I'm not a psychologist, but am very interested in psychology and politics and am constantly contemplating questions like this one. So I think your story and question is very interesting.

What I believe to have discovered so far is what you yourself also already seem to allude to: Competition occurs both because of environmental factors and because of genetic predisposition.

The most commonly accepted reason (and also the most likely one) why bonobos are much more peaceful and are much more socially egalitarian compared the common chimpanzee is because of their constant access to and the availability of an abundance of nutritious vegitation in their natural habitats. Or in other words they don't know scarcity, while the chimps on the other side of the Congo River do.

This does not mean however that if we completely eradicate scarcity people will just stop competing from one day to another. Some social hierarchy even still exists between bonobos, but it's logical to assume that given enough time, the less scarcity, the greater the chance our drive to compete will begin to fade. But a dog which has been bred solely for the reason to fight other dogs will statistically be more likely to show agression and competitiveness even in an environment where it has no apparent reason to do so, because of its genetic makeup.

Considering that the systems we live in and take part of are for the most part far from egalitarian (unfortunately), and things like competitiveness and greed are encouraged from a young age (even if it's at the cost of someone else's wellbeing) and scarcity is being artificially maintained (so the most powerful among us can become even more powerful even though it means that the people who are less well off are going to be exploitend and are going to be the first to bare the burden and thus risk losing the little they have left), we remain like chicken in a henhouse: pecking each other to death for fear of the other chickens laying more eggs than we do - meaning that if we don't peck those other chickens to nuggets it will be probably be us who are the first to end up in the frying pan.

So yes we are breeding ourselves a dangerous species this way and should we ever be able to establish a world wide society devoid of scarcity and a system at least resembling egalitarianism at least somewhat, like with dogs, it would probably take a couple of generations before our genes begin to adapt and people feel less of an urge to beat "elderly" people swimming laps in the swimming pool. Even if those genes would still exist, those genes would probably remain dormant in such a probably more ideal society.

I'm not against competitiveness however and think that it can also be a good thing. Even though I'm of the opinion that all the good things we achieved as a species, we've achieved mostly because of cooperation, this does not mean that I think that competition cannot also lead toward progress. It obviously can. What I think to be important though is that if we compete, we should compete consensually and in a manner of good sportsmanship and under no circumstance should my "win" ever be a threath to someones life or livelihood.*

But all this aside.. playing games can be fun. Even if I know I'm probably going to lose this does not mean that I might not try to challenge someone to a friendly match, and if all I intend is to see if I can swim faster than the guy who's swimming next to me, out of sheer curiosity, then there is hardly any malice involved. I might find reasons to admire that guy regardless of whether I win or lose. This is probably not exactly the same situation as you present it in the story above... but still... kids be kids? maybe?

*At least not in the society I imagine since in our current state of affairs regular people often don't have much of a choice if their desire is to survive.