Boil it down to it's simplest form. People tend to use what they have around and make a sport out of it. Maybe they have reverted to soccer played with an animals stomach or a bunch of plastic bags rolled into a make shift ball.
The dystopian future you give us has the constraint that there are no more resources and the world is fighting. This means get rid of any sports that require manufacturing (no killer death robots or F1 racing). If most of the world is fighting then get rid of international events and some/most/all government organization.
Sports would probably be very similar to sports you would find in the third world / developing countries (don't know which is the pc term). Simple pick up and play games that are not much different than anywhere else other than nothing is manufactured. Kids grab anything and start playing (a pipe or stick makes a good bat).
Since we have not enough constraints I'll just throw one of many possibilities out there.
Because safety in sports is almost directly linked to manufacturing and manufacturing is no longer used for recreational reasons (my assumption) sports are now much more violent. As safety products stopped being used people didn't adjust the intensity down but like normal humans started increasing intensity.
Almost all sports started degenerating into a form of gladiatorial game where full contact was allowed and fighting was encouraged. Jake Dow was the first world champion in chess to win by knock out.
Not long after, sports continued with the flow and some sports are now glorified versions of Russian roulette. Players pay an entrance fee to compete and claim a portion of what is left from the players who didn't make it.