Here's the basic problem, familiar to those of you who have heard me lecture on the topic. Consider three people, with preferences over three options ranked as follows. Person 1 ranks the options as follows: Option A, then Option B, then Option C. Person 2 ranks the options as follows: Option B, then Option C, then Option A. Person 3 ranks the options as follows: Option C, then Option A, then Option B.
Each individual person has "transitive" preferences. Person 1, for example, prefers A to C. What about the group? Nope. Groups are screwy, and screwy in a way that makes the concept of "the will of the people" a mathematical impossibility. "Democracy," for all we use the word, is a fuckin' joke.
Consider Options A and B. Does a majority prefer A to B, or B to A? Note my wording. I didn't say jack fucking shit about how anyone would vote because I ain't talkin' 'bout votin' yet. Why not? Because I am making an argument that will apply regardless of what voting rule you construct, and the way to do that is to argue independently of voting. So, A or B? A majority prefers A. Persons 1 and 3 prefer A to B, so a majority prefers A to B. What about B versus C? Persons 1 and 2 prefer B to C, so a majority prefers B to C.
So, a majority prefers A to B, and a majority prefers B to C. That means, if the transitive property were to hold, that a majority should prefer A to C, right?
Does it?
Nope. Persons 2 and 3 prefer C to A. A majority prefers C to A. That's an "intransitivity" in collective preferences, even though every individual has transitive preferences. If a group has intransitive preferences, as they will any time there are more than two options and preferences are multidimensional (lots of math here...) there is no such thing as the "will of the people." That means that the outcome of any decision-making process is determined entirely by the decision-making process.
Put another way, the winner is determined entirely by the voting rule. Richard McKelvey's "chaos theorem," not to be confused with chaos theory, shows that when a group's preferences are sufficiently complex, any outcome is possible with the right voting rule.
The outcome depends on the voting rule. Period.
If there are more than two options, there ain't no such fucking thing as democracy. This is, in fact, a very old mathematical observation, that has driven scholars nuts for centuries. It was best formalized by Kenneth Arrow, who won his Nobel for Social Choice and Individual Values. His formalization is popularly called the "impossibility theorem" in scholarly circles, because it states that you cannot devise a voting rule that meets his basic conditions for democracy.
Oh, and as a side-note, one of the famous mathematicians who tried? Lewis Carroll. I've mentioned that creep a few times lately. Did you know he was a mathematician too? Apparently counting to "age of consent" was a little too advanced a concept for him.
Anywho, the basic observation that you need to take from this is that you cannot devise a voting rule that solves the problem I have laid out because it is built into the preference structure of groups.
The Maine voting system is an attempt to devise a voting rule that solves a problem built into preference structures. You can't do that. It's like saying, hey Einstein! You're wrong! You can go faster than light! All you have to do is get on a train going near the speed of light and throw a ball forward while you are inside! That ball goes faster than light!
No. Dude, you just missed the point of relativity completely. Time dilation, dude. Time dilation.
The intransitivity of collective preferences is built into the structure of group preferences over multiple options. That means you can't solve it with a voting system. If you try, you are missing the point of Arrow's impossibility theorem, and missing the point of why McKelvey is right.
Yeah, Maine has a weird voting system. What does it do? Main(e)ly, this. You can put a third party as your first choice, and that choice gets knocked off in the first round, so you don't waste your vote. Voting rules do affect outcomes! This can change outcomes! From the perspective of "democratic" theory, though, don't fool yourself. There's no such thing as the "correct" outcome.
OK, there is an "incorrect" outcome. Trump winning. That's an incorrect outcome. Seriously, fuck that guy.