In The Vault your a new recruit and your thrown into a training arena against robots. You are given specific objectives that you need to complete within a certain time frame. The robots you face Come at you from all directions. It's quite intense.
In Bullseye your given weapons training on a range. You square off against cardboard or metal targets. Again, you are timed. You have to hit stationary, moving, concealed, and "hostage" targets.
I'm not entirely sureness Running Man is about as I haven't done it yet. If I had to guess though it probably has to do with running through an obstacle course while avoiding robots.
The game is still currently in development but from the demos I've played it's going to beawesome. They are currently working on the Multiplayer demo. From what I've seen it might be three player co-op and you're given a choice of the marines or the opposition. Again I'm not sure about this.
Check this game out. I thought I saw somewhere that it major also end up on Kickstarter.
http://www.interstellarmarines.com/