Win Probabilities in the First Ahead by at Least k Multinomial Game

Multinomial trials having probabilities p₁, … , pₙ p 1 , … , p n are observed until one of the outcomes, called the winning outcome, has occurred at least k more times than each of the others. With P(Aᵢ) P ( A i ) being the probability that i is the winning outcome, we give a new approach to proving that P(Aⱼ) ≥ (pⱼ/pᵢ)ᵏ P(Aᵢ) P ( A j ) ≥ ( p j / p i ) k P ( A i ) when pⱼ > pᵢ. p j > p i . We then utilize this approach to obtain very efficient simulation estimators.