There's the notion that "playing a beautiful game" is a better outcome than "winning". This can be part of chess too. A child can start winning pretty fast by memorizing opening traps, but that gives small satisfaction. And when playing a beautiful game, you're actually cooperating with your opponent, even while competing; these aren't mutually exclusive.

Think of race car drivers who, of course, all want to win. But winning by forcing your opponent off the track is less satisfying and less admirable than racing "cleanly" (i.e. cooperating with your opponent to prevent crashes).

Competition, within a well defined set of boundaries, is cooperative.

Edit: I'm not sure whether the competitive impulse is a general feature of humans, or just a byproduct of our age, but it nevertheless is a factor currently, so it has to be accounted for in any vision of how we get from here to a more cooperative future. I'm specifically thinking of Cory Doctorow's Walkway, in which people (implicitly, and with some measure of shame) feel a competitive drive to do more for the greater good than others.