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

Abstract Multinomial trials having probabilities $$p_1, \ldots , p_n$$ 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 $$P(A_i)$$ P ( A i ) being the probability that i is the winning outcome, we give a new approach to proving that $$P(A_j) \ge (p_j/p_i)^k P(A_i) $$ P ( A j ) ≥ ( p j / p i ) k P ( A i ) when $$p_j > p_i.$$ p j > p i . We then utilize this approach to obtain very efficient simulation estimators.