Random voting is stupid for one reason: there is little incentive for any individual to vote for a competent candidate. You have the same chance of choosing the next president whether you vote for Obama or Joe the Plumber. Yes, reasonably-minded citizens will vote for one of a handful of competent candidates, but a vast number of people will simply vote for themselves.
The equivalent in a plurality system would be if all write-in votes were counted together, and then a write-in chosen at random should the write-ins outnumber the votes for any other single candidate.
I guess the "solution" would be to require a minimum # of signatures for a candidate to be on the ballot (and votable). But that's just starting to sound like approval voting, so you might as well just go all the way with it!
I hadn't realized this, and your comment is too late to gain many upvotes, but yes - very good point.
Couldn't you solve this by requiring any randomly-elected candidate to have at least, say, 1% of the popular vote, though? (This means that you don't entirely eliminate strategic voting, unless people can specify a fallback option; but it reduces it by a lot.)
The equivalent in a plurality system would be if all write-in votes were counted together, and then a write-in chosen at random should the write-ins outnumber the votes for any other single candidate.
I guess the "solution" would be to require a minimum # of signatures for a candidate to be on the ballot (and votable). But that's just starting to sound like approval voting, so you might as well just go all the way with it!