From your description and the fact you've said no democracy most people are picking up the anarchy angle. There is another option, direct democracy.
Any time a group of people with differing options have to agree a course of action, either the whole thing falls apart or direct democracy applies. The group come to an agreement by voting on which course of action to take.
As with anarchy, this falls apart with groups over a certain size without leadership. See Dunbar's number for details on your approximate limits for anarchy, direct democracy can cope with larger groups, but not much larger without developing a bureaucracy.
Even people who are all attempting to work towards the common good, will sometimes disagree on what the common good consists of and/or what course of actions best achieves it.
You will need some form of democracy simply to allow dispute resolution, otherwise any dispute will eventually lead to either the collapse of your civilisation or a return to dispute resolution by combat in other words, collapse of your civilisation.
As has already been mentioned, anarchy works as long as there's no more than one anarchist in the group. That's not entirely true, it works as long as there's no more than one opinion in the group. So small groups only.
What is the group though?
Family groups will naturally tend to have a leader, a head of the family, whether the senior male (at least he thinks it's him) or the senior female (more likely in practice) there's one person who is in charge. Under that person's instruction everyone acts towards the common good of the group, the family. This is patriarchy in its purest sense.
Groups of peers will often develop a natural leader, as we all know there's one person who sees the washing up before the others, and at least one who just can't see that the bin is full. A leader will step forward to help the group stay cohesive, meritocracy.
Sometimes the group will more directly acknowledge that a leader is required, someone to coordinate the cleaning and bin schedules and they appoint someone to deal with this, democracy.
Occasionally one will just be making more money than the rest and end up with the largest room and paying more of the bills to help the others out and eventually you end up with oligarchy.
It's actually very hard to maintain anarchy without slipping into another form of "government" within any group of people. The larger a group gets the more likely you are to end up with a leadership group of some sort, whether natural leaders (they do exist) or natural busybodies (them too) leadership is inevitable.
Whoever had created humanity had left in a major design flaw.
It was its tendency to bend at the knees. - Terry Pratchett, Feet of Clay
In short, any group of people will tend to appoint a leader of some sort quite quickly, if one is not appointed, one will arise.