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The superiority of competition or cooperation is a debate that spans the left and right, individualists and collectivists, socialists and capitalists, and beyond.  But even if these are two opposite ends of the spectrum of human interaction, it is not the absolute pure ends that anyone advocates.  Free-market anarchists argue that the ethical element of the debate is over voluntariness.  It is peaceful competition and peaceful cooperation vs. coercive competition and coercive cooperation.  

Unfortunately, as simple and as clear as I think this solution, people here our argument and then go back to arguing against competition or cooperation.  Perhaps a reframing of the issue can help others share the libertarian epiphany.  

The most unrestricted level of human competition is unlimited war.  Within this state there is no obligation to grant mercy, accept surrender conditions, fight fairly, limit pillaging, rape, or torture, or other malevolent acts.  Sure, this may be called pure competition, but it is not a competition meant to allow anyone to enjoy life or have any liberty, especially in the long run.  It is also worthwhile to notice that at this extreme, there is no obligation to limit such acts to opponents in such war.  One could equally turn on allies and do the same to them as to enemies.  There would then be no safety in numbers where there is no honor among thieves.  

The next step up from this abyss is the limited war.  In this stage, both sides cooperate by agreeing to certain rules of war.  They may be quite ruthless and kill thousands or millions, but they do gain a little security in knowing they are less likely to be tortured for pleasure, and receive certain rights in case of surrender.  It is not just the weaker party that benefits by the cooperation of limited war relative to the unlimited war.  If the likely victor can assure the likely loser that the loser will "only" receive some specified maximum punishment if he surrenders compared to expectations in unlimited war, then the weaker party may surrender.  In the previous state, no such assurances would be trustworthy.  In this case, the loser receives less harm than otherwise expected by surrendering, and the victor obtains more power than if the loser would have fought to the death.  

The second step up from this is that of the slave caste of a society/state not at war.  The slave has his liberty restricted, and some portion of his labor is taken from him and is "paid" a certain allotment, neither of which he can control.  The slave-owner is free to maximize what he takes from the slave, and will then provide only so much as he thinks will maximize the net of slave profit vs. potential cost of slave rebellion.  This system is still better than the previous one.  From a moral perspective, there is an increased restriction on injustice, and from a utilitarian one, there is less hindrance of capital growth.  

The modern nation-state citizen is not really a separate step from such outright slavery in the examples above.  Rather, it is more of a fulfillment of these steps.  The most powerful coalition in each step realizes that granting a little more liberty, citizenship, and rights to the weaker classes both allows and encourages them to be more productive so that the powerful can extract more money, profit, goods, or labor from them.  But at this level, this system reaches maximum profit for the ruling class.  It will not voluntarily advance to another social structure that would increase either net freedom or profit to the whole society if it comes at a cost to the ruling class, because the next step up involves the concept of restitution, which the ruling class does not want to pay.  

Considering other combinations of competition vs. cooperation require stepping into a completely new series with no natural or pareto-optimal passage from the former stages - unless the rulers were to adopt a more libertarian moral code and pay for previous acts.  

The new social structure could be considered unrestricted anarcho-capitalism.  It is not pure competition because it begins with the foundation of cooperation by acceptance of anarcho-capitalist law.  Competition that would harm another person, their property, or their equal liberty is prohibited, but every other dog-eat-dog competition is allowed.  This is a strong "buyer beware" society.  A used car dealer sells a car "as-is," you buy it and discover it is a lemon.  Tough luck.  Everything is sold "as-is" with no guarantee unless otherwise stated.  You can't trust anyone without a contract.  Everyone tries to get every penny of profit from everyone else.  Everyone spends a large amount of effort to evaluate the maximum the market will bear for his goods and labor.  Every stupidity, ignorance, or mistake is "exploited" to the extent that does not violate the rights of the system, and so people must try very hard to avoid such receiving such exploitation.  

Instead of taking a one-step up look at more cooperation, compare the final end of pure cooperation.  Everyone always tells not only the truth, but also every relevant detail with no trade secrets.  Everyone tries to work equal amounts of labor and take out equal amounts of value.  Minimal time is spent in pricing goods because the profit (the difference between each party's subjective values) is split about equally in any trade.  Given such, prices are not precise measures of value.  

Now these two examples may seem like opposites, but neither is accurately representative of anyone’s complete belief system.  The anarcho-capitalist situation above is merely what boundary the market anarchist believes should be punished as a crime.  It does not imply endorsement of the vices that are not crimes.  Even market anarchists practice every day the maximum cooperation situation above . . . in marriages.  Every successful marriage (assuming a voluntary marriage of equals) requires that each spouse give and take about equally, act truthfully, accurately represent the labor they contribute and the expenses they create, etc.  

The problem is that such cooperation is oftentimes hard.  We may think that our labor is harder than our spouse's.  We may think that some of our spouse’s labor is more leisure than real labor, a different degree of “toil and trouble” than we do, have a different time preference towards thrift vs. current consumption, etc.  Partners either struggle to evaluate each other's net labor contribution to the marriage, or they just trust the other person, which tempts the other to free ride off of the other.  When one spouse feels such, there is distrust and frustration.  Yet, these are the challenges for pure cooperation for just two people who spent much time in trying to find the perfect mate with compatible goals.  Marriages fail all the time, and even frequent sex as a reward for success is apparently not enough to keep many from failure.  As the size of a totally communal/cooperative group increases, the difficulty of efficiency increases exponentially as it is harder to reach unanimity on goals and evaluation of the use-value of each other's labor.  

So what degree of cooperation is ideal?  The near tautological answer is to cooperate to the degree the expected benefits of cooperation outweigh the expected costs.  The real world answer is that we cooperate with those we trust, but only to the degree the trust will not tempt them to free ride too much.  We cooperate least with those who have goals contrary or hostile to our own.  I hope you cooperate minimally with any nation-state because they do not cooperate with you except with you as citizen-slave.  We cooperate more with those whom we have ideological, religious, and ethical agreements.  We cooperate more with someone whom we can hold responsible and see every day than the transient stranger.  I will cooperate more with someone who, when in trade with me, seeks to split profit (or "surplus value") equally with me than with someone who wants to try to get every penny of profit from me he can.  

In short, no one should attack market anarchism for opposing cooperation.  We don't.  We don't care, at least from a legal perspective, if you want to form a commune where everyone is married to everybody in complete and permanent income sharing.  Consider the debate about competition vs. cooperation as one of minimal detail for every individual to work out in each of his own interactions with others.  Outside of cooperation on the minimum of opposing crimes, it is not a significant matter of political theory.  It is those who want to coerce cooperation that should realize that doing so takes your beliefs out of the voluntary sphere back into the hegemonic social stages of unlimited war to class war.  

Nevertheless, there is a point where such issues appear to make a difference.  It is not an issue of principle, but of the temptation to sell out our principles.  When anarcho-capitalists sell out, they sell out in favor of state-capitalism, and when anarcho-socialists sell out, they do so in favor of state-socialism.  We shouldn’t fear only the other side selling out, but proactively work to keep our own sides from the temptation of the ballot, the parties, think tanks, academia, trade unions, corrupt businesses, or whatever may tempt us.  Speak out harshly against the state today so that they will not want you in a future moment of weakness.