Let's consider some alternate problems.

- Let's consider the case where A and B play a sequence of games and the first person to win a game, wins the match-up. In such a case, the best strategy for A to win is to play aggressively in the first game itself which gives A the probability of winning as 45%. Playing conservatively does not help A's cause as A HAS to play aggressively to win.
- Interestingly, if the problem was stated as what is A's strategy to engage B in as many games as possible then the A's strategy to play conservatively all the time.

Essentially, if A has to win, A has to play as few games as possible.

Now, to our problem.

There are three possibilities after two games:

- A loses
- A wins
- A ties with B - In this case, the only option A has is to play aggressively in the third game

Essentially, A has to adopt a strategy where the probability of A winning or tying the game is maximum.

A has the following strategies for the first game to minimize loss in the first two games.

- Play aggressively - if it results in a loss, play aggressively in the second game to stay alive. If it results in a win, play conservatively to play for the win. Probability of winning = 45% X 90%, Probability of a tie = 45% X 10% + 55% X 45%, Probability of losing = 55% X 55%
- Play conservatively - if it results in a loss, play aggressively in the second game to stay alive. If it results in a draw, play aggressively for the win. Probability of winning = 90% X 45%, Probability of a tie = 10% X 45%, Probability of losing = 10% X 55% + 90% X 55% = 55%

Based on the probabilities above, we see that A's strategy is to:

- Play aggressively in the first game
- Play aggressively in the second game, if the first results in a loss and play conservatively if it results in a win
- Play aggressively in the third game, if the first two games result in a tie.

Therefore, total probability of winning = 45% X 90% + 45% X 65% X 45% = 0.536625

Probability of losing = 55% X 55% + 45% X 65% X 55% = 0.463375 = 1 - Probability of winning.

Note to make sure that all probabilities add up to 1.