Tail Risk, and Featured-Target Extensions

Randomized reward mechanisms are often described as repeated trials with a fixed success probability. This constant-hazard reference case is symmetric in the limited finite-state sense that, conditional on non-absorption, the next-draw success probability is invariant with respect to the current draw count. Pity and guarantee rules break this draw-count homogeneity by making the hazard depend on the current state. This paper studies that state-dependent asymmetry for a finite soft-pity waiting-time model. The waiting time for one rare item is represented as an absorption time of a Markov chain whose transient state is the pity counter. We write the corresponding absorbing transition matrix explicitly and then derive the equivalent first-step recurrences for the expectation, variance, and full probability mass function. A simple stochastic-ordering proposition shows how increasing the statewise success probabilities decreases the waiting-time distribution in the usual tail order. Repeated convolution then yields the distribution for multiple independent stages. The numerical section reports quantiles, tail probabilities, VaR/CVaR-type summaries, expected excess values, sensitivity analyses, normal-approximation diagnostics, and distributional asymmetry indicators. A featured-target variant with a binary guarantee state is also included. Throughout, the reported quantities are consequences of the stated transition rule; Monte Carlo simulation is used only as a numerical check.