As Alfie relays, the idea would be to come up with structural frameworks that don't require competition in its ancient form. A way of conflict solution that does not involve mutually exclusive goal attainment, namely finding a way that me reaching my goals does not come at the expense of you meeting your goals (as is sadly still the standard in our "economy" today).

Your question is very very important though. From my own experience I have to say there aren't many things I could name at this time that come in a non-competitive form. The closest thing would be a good dancefloor, where there is no competition and where we reach states of consciousness and happiness because we simply allow everyone to be exactly as they are without ordering them around or judging them for their choices unless it violates our own choices and well-being. It is the greatest blast ever living for such an environment in a week, where everyone shares not because they HAVE TO but because they choose to.

As for economic models, there are great ideas put forth by Franz Hörmann for example, years ahead of its time. Now that I mention it, I should really write a post about it, not sure there is anything yet in English language space (I feel a new dub coming up ;))

To put it simply: There is no neeed for the maintaining of artificial scarcity in our structures and systems today other than conditioning. If we used technology for overcoming scarcity (everyone to the degree he freely chooses, no obligations) and we would get off the hoarding mode, scarcity and hence competition would be overcome all by itself in a generation. That said, things like Chess, soccer or whatever would probably still be enjoyed by those who dig it, simply because they voluntarily choose to do so. Like some people still dig to act out medieval battles in roleplay or whatever ;)

The issue with competition is that we have swallowed the idea that it is mandatory in order to survive, which is nothing but an ancient hangup humanity has yet to see through. 

The healthier and happier your neighbors are, the less you need to protect yourself from them - it's very simple. We just have to find a way that this can develop organically without any coercion or mandatory guidelines (because *that* would be 1984 and soviet communism all over again).

There are many paradigms in between we never hear or talk about, and blockchain would be one of the many actual pathways of finding a way there. 

If you haven't, listen to the lecture. For me, it will address a lot of misconceptions I had before getting into this topic with an open mind. I always liked to win, but never had considered other possibilities that don't involve the others loosing. Multiplying abundance is so much more fulfilling than to hoard and win at other people's expense, we have just never been shown that that is an option.

Inventing new systems, games and approaches could become some of the most important tasks of our generation(s).

Thanks for your comment!