Gladiator Battle

Been some time since I had a new puzzle, and there are still two left in the paper "Games People Don't Play", though they are basically the same with a slight variation. Here is the first one:

Alice and Bob are competing team managers in a gladiator competition. Alice's gladiators have strength a₁, a₂, a₃...a_n and Bob's have strength b₁, b₂, b₃...b_m. Each round, Alice will select a gladiator from her team and then Bob will select one from his team and the chosen gladiators will fight.

If a gladiator of strength x fights a gladiator of strength y, the chance the one with strength x wins is given by x/(x+y). The losing gladiator is eliminated and the winning gladiator gains confidence from the fight, increasing the winning gladiator's total strength to x+y. Then a new round begins with Alice and Bob selecting gladiators to fight. The competition ends when one of the teams has run out of gladiators.

What is the optimal play? In particular, suppose Alice always chooses to send in her strongest gladiator, how should Bob respond?

It seems needlessly elaborate, especially given that in theory you have to solve for all possible values of a₁ through a_n and b₁ through b_m. It actually has a fairly elegant solution, but requires something of a mathematical transformation of the puzzle to reduce it to an easier setup.