No "market rent" is set by the market.
Some participants in the market try to get apartments for below market rent (those are the tenants) and some try to get above market rent (those are the landlords) - when they reach agreement, that number becomes market. Obviously if the landlord gets overly aggressive and sets the price at 5x market, their demand doesn't become "market" just because they are asking it -- in fact, in that case, they will probably never rent the unit. However, if they price it 5% over market, they will likely eventually rent it (each applicant's desires, price sensitivity, and urgency vary) but on the average it will take longer to rent. So, it's probably most accurate to talk of "market rent" as being a price that, on the average (or perhaps median), results in the unit being rented within some fixed period (perhaps two weeks). If the units are being snapped up the moment they are marketed, the landlord will generally raise the price even if it means it now takes a day extra to rent, if they are sitting vacant for months, they will generally lower the price.
Of course, unlike a commodity such as table salt, every apartment is different so the price discovery process is a bit ad hoc and imprecise. Although the differences, for a given size/bedroom count, is usually greater across complexes than across apartments within a complex, there are often substantial differences between seemingly similar apartments within the same complex (for example 1st floor vs. top floor, overlooking courtyard vs. overlooking parking lot, near the pool vs. far from the pool).
The landlord needs to come up with some number for rent that they think is market -- but they don't really know if they are right (regardless if the process involves a seat of the pants guess by the on-site manager or a statistical market research data driven approach done by corporate using analytics). This number is often non-negotiable, esp. at the big complexes -- but it also sometimes changes quite often as price discovery proceeds to figure out what market rent is for that particular unit.
I've lived on a large complex (about 3000 residents) where their web site listed every available apartment by unit number, the price, and the availability date. I created a cron job to pull all that information every morning at 3AM and massage it into a database. It was quite interesting, one unit sat vacant, for example, for six months and the price vacillated all over the map during that time from absurdly high to comparatively cheap. Prices often changed daily - sometimes by 0.1% and sometimes by 15% -- probably mostly driven by some algorithm that changed some of the prices for the purpose of price discovery. Due to local laws, this particular property had some odd constraints on it, but a more casual analysis of other properties without those constraints seemed to yield rather similar results.