This would be where players can compete against each other not in PvP, but in PvE. My idea is essentially, clockworks/mission racing. say you're doing a vana run, and your friendly SK rival, or a friend, guildmate, or just a random player, is also doing a vana run. And you want to see who can successfully complete the mission faster. In this case, when you go to select a mission, there is an option "compete". when you click it, you can either type in a knight's name who you wish to compete against, or choose the option; challenge random. in the case where you type in the knights name, they are sent a challenge request, which they can accept or decline. If they accept, then they are sent to a mission lobby of the same mission and difficulty as the one who challenged them. the competition could be one on one, or you could organize teams (they numbers wouldn't have to be even, but they could). the missions start simultaneously when everyone on both parties are on their respective elevators. After that, you do your mission like you normally would, accept there is an icon showing you how far the other team is into the mission. if anyone leaves either party at any time, they leave the competition, but the party leaders will always be competing against each other. whoever finishes the mission or exploration first wins. if one team's party leader quits the mission, the other team will win by default. This concept works with clockworks explorations as well, in which case whoever delves the furthest wins.

I also thought that people could watch teams compete. like, someone is hosting an exploration tournament, in which the winning team gets a reward. This ties to a previous post of mine:
http://forums.spiralknights.com/en/node/108041

Please let me know what you think!