The gladiators who absorb their opponents' strength

Question

Two teams of gladiators fight a tournament. Each gladiator has a strength, a positive number. When gladiators of strengths a and b fight, the first wins with probability a/(a + b). The winner then absorbs the loser's strength, becoming a gladiator of strength a + b, and fights the next opponent. A team loses when all its gladiators are gone.

Each team chooses the order in which to send out its gladiators, and may change its mind as the tournament proceeds.

Does the order matter? What is the probability that a team wins?

Accepted Answer

The order makes no difference. A team's probability of winning the tournament is its share of the total strength: S divided by (S + T), where S and T are the two teams' total strengths. A fight is a fair bet. Suppose a gladiator of strength a from team 1 fights one of strength b from team 2. With probability a/(a + b), team 1 gains b (the absorbed strength); with probability b/(a + b), team 1 loses a. Team 1's expected change in total strength is (a/(a + b)) × b minus (b/(a + b)) × a = 0. So every fight, whoever is sent out, leaves each team's expected total strength unchanged: the totals for…

The gladiators who absorb their opponents' strength

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