Two players, A and B, are each given 10000 dollars. They'll play a 10 round game. In each round both have to gamble some of this money (can be zero); in the 10th round, they both must use all their remaining money. The winner of each round is the one who gambles less. He gets both sums gambled in that round; these winnings are put away and not added to the money available for gambling. The winner of the game is the one who won more money.
What is the best strategy to win this kind of game?
Example for a 3 round-game: round 1, A gambles 5000, B gambles 4000 --> B gets 9000. Round 2, A 5000, B 4000 --> B gets 9000. Round 3, A 0, B 2000 --> A gets 2000, B wins getting 18000.
This seems like a zero-sum game theory, but there's something different, any help will be appreciated.