The description was intended to be conceptual. You wouldn't want to literally pick a random ballot paper. It's relatively easy to solve in a way that lets you make the system deterministic but based on a secret that is distributed before the election.
Example: You distribute a random seed to each constituency to be opened after the ballots are tallied. When the elections are announced closed you announce a second random seed nationally. You combine these in each constituency to produce a fixed number N between 0 and number of votes in that constituency. You order the votes alphabetically by name of candidate they voted for and pick the Nth. It's deterministic, reproduceable, and fairly stable against recounts because most of the time you'll be deep inside a block of votes.
]]>Basic premise is simple: You take exactly the current British system. Still have local MPs, everyone still votes in general elections and votes for a single candidate, etc. You just change one detail:
Once everyone casts their vote, you pick a voter at random and use their choice.
This sounds like the "Just elect the house by lottery" proposals but is in fact completely different: Your probability of winning the election is directly proportional to the number of people voting for you.
When I proposed it initially I highlighted that it removed a lot of the pathologies of voting - the system has no spoilers, no incentive to vote strategically, etc.
What I've since decided is much more interesting is the fact that despite the fact that it's based entirely on local constituencies, it achieves a very strong form of proportional representation: For any property you care to name (party, religion, race, gender, views on one specific niche issue), if x% of the votes across the country go to someone with that property, you expect x% of the parliament to have that property. This makes it very powerful for getting independents into power (if roughly x% of the country want an independent candidate, roughly x% of the parliament will be independents) and helps parties who are thinly spread over a large number of constituencies.
Hard to say whether it would completely avoid this failure mode, but I think the dramatic increase in diversity of the house can't but help.
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