aaqucnaona,
You ask why females dont compete?
As James R has already stated, females (in most but not all species) contribute far more to the raising of the offspring.
ex. more energy put into an egg than a sperm
ex. females incubate or gestate etc, males mostly dont
ex. females often contribute the most in rearing. the theory is because they have already put more into raising the young at this stage, therefore the male can adopt (via natural selection) an abandonment policy (less to lose) forcing the female to stay to do all the rearing work.
Furthermore in a population where the sex ratio is 50-50, females that are pregnant are not in the mating game, but males alwalys are, creating an imbalance at any one time.
In addition, males have greater capacity to breed, millions of sperm, while females have a limited number of eggs in a lifetime, and can carry only so many offspring at once. An addition potential bias.
This all makes females rarer. therefore males are forced to compete for the rarer. Females can therefore choose the best, and its vital to choose, to ensure the best chance of passing on one's genes.
So as you can see there are a multitude of factors to consider, and in depends on the specific species.
However, there are examples of the complete opposite. In seashorses the males carry the eggs in the pouch (reason unknown), but whatever the reason in results in the females being ornamental (to impress the males, while this isnt physical, it is competition nonetheless) and the males choose who they partner with, Iam guessing because they make the larger contribution. There are other examples.