Further elaboration on the cake problem's discrete case:

Suppose there are two slices of cake, and three people who can chose how these will be distributed, by majority vote. Nobody votes so that they alone get both slices, since they can't get a majority that way. So everybody just votes to get one slice for themselves, and randomly decides who gets the other slice. There can be ties, but you're getting an expected 2/3 of a slice whenever a vote is finally not a tie.

To get the continuous case:

It's tricky, but find a way to extend the previous reasoning to n slices and m players, and then take the limit as n goes to infinity. The voting sessions do get longer and longer before consensus is reached, but even when consensus is forever away, you should be able to calculate your expectation of each outcome...