No closed society can sustain anarchy for prolonged periods of time

I would like to open this debate by defining the terms that the debate is based on and explaining my exact position. It is my contention that any closed society based on anarchy cannot continue in that state for any prolonged period of time. First, a few definitions, and then we can proceed to argument. Anarchy: the absence of government (courtesy of merriam-webster online dictionary; I realize that there are other definitions, but this debate is about this specific definition) Government: we define "government" here to be a centralized authority within a society that sets rules that govern the society as a whole and that has the power to punish in some form the members of society who break these rules. "Closed" in this case means that the society does not recieve any additional citizens from elsewhere and cannot send its citizens out to governmental societies where they will be accepted. An anarchy-based society is a society that lacks the centralized authority I described as "government". Therefore, there are three possibilities: (1) the society simply lacks rules; (2) the society is conditioned so that everybody would work with society's best interest in mind; (3) society would informally and collectively punish 'transgressors' (of universally understood and unofficial rules) without any need for centralization. I would like to address each of these three possibilities seperately and demonstrate that they indeed must lead to the creation of a government. (1): If the society simply lacks rules, there would either be no incentive not to immediately turn on your neighbors for personal benefit, in which case society would collapse and splinter from the infighting (for evidence, see what happened in the Montreal Police Strike of 1969; after one night of strike, mobs had rampaged downtown causing $2 million (10.5 million in 2005) dollars worth of damage - although this might not seem like much given the circumstances, remember that this was after only 16 hours of government non-control: http://archives.cbc.ca...), or one would have to value their neighbors enough not to turn on them. However, there will always be some taking advantage of some people by others, and only loss of trust and personal watchfulness will stop that, leading to groups of people with mutual trust forming within the society. These informal groups would develop their own rules for conduct, thus establishing governments within the society and fracturing the society into a set of mini-societies, each with its own government. (2): This state of affairs is impossible unless free will is removed, in which case I contend that it loses the aspect of a 'society', the same way that a hundred different computers networked together is not a 'society'. If free will is retained, then at some point (statistically speaking) someone will take an action for reasons other than the overall benefit of the society. (Removal of free will removes the entire point of anarchy as well, although this does not directly relate to my argument). (3): If society could collectively punish transgressors, then society by definition must have rules, whether formal or informal. However, any set of rules has gray areas; certain actions under certain conditions might remain doubtful, or it may not be completely clear what someone did and therefore also whether that person broke society's rules or not. If society has a clear and consistent way to deal with these gray areas, then the administrators of said method are by my definition the government. If not, then society will remain divided in its punishment of said individuals. Consequently, those individuals would be punished by one side but not by the other. If this punishment is forbidden except as punishment under the rules of the society, the portion of society that believed that the protagonist was innocent would view the punishment meted out by the others as 'illegal' by their definition and would punish the punishers. The original punishers would retaliate, leading into a feud that would either (depending on the nature of the punishment) split society into two pieces (which would then split again or develop a government, by the exact same argument) or dissolve it entirely. If the act that is used as punishment is allowed in uses that are not explicitly punishment, then it is either completely ineffectual or will be abused by the citizens (they will 'punish' people not for breaking a rule, but simply for dislike or for personal revenge) prompting either the creation of a controlling body (a government) or the continued dissolution of the society. Therefore, a society based on anarchy will always either splinter into smaller societies or a government will emerge and take control of the society.