Alice and Bob both have N warriors under their command, numbered 1~N, and $1$ point of fighting power at their disposal. Before the game, they privately distribute the power between their warriors. When the game begins, both send their warrior #1 to a 1v1 fight. If a warrior with power x fights one with power y, the former wins with probability $\frac{x}{x+y}$, and the latter with probability $\frac{y}{x+y}$. If #1 is defeated, #2 is sent to continue the next round of fight, so on and so forth until one party has all of their warriors defeated and loses the game. The more battles a warrior wins, the stronger he becomes: after a warrior defeats an $x$ power opponent, his power will increase by $\frac{x}{2}$. Both players want to maximize their winning probabilities.
How should Alice distribute her fighting power?