Anarchy-No rules. Society works on peer per pressure and people being both 100% smart and somewhat nice-Crazy.
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Socialism-Everything is owned by the government. You get your stuff from the government. The people are supposedly the government. All hail the chairman-Not crazy per say, but I do not think it is a good idea.
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Communism-Anarchy take two. There is no need for rules or laws because everyone will help each other and pool resources communally-Crazy.
That's Nazism(sorta)/Stalinism you're describing there as Socialism, which is something 19th century socialists were dead opposed to. The Socialist/Communist term was appropriated in the 20th century by totalitarians as a facade.
"anarchist ideals" of individual liberty ("no rules") are not the same as "anarchist political ideas" which are about society as an amalgam of voluntary-membership democratic local councils. e.g. participatory democracy. There's still regional councils made up of delegates from the smaller councils etc, so there is a definite structure, and a natural market economy. This no more or less requires us to trust our neighbour than any other similar system, such as representative democracy (though anarchism still uses representative democracy at the regional and higher levels). Conservatives actually used to claim that giving the common man the vote would lead to stuff like killing in the street, rape
I would say that society works on peer pressure right now, we delegate much of that pressure to a government. So it's a matter of can you trust the appointed bully or not?
Communism (Marxism) had two stages. I'll explain my understanding of Marx's thinking:-
1 - Socialism. There's a constant struggle between "owners" and "employees". If the employees were themselves the owners, then where's the conflict? Marx's first idea is for the employees to become the owners/shareholders themselves (on a local company level which he called a Commune), and not be exploited by an "other" class. They would still be part of a market economy, and market incentives still exist with profit shares, competition between the communes etc. Central wage controls and taxation rather defeats the point of the whole exercise. Note the Unions would cease to serve any purpose under true socialism, so some union leaders have been caught working with the ruling class over the years, in a form of self-preservation.
2 - Communism. Marx's second target is the dual hierarchy of economics and politics. The collectives created in stage 1 gradually take on the role of organizing society. They co-ordinate their actions through councils of delegates etc to a senate. This is certainly not "anarchism" in the "no rules" sense, nor is it the end of free trade (the communes will have to trade etc)
All these theories were formulated in totalitarian 19th century european states, basically they're just different ideas about how to organize a democratic state.
I find it scary that what you label "Socialism" - a form of Nazism/Stalinism by your description is not
utterly rejected by you (you're lukewarm on it), whereas systems based on local councils you find totally repulsive.