The developers can host a few servers, sure, that’s an option. If that’s the method they take, they also release what’s known as a dedicated server utility, that allows anyone to launch a dedicated server on their machine, or to rent out a server in a hosting center. You can find this model in games such as Counter-Strike, Quake, Unreal, and some of the Battlefields.
This allows for the community to self police, and people will naturally end up in a community that fits their preferences, and rude or toxic players will quickly find themselves banned from the majority of servers and be forced to change their behavior or play a different game. Players can modify server settings, or make entirely new game types that the developers may not have thought about or wouldn’t have the resources to create, and people can create tools that allow servers to easily moderate their servers, and elect moderators and admins from within the community for when they’re not online. This also allows for developers to negate the need to be able to host millions of players, and when the game dies, if it does, all they have to host is a Master Server list.
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Another option, especially for games with small groups of people is to allow the game to be hosted live by one of the players in the squad or group. This is called peer-to-peer servers. In this case, and can either be done by “hosting” the game server and waiting for or inviting players, or by having the game monitor latency and automatically migrate to the best host based on connection and distance. Deep Rock uses the first of these two options, whoever starts the game becomes the host, and stays that until they close the server or quit the game. In this instance, devs host no servers except the master server list, allowing even the smallest of devs to be able to handle millions of people playing their game simultaneously without any real increase in their server costs.
Typically, for smaller squad based games, like Deep Rock, this is the better option, while for larger player per match games like battlefield, the former is the better option. In both instances, players choose from a list of available servers in a menu and load in from there. You can check out Deep Rock Galactic or the Diablo 2 Remaster to see what a server list looks like.