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Welcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: Voting Theory Introduction, published by Scott Garrabrant on October 17, 2022 on LessWrong. Sequence Introduction This is the first post in a sequence in which I will propose a new voting system! In this post, I introduce the framework and notation, and give some background on voting theory. In the next post, I will show you the best voting system you've probably never heard of, maximal lotteries. (Seriously, it's really good.) After that, I will make it even better, and propose a new system: maximal lottery-lotteries. Then comes the bad news: I can't prove that maximal lottery-lotteries exist! (Or alternatively, good news: You can try to solve a cool new open problem in voting theory!) Thanks to Jessica Taylor for first introducing me to maximal lotteries, and Sam Eisenstat for spending many hours with me trying to prove the existence of maximal lottery-lotteries. Generalizing Voting Theory A voting system is a function that takes in a distribution on utility functions on a set of candidates, and produces a distribution on that set of candidates. This is not what a voting theorist will tell you a voting system is. That is because they like to make a bunch of extra assumptions: The set of candidates is finite. The function only uses the preorders on candidates implied by the utility functions. The output distribution assigns probability 1 to a single candidate. The input distribution is the uniform distribution on some finite set. I am going to play along with some of these assumptions, but I want my types and notation to treat them as explicit assumptions, so I will use notation that will make it easy to remove these assumptions as needed. Technically, I also make one assumption that voting theorists usually don't make. That is that the voting system is homogeneous. Homogeneous voting systems are only a function of what proportion of voters have each preference, not on the absolute number of voters that have each preference. Thus, when I said the input was a distribution on utility functions, rather than a multi-set of utility functions, I threw out the information that would have allowed for non-homogeneous voting systems. I don't know of any seriously proposed voting systems that are non-homogeneous, so this is not a significant extra assumption. Throughout the sequence, I will take for granted that all voting systems are homogeneous. Assumption 2 is most likely to be violated by voting theorists, as in approval voting and range voting. Next is 3, and there is a small subset of voting theorists who think about non-determinism. Assumption 4 is not very important, and while it is usually made, it also usually does not matter. Assumption 1 is almost never violated, or at least when it is, the field is called something other than voting theory. A utility function on a set S is just a function from S[0,1]. I will write Δ(S) for the set of distributions on the set S. (I'm not going to worry about the sigma algebras; usually it will be obvious/not matter.) If f is a voting system, C is a set of candidates, and V∈Δ(C[0,1]) is a distribution on utility functions on C, I will write fC(V) for the output of the voting system on V, so fC(V)∈Δ(C). Some Important Criteria To get more comfortable with this formalism, we will translate three important voting criteria. Condorcet Criterion If a candidate would defeat all others in one-on-one elections, that candidate should win. Translated to our formalism, f satisfies the Condorcet criterion if whenever there exists a c∈C such that for all d∈C, we have Pv∼V(v(c)>v(d))>12, we havefC(V)(c)=1. We can think of Pv∼V(v(c)>v(d)) as saying when we randomly choose a voter v, what is the probability that v prefers c to d. If this probability is greater than 12, then a majority of voters prefer c to d. Consistency Criterion If two disjoint el...