Lottery Voting

I’m reading Allan Gibbard’s paper on Lottery Voting, i.e., the “Random Dictator” rule. According to this rule, people vote for candidates the same way they normally would, but instead of the votes being counted, one vote is selected at random, and the outcome of that vote implemented. (“Manipulation of Schemes that Mix Voting with Chance,” Econometrica 45, April 1977). There have been a number of philosophical discussions of the idea over the years–most notable Akhil Reed Amar’s paper in the Yale Law Journal (1984)–but Gibbard’s paper is the central paper on the mathematics of the rule. Unhappily, the paper is very technical, and I find myself stuck at one point in the argument. Does anyone know the paper particularly well? I could use some guidance here.