Does society place more value on competition or cooperation.
I don't think there's a more important question than this, but it needs to be rephrased, because there's no such thing as a single, all-encompassing human "society" and never has been. The biggest form of human association we have today that could conceivably be called a "society" is the nation-state (a "country"), which first appeared only about 400 years ago (the passport, for example, was first invented in the 17th century).
With that said, the answer to the question is this: Over the roughly 200,000 years that fully modern people (homo sapiens sapiens) have been around, the relative importance of competition vs. competition has varied very widely, through time and from society to society. Physical fossil, archaeological, and written evidence, along with carefully-developed and -refined hypotheses to explain this evidence, suggest that the balance between competition and cooperation that had been developed over most of those 200,000 years within the typical societies of most of those years ?that being a group of 20 to 40 individuals ?began to be disrupted at, or shortly before, the end of the last glacial period (about 11,000 years ago) when climate change (and changes in plants and animals accompanying it) led to the discovery and development of agriculture, which led, over the next 5,000 years, to the human invention of "improvements" in farm land ?the key example is often thought to be irrigation systems ?with such improvements either leading to or supporting the entirely new concept of the ownership of land. The concept of land ownership seems to have been well-established at, or soon after, the beginning of "civilization" (literally" "cityization"), defined by very small cities (sites of the food-storage that agriculture enabled, and eventually of palaces, temples, the offices of managing bureaucrats) that were surrounded by and supported by the agricultural fields. In such places ?which by 5500 years ago had become the first true states ?the balance had unquestionably shifted in favor of competition: a managing elite minority competed with a physically-working majority, and won the competition (part of the time at least) by either compelling or persuading (or both) the majority to give part of the (literal) fruits of its labor to the ruling/managing minority. A thousand years later, at the beginning of the Bronze Age (bronze being the first metal used to develop superior edged weapons, by those who had access to the materials for making it and knowledge of the technology) aggressive warfare for profit (as opposed to border conflicts) began, and one result of this was a further tilt in favor of competition, in the form of the routine enslavement of prisoners and their use as "instrumenti voci" (a Classical Roman term for slaves, which means literally "talking tools"). Within this framework of "civilizations" ?which co-existed with (an increasingly-small world-wide number of) band societies (most of them hunter-gatherer: food-collecting) ?until less than one hundred years ago (! : a very few remain, but are, as far as we know, at the point of disappearance)! Within both the group of societies at all the various stages of "civilized" development (from primitive agricultural to advanced industrial), there's variation from society to society (and also within sub-societies within the society: religious or utopian or specialized-industrial or academic communities, for example) of the degree of competitiveness versus cooperation. Just the widely differing North American Indian societies that existed at the time of Columbus's arrival ?the "500 Nations" ?displayed a range of competitive versus cooperative behaviors, including one that gives an example of the difficulty of precisely defining competition versus cooperation: the fierce competition to be the most generous giver to the society known as "potlatch". I think the real answer to your question lies in understanding why small societies were successful for a total of more than 3 million years of human-ancestral (hominid) life, and 195,000-plus years of completely modern human life; and why the much larger societies that began to appear six thousand or so years ago have begun to seem to be wildly unsuccessful, both in terms of the human happiness they provide, and in their adaptiveness to the continuing existence of the human species. Thinking about this shouldn't produce despair, because movement through time ?deep history ?always proceeds in a wave-like way, with experimental excursions from side to side, some in maladaptive directions that are then followed by corrections, always accompanying forward progress.