I'll start with my proposal, and then come back to explain why it is necessary.
If someone joins a group after the start of a level or goes solo during the level, then start a 10 minute timer that is invisible to players. If the group completes the level or activates a party button, then get rid of the timer. Once the timer expires, the group is restricted so that no new players can join, just like in boss levels. Players already in the group and return to Haven, go solo, or stay in the group as normal. Players fighting mobs continues as normal, and they wouldn't be kicked out of the level. Once the group moves on to the next depth or activates a party button, players can join again as normal.
Furthermore, if someone goes solo, the new group inherits the timer from the old group. If you're in a group that has already had the 10 minute timer expire, then you can go solo. But no one will be able to join your new solo group until you finish the level or get on a party button. If the old group didn't already have a timer active, then going solo would start one, both for the old group and for your new group.
The timer would not allow players to do anything that they can't do now. If you lock your party to intentionally not allow people to join, then the timer wouldn't allow them to join a party that they wouldn't otherwise be able to.
In addition, there should be no timer at all in non-combat levels, that is, depths 0, 4, 8, 13, 18, 23, and 29. This system would not impose any new restrictions on joining at those depths.
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First, let's consider the impact on normal gameplay. This would have no impact on the PUG system at all. That only adds players to a group shortly after the start of a new level, which is guaranteed to be long before any 10 minute timer could expire.
Next, it would have no impact whatsoever on groups that don't spend more than 10 minutes between party buttons. That is, it would have no impact whatsoever on the overwhelming majority of parties at the overwhelming majority of depths, as even if a 10 minute timer does start, it would never expire.
Even if a group does spend more than 10 minutes between party buttons or elevators, it usually won't be very much more than 10 minutes. It would be very rare for someone to join a group or go solo near the start of the segment, and then for someone else to try to join through a guild, friends list, or direct invite more than ten minutes later.
Even in those rare situations where it does block someone in the course of normal gameplay, it's probably a case where someone went AFK for a while, and everyone is waiting for him to get on the party button. The late joiner wouldn't actually miss any combat, but would have to wait for someone to come back and get on the party button.
So as you can see, the 10 minute timer mechanic would only rarely impact normal gameplay. People could easily spend hundreds of hours in the game without ever once seeing its effect directly. It's very narrowly targeted.
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After that whole discussion, it may not be obvious to some readers what this is intended to restrict. Let me explain the activity that I want to see stopped.
What happens is that a player will pick an easy level with large payouts (typically an arena in an even stratum; the tier can vary depending on the desired difficulty and payouts) and kill all of the mobs up to the first party button. He'll intentionally avoid picking up any of the loot, however. Rather, he'll go AFK, but not let the game kick him out for idling too long. Presumably this is a simple macro. He'll leave his group open so that guildmates and/or friends can join it. And then he'll stay like that for hours or days at a time.
What then happens is that a guildmate joins his group, and then immediately goes solo. In the new solo group, he can pick up the loot that was dropped by everything up to the first party button, without having to fight anything himself. He then finishes the level and goes back to Haven, ending his group. Because it's a level that gives very large payouts, he gets the large payout for the full level, while only clearing some of it himself.
Then he joins the group with the AFK guildmate and goes solo again. And then does it again. And again. And again. This doesn't just let him do one high-payout endlessly. It lets him get the full loot while only doing the work to clear part of it. That's allowed by the game mechanics as they stand, but surely shouldn't be encouraged.
So how does my proposal break this method of looping? As soon as someone joins the original group once, people can only join it again for another ten minutes. After that, the AFK person will have to start over and get back to the level again. Trying to kill mobs without picking up loot is something of a pain to do. So is doing other depths, and waiting for the "right" level to cycle in. It's would also a hassle to sit in Haven and wait for someone else to do the work to set up looping, and then have to jump on it as soon as he's ready.
Furthermore, the work to set up the looping system would no longer allow a bunch of guildmates to loop the level dozens of times each. They'd only get ten minutes. Maybe they'd get to loop it once. Maybe if they hurry, they could do it twice. But no longer would it be dozens of times. This wouldn't completely kill looping levels, but it would sure make it a lot harder than it is now.
Now, I don't know if dropped loot stays forever. I know that crowns, heat, materials, and vials stay for a long time, at least; I don't think I've ever seen them despawn. Hearts despawn much more quickly. But even if everything does despawn, a group could still loop a level endlessly from the very start, where the AFK person doesn't kill anything before going AFK. My proposal would greatly restrict that. Again, this wouldn't completely kill looping levels, but it would make it a lot harder to do than it is now.
"He'll intentionally avoid picking up any of the loot, however...It lets [his guild mate] get the full loot while only doing the work to clear part of it"
Just a note: What you are talking about there is a bannable offense. It is allowed by the game mechanics, but it will get you banned, like various other things you can do in the game.
Other than that, while looping is not encouraged by OOO, they are well aware of it and have not yet acted to prevent it. In any case I agree with your sentiment that looping should be stopped.