Alice and Bob are in Roman times and have 4 gladiators each. The strengths of each of Alice's gladiators are 1−4, while Bob's gladiators have strengths 4,5,9, and 12. The tournament is going to consist of Alice and Bob picking gladiators to fight against one another one-at-a-time. Then, the two gladiators fight to the death with no ties. If the two gladiators are of strengths x and y, respectively, then the probability that the gladiator with strength x wins is x+yx. The winning gladiator also inherits the strength of its opponent. This means that if a gladiator of strength x wins against a gladiator of strength y, the winner now has strength x+y.
Alice is going to pick first for each fight among her remaining gladiators. Afterwards, Bob can select his gladiator (assuming he has one) to go against the one Alice selected. The winner of the tournament is the person who has at least one gladiator left at the end. Assuming Bob plays optimally, what is his probability of winning the tournament?
