Banks do compete but it's not really an iterated game - a company only goes public once.
Because in almost all countries politicians have been bought off one way or the other by banks.
This is why dilution happens too - new shares are created so that they can be sold.
You're correct that the whole company "goes public" at the same time. But that just means it becomes legal to sell shares to small investors, and the company becomes subject to the reporting requirements, etc. The previous owners still typically own most of the shares, and (almost always) are forbidden to sell them for around 6 months after the IPO.
[0] https://www.nytimes.com/2020/12/09/business/airbnb-ipo-price...