How many competing papers can be expected for a topic?
The answer entirely depends on the topic and could be anything between 1 and several tens or hundreds. Generally, there are "hot" and "cold" topics.
Assuming that papers should be competed for, since as in a free market, the competition should lead to better outcomes, right?
The main hurdle that researchers have to face is to get their paper accepted in good venues. This is not a question of competition, but of meeting the standards imposed by the reviewers in the peer review process.
Once that papers are accepted and published, there is somewhat of a competition about impact (often measured in terms of citation counts, a somewhat problematic metric). Generally, one strives to have impact, but a high-quality paper is not guaranteed to have impact. On the contrary, an otherwise mediocre paper can have high impact if it has a unique aspect to it, like being the first to observe something that turns out to be important. Marketing also plays a role in impact.