Board games just aren't enough for these super athletic people so they also have their own little sports days. Except just like them, the sports are also on steroids.
Normal sports just aren't enough for them, so from the early days of hyper-energetic Robin Dick to Jason Robin contributing his ideas to Nightwing to whatever their crazy ideas have built up into now, they've created an entirely new mashup of sports complete with a strict set of rules. Basically it's the culmination of Dick's crazy idea being added onto by all of his siblings at some point or other until it turned into this.
Before they just named it BatBall - patent pending - it was called Hockfoot-bolleyccer but that was a mouthful so they changed it. Essentially it mushes up the rules and play of (field) hockey, American football, soccer, basketball, and volleyball. They just play with a normal bouncy ball so it works to some extent.
The way it works is that if the ball bounces, the rules of basketball apply, if it's in the air then volleyball hits work, if it's on the floor then using your foot or a hockey stick both are valid and tackling is allowed at all times because why the fuck not? Basically because everyone's good at different sports, it's really just a chaotic free-for-all and any of the batfam (and for one day a year it's extended to their superpowered friends with adjusted rules). One constant rule though is that there is always gonna be extra points for showing off, so if you score in an unusual way like kicking it into the goal while doing a handstand or something will get an extra point.