In order to understand game design, you must first understand play.
Play is experimentation. The concept of playing with your food and the concept of eating food are distinct because playing with food involves using it not for its primary purpose, but experimenting with alternate purposes. Building a tower out of mashed potatoes is an experiment that shows you the structural properties of mashed potatoes; rolling carrots around on your plate shows you how their conic shape leads to a unique rolling pattern, and how their irregularities in form lead to them rolling at different speeds and bumping around in different ways. Once you establish the physical properties of the carrots and mashed potatoes through experimentation, you can continue your play by further experimenting with extremes: how high can my mashed potato tower get? Can I get my carrots to roll to this specific spot on my plate? Experimentation is fun, and we have to be stopped from doing it by force--either by force of the displeasure of your parents, or by physical restraint--or complete (enough) understanding.
Play is a natural behavior exhibited in mammalian species. There are myriad reasons why those who tend to play would not have been evolutionarily weeded out of the gene pool. Compare the fitness of two primordial people: one of them lazes around when not hunting; the other spends a portion of his non-hunting time throwing spears at a circle he has carved into a distant tree. Who will be more socially prepared and well-adjusted: children who play at being mothers and fathers and mime responsibilities they'll have in adulthood, or children who spend that time eating or sleeping or sitting quietly? Play prepared our ancestors for the rigors of life, both social and physical. Play seems to fill the role of simulating future experiences so that those who perform well in play will perform better than average on those activities when they must be done for real.
Children will enjoy playing solo when they're learning about their environment and capabilities. They will play to test their abilities and mimic the behaviors of those they look up to. Play is often free-form and solo, but can involve cooperation, even at a very young age. Usually the play of children is not obviously confined by any rules but those of the physical universe. Rules become apparent and emerge naturally when play becomes a social activity. Children arrive at rules while playing in order to express their will and establish conventions so that the play of other children can interact with shared resources amenably. Rules are social conventions. Rules mentally communicate the imaginary "laws of the universe" that the child tacitly invents for his toys. Now, of course, the child won't necessarily keep these rules constant within a play session, but clearly the child's decision of who can and does do what action isn't entirely arbitrary. In play there seems usually to be a sense of world-modeling: the child represents in his play world ideas and processes he's been exposed to, often recombining those ideas in novel ways instead of merely repeating them as practice.
Social cooperation leads playing children to attempt to establish rules, but these rules aren't all we need to arrive at what we would today call games. In order to solidify the experience of play within a system of rules, those rules need to be institutionalized and recognized by players--not merely ad hoc created to resolve social conflicts as they arise during play. In my next post, I will address this process of institutionalizing play, and discuss how an individual game is born.