The baddges are for gyms, but these gyms have not been as successful as our previous gyms had been. If you win 4 badges you can be a gym leader. The gym leader has to use at elastthree of their type pokes, it does not have to be a mono type gyms.
We have three other battling type games. Two have the same rules except one is for wifi and one is for showdown. They are king of the Hill and Mayor of the hill. They are like an ongoing, self administerred tourney in that the king or mayor is the previous champion. anyone can challange them to their favorite tier or mode of play (sinlges doubles), and if the challanger winms they are the new king or mayor. If the king does not battlewithin 48 hours, they are kicked off the hill and any two people cna battle for the title in an OU match
The third battle game fizzled out - subway matches. See that thread for the rules.
As far as tournaments, we are trying to get one going based o nthe card game war where if you win, you have to switch out any pokes that feinted for your opponents for the next round. We had a previous tourney like this that was loads of fun, but partiicipation was low.