Yes, it should be noted that if you live in somewhere like the UK or the US, the "bad" lines, 9 and 2, are technical and organisational marvels with no parallel in your country. From the end user standpoint they operate amazingly well. They're cheap, clean, safe, reliable, convenient. They're crowded at rush hour. Line 9 is so crowded at rush hour that there was this one time, a few weeks ago, that I wasn't able to get on a train and had to wait 3 minutes for the next one. That's how bad it is. ¯\_(ツ)_/¯

Waiting for the next one is a common thing in most countries. At least once or twice. Korea included.

in beijing at rush hour you line up at a train door, and you see two or three trains stop, allowing the first few people at the head of the line to enter, before you make it to the head of the line so you then can enter the third or forth train.

I don’t mean to be rude, but your theoretical “1 monopolistic player” goes against practical experience.

Let’s look at how privatization works in for example Europe.

One (private) company owns the infrastructure (the actual railways) and private companies pay them for usage.

The same private companies make bids on operating a particular bus or rail system, in some cases competition is open, and anyone can start a new long distance bus service.

So you have (for example) Dutch companies running bus systems in the UK or Sweden, or German railway companies running trains in Denmark or the U.K.

Japan has multiple lines that go between the same stations that are privatized. Works perfectly fine there.

Japan's metro is an actual mess. Doesn't work perfectly fine, but it is definitely good for a private network of old lines.

Well Singapore also has privatized subways (though recently one of the operators has been re-nationalized) so we will see what happens as a "natural experiment"

If network time were bid on at time of usage, would that begin to move things in the right direction?

This is the idea of introducing a complex bureaucratic layer in order to introduce elements of competition into the system. In principle it can work, but such a system was tried with British rail privatisation and resulted in lots of situations where the incentives for the private operators led to really bad service on key lines for many years.

Britain is a perverse example of staggering incompetence and corruption.

Argentina tried to do this, lost 20 years of maintenance and development.

And remember that you can loosely correlate price with energy usage. Bid at time of use would be energy hungry. I also have exemples of why consolidated actors coming into a new market is actually worse than a govt taking care of the market, and that's make bidding a poor way of privatizing networks.