Nuclear Hoagie's remark is a good start.
I evaluated the best strategy with a computer and here is what came out:
The images show the best bet for Alice for each chip count.
Alice's chip count is on the x-axis, Bob's chip count is on the y-axis, from 0 to 99.
The origin is at the bottom left.
The color indicates the best bet: red=1, yellow=25, green=50, cyan=75, blue=99.
Sometimes multiple bets are equivalent.
The first images shows the smallest bet that is optimal.
The second image shows the largest bet that is optimal.
As Nuclear Hoagie saw, as soon as Alice has 50 chips and is
behind Bob, she should bet it all for a 50% chance of winning.
That is the blue part on the 2nd picture.
Note that it is enough to bet just enough chips to reach 100.
If Alice and Bob have 50 chips but Alice is leading, she knows Bob will bet it all on the next move.
So she should try to take a chance at winning the game first by aiming for 100.
In short, if both player have >= 50 chips, an optimal strategy
for Alice is to bet just enough to reach 100.
On the picture you see that in the lower left the optimal bet is 1. This is because on average a bet wins nothing for the player but half the bet for the opponent.
It is better to minimize it.
OK, near the right and top borders it becomes messy.
I won't bother explaining what happens there. I don't know in fact.
But considering that players start by betting 1 on each move, they will inch forward pretty much along the diagonal and have little chance to reach the messy parts.
So a good approximation for the optimal strategy is:
- If one player has <50 chips then bet 1.
- else bet just enough to reach 100.
There is a small exception for the case where Alice has >50 and Bob has 49. If Alice plays 1, Bob might reach 50 and might go for all-or-nothing.
That is why in this case it might be better to play everything. It seems to be the case only if Alice has 60 chips or more.
Below 60 it seems that it is interesting to bet just enough so that when Alice looses, her chip count goes below 50 and Bob won't play it all.
And here is the winning probability map for Alice.
Red is a 0% win, green is 50%, blue is a 100% win.