[RocketManJames]

Just thinking out loud here... I don't really have an answer.

If you bet 0 for both rounds, then a thinking opponent wins 75% of the time, like you said. Because he can bet 10 in round 1, which gives him 108 or 90. And, he can then choose to wager 20 if he has 90 and 0 if he has 108. So, you've got the four outcomes: (100, 108), (100, 108), (100, 108), (100, 70).

What if you bet 0 and your opponent bets something greater than 0 in Round 1. Let's say he bets 10 in Round 1, and you bet 0. Then, after Round 1, you have 100 and opponent has 90 or 108.

He has 108. At this point, he will see you have bet nothing. If he chooses to stay at 108 by betting 0, then you will be 50% to win by betting more than 10 in this round. But, if he bets 100 at this point and have an outcome of 8 or 188. This would ensure that he'd be at 50% to win at worst or 75% to win at best (if you bet too much).

If he has 90, then you can bet 90 and be at worst 50% to win or best 75% to win (if he bets too much).

I guess I'm pretty stuck, so far in my line of thought, it's clearly not better than fifty-fifty against your opponent. I will keep thinking about this.

In Edit: I'm not sure how one player can get an edge over another player without knowing the other player's strategy or the probability the other player will employ a given strategy. The players make equivalent decisions and there's no postional advantage. Maybe I'm misunderstanding this a bit.

-RMJ