All of these theorems show how one or another single-winner voting method can be regarded as the "optimum" single-winner voting method under some particular (perhaps somewhat artificial) quality measure. All have very easy (usually trivial) proofs, but I nevertheless think they have some interest.
Quality measure A: let A be the probability, after some election system elects W, that a random voter ranked W top. Supposely the greater this probability is, the "better" the voting system.
Theorem A: The voting system which maximizes A, is the Plurality voting system.
Quality measure B: let B be the probability, after some election system elects W, that a random candidate R (R≠W) would be preferable to W in the view of a randomly chosen voter V (based on the ballots as cast). Supposedly, the smaller this probability is, the "better" the voting system.
Theorem B: The voting system which minimizes B, is the Borda voting system.
Quality measure C: let C be the probability, after some election method elects W, that a random candidate R (R≠W) would beat W in a head-to-head simple majority voting contest (again based on the ballots as cast). Again, supposedly, the smaller this probability is, the "better" the voting system.
Theorem C: The voting system which minimizes C, is the Copeland voting system: voters supply rank orders of the candidates as their votes, we compute the pairwise table of who beat who pairwise (ties count ½ point, victories count 1 point), and the candidate with the most points wins.
Quality measure D: let D be the probability, after some election method elects W, that a random voter would "approve" or "consent to" W (again based on the ballots as cast). Supposedly, the greater this probability is, the "better" the voting system.
Theorem D: The voting system which maximizes D, is the Approval voting system.
Quality measure E: let E be the probability, after some election method elects W, that a random voter who generates a random number R uniformly in the allowed-score-range, had scored R above W. (Supposedly, the smaller this probability is, the "better" the voting system.)
Theorem E: The voting system which minimizes E, is the Range Voting system.
Quality measure F: Suppose every voter has an opinion about who is the better candidate in any candidate-pair and says it in her vote. But suppose, due to random mental or writing mistakes by voters, that each one of those opinions is wrong with (independent) probabilities ε>0 (in the limit ε→0+), Let F be the probability that the election winner W would win a head-to-head contest against every opponent (based on the cast ballots but with all the mistakes corrected). (Supposedly, the greater this probability, the "better" the voting system.)
Theorem F: The voting system which maximizes F, is the "Condorcet least-reversal system" in which the minimum number of pairwise-comparisons are reversed so that some candidate now pairwise-beats every other.
Remark: Although all have their intuitive appeal, we do not believe either quality measure A, B, C, D, E or F is the right measure to use when comparing voting systems. Bayesian Regret is.
Note that the weasel words "based on the ballots as cast" mean "as opposed to what the voters truly feel" (which might be different due to strategic voting). In that sense, all of these theorems are only about honest voting.
Proofs: The proofs all are trivial. These theorems really are just an immediate consequence of the definitions of these voting systems. However – since some people don't think it is so obvious – we shall provide a proof for theorem B (since it is the hardest).
The probability that a voter will rank a random candidate R above W, is proportional to the number of candidates she ranked above W on her ballot, which is just her rank of W minus 1. (E.g. if she ranked W top, i.e. 1st, then 1-1=0, correctly giving zero probability she ranked a random candidate above W.) The probability that a random voter will do this, thus is proportional to the average rank-minus-one of W on all ballots. But this is precisely N-1-S where S is W's average Borda score and N is the number of candidates. So minimizing this probability is precisely equivalent to maximizing Borda score, i.e. to using the Borda voting system to choose the winner. Q.E.D.