How to Make Elections More Representative of the People's Ideals
Current voting systems discourage the emergence of third-party candidates. Under the electoral college or a plurality system (where the candidate with the highest number of votes wins), the natural tendency will be to let go of the candidate who you believe is best-suited for office, and vote for one of the top candidates who embodies your ideals more than the other. This gives rise to a two-party system, and it's very unlikely that one of only two very popular candidates would be the person best embodying the ideals of the nation. Having more candidates gives us a higher chance that the winner embodies the ideals of everyone in the nation, compared to having only two candidates.
Let's work through an analogy, and then we'll return to the point at hand.
Imagine we have N villages at various points on a line, with different populations, and we want to dig a well in one of the villages, where everybody from all N villages will have to go to in order to get water. One of the N villages will inevitably be better than the others as the well's location, meaning that placing the well in one of the villages will require all N villages' populations to travel a lower distance than had the well been placed in any of the other villages.
Now suppose that the villagers believed in democracy, and didn't know how to mathematically determine the ideal village to dig the well, so they wanted to vote over the well's location. All villagers only want to minimize the distance they personally have to travel in order to get water, so they are going to vote for their own village, and if they were to rank their preferences, their first choice would be their own village, then the closest village, then the next closest, and so on. Everyone votes out of self-interest.
After polling all villagers for their ranked preferences, the council of vote-counters tries to decide how to go about counting their votes so that the outcome is best for everyone. Here are the options they consider:
In a plurality voting system, the village who is the top preference of the highest number of villagers wins. Inevitably, the village with the highest population will win under the plurality system.
With instant-runoff voting, everyone's top choice will be counted. If no village is the top choice of more than 50% of all villagers, the village who is the top choice of the least number of villagers gets eliminated from the race, and the votes of everyone who had picked it as their top choice goes towards their second choice candidate. The process repeats until 1 village has more than 50% of the votes.
Under a Borda count system, villagers would list their preference in ranked order, and each village receives a number of points from each villager equal to the number of villages the villager prefers less. So if villages A, B, C, and D were ranked 1, 2, 3, and 4 in a voter's ballot respectively, village A would get 3 points from the villager, village B 2 points, village C 1 point, and village D 0 points. After the points from all villagers are added up, the village with the highest number of points wins.
Fraction-Based Positional Voting
This is similar to the Borda count, but (from the example above:) instead of villages A, B, C, and D getting 3,2,1, and 0 points respectively, they'd get points equal to 1 over their ranking. So A would get 1/1 (equal to 1) vote, B 1/2 vote, C 1/3 vote, and D 1/4 vote from the aforementioned villager's ballot. Similarly, the village with the highest number of cumulative points wins.
Here, the only candidates on the ballot are the top two most populous villages. Thus, everyone votes for whichever of the top two most populous villages is closest to them.
My girlfriend Caroline and I simulated the different voting methods. Our simulation, given N (the number of villages), creates N villages at random locations with random populations. We then determine the mathematically ideal village for the well's location (i.e., the village where the least total amount of distance has to be travelled for everyone to obtain water). The simulation assumes that all villagers will vote only in self-interest (so the ballots of everyone in the same village will be the same), then counts the votes using the different systems mentioned above, and finally determines what percent of the time the winner using a certain voting method is the same as the mathematically-ideal village. Clearly, the higher the percentage, the better the voting method.
As you can see, if everyone votes unschemingly in self-interest (i.e., they don't have considerations for "wasted votes" and they don't vote for a non-ideal village in order to prevent a less ideal village from winning), the Borda count method is always superior to other methods of voting for picking the ideal candidate.
The villages are an analogy for political candidates, and their locations analogous to where they place on a political spectrum. Of course, politics isn't one-dimensional like the simplified villages in the analogy, but you can still determine mathematically ideal results on higher-dimension shapes. The point is, if you could place every candidate and voter's political standing on a plane, there would be one candidate whose ideology is least distant from all voters compared to all other candidates, and the way to find that candidate is not through a plurality system (highest vote wins) or majoritarian voting (>50% of votes wins), but through more complex methods which take into account everyone's ranked preferences -- namely the Borda count.
In a democracy, lacking the foresight to know which candidate will be "best" for the country, the winning candidate needs to be the one most representative of all voters' political stances, and in order to find that candidate, we need ranked voting systems and specifically, the Borda method.