I've made my mind up on the setting now. The date is thousands of years in the future, where Mankind has discovered that there are endless alternative dimensions that can be accessed with relative ease, differing from their reality and timeline by varying amounts. Travel to these dimensions is easy at any point in their timeline, but, unfortunately, while the physical bodies of people survive the trip without harm, their minds do not.
However, another invention allows people to project their minds through to the other dimensions and take over existing beings, which protect the mind inside their bodies. This kind of 'body-surfing' has become immensively popular, allowing those with enough money to pay for the trip to do anything they wish and could not do in their world. The ethics of this are questionable, but hey, they're not real people, right?
While officially illegal, body-jumping dens are easy to find for those who know where to look. Recently, a certain corporation has offered to host a contest broadcasted live to all around inhabited space outside government space, the reward more money that the average citizen is unlikely to ever need...
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So the players aren't psychics anymore, but the main idea is the same. The players will try to complete the three goals assigned to them by the Overseers, gaining one point per completed objective. If someone completes all three before the end of the 'episode', he's sent to the next world ahead of anyone else. There is a time limit otherwise. The Overseers can transport over physical items, such as weapons or vehicles, or punish the contestants, according to audience vote.