Suppose you have ten ideal game-theoretic selfish agents and a pie to be divided by majority vote....

...Every majority coalition and division of the pie, is dominated by another majority coalition in which each agent of the new majority gets more pie. There does not appear to be any such thing as a dominant majority vote.

I suggest offering the following deal at the outset:

"I offer each of you the opportunity to lobby for an open spot in a coalition with me, to split the pie equally six ways, formed with a mutual promise that we will not defect, and if any coalition members do defect, we agree to exclude them from future dealings and remain together as a voting bloc, offering the defectors' spots to the remaining agents not originally aligned with us, for a 1/6 + epsilon share, the cost of the excess portion divided among those of us remaining. I will award spots in this coalition to the five of you who are most successful at convincing me you will adhere to these terms."