Voting Systems
Many countries employ many different voting systems to elect candidates in an attempt to make elections fairer and to not bias the vote in any particular direction. Not all voting systems are equal, and perhaps an easy example of a flaw in a voting system would be the phrase: “Don’t vote for them, there’s no way they’ll win anyways.” In the US and other places where a winner-take-all plurality voting system is used, people are incentivized to vote for candidates which are not their top choice because voting for third-party candidates is considered a “waste of a vote”. Such a system could be considered a dishonest mechanism, because agents’ best response is to lie about their preferences.
So, you might think that you would want to impose a few requirements on your voting system to prevent bad incentives like above, and to make sure a reasonable candidate is selected. Two requirements you might want are IIA (independence of irrelevant alternatives), which states that removing a candidate which is not elected should not affect the outcome (preventing the effect above); and unanimity, which is kind of like pareto-optimality for voting—if everyone prefers one candidate to another, the voting system should prefer that candidate as well.
These seem like pretty reasonable, modest requirements, but interestingly, Arrow’s Theorem shows that they are impossible to meet, except in a dictatorship. This might be surprising, but if you think through it you will realize that if the dictator votes for himself then at least one person votes for him meeting unanimity and it is IIA because it does not even consider other candidates.
The proof for why other systems cannot work is a bit tedious but effectively consists of showing that in a voting system meeting these requirements, if voters list out their preferred candidates in order of preference and all start by preferring one candidate, then by unanimity this candidate must be chosen. If they then switch one-by-one, there must be some special voter which flips the whole vote. Then by carefully changing how the voters choose their preferences and applying the rules of IIA and unanimity it can be shown that the voting system’s output is actually just this special voter’s preferences regardless of how other voters change their votes.
A proof can be found here: https://www.rochester.edu/college/faculty/markfey/papers/ArrowProof3.pdf
Fortunately, according to this article: https://www.princeton.edu/~cuff/voting/theory.html a Condorcet Winner can almost always be found in any real-life situation. A Condorcet Winner is defined as the candidate which would beat any other candidate in a head-to-head match-up, which can be computed if an ordered preference list of candidates is gathered from each voter. Although there are cases in which a Condorcet winner does not exist, the article claims that these cases are unusual based off of empirical evidence of how voters actually vote and not just using probability theory for random voters.
It seems to me a good way to reason about why this is true is to imagine that voters could be broken down into a network of people (described below). The conditions under which a Condorcet winner does not exist are some kind of voting similar to the following:
Voter 1: A –> B –> C, Voter 2: B –> C –> A, Voter 3: C –> A –> B.
There is no Condorcet winner because A would beat B, B would beat C, and C would beat A.
However, consider how actual voters usually pick a candidate. We can imagine each voter as a part of a network of voters which exists in some (likely higher-dimensional) ‘political belief space’. Then the strength of edges between nodes is, in general, inversely proportional to their distance in this space—perhaps there could be found a dividing surface at which point the edges become negative. For example, political parties would show up as clusters of voters with strong positive ties, and it would be expected that people will form these groups based off of triadic closure, since people near a group will begin to develop strong ties and will be ‘pulled’ closer to the political belief of the party. People would select their preferences when voting simply based on how strong their edge is to that candidate’s node in the graph.
In this model for how people vote, it is difficult to imagine the cyclical case outlined above. To get the cyclical pattern described above you would need an odd situation in which large numbers of people begin to choose people far away from them in political belief over people close to them in political belief. However, I suppose this is possible if these voters came from a much different world view and saw the political space differently, or if you accept the theory that the extremes of the political spectrum are similar to each other (making it an elliptical space).
It is interesting that although there has been a fair amount of research into the mathematics behind developing good social preference functions, there has been little to no implementation of a better voting system.
The reason voting systems have an impact not only on the outcome of the vote given some set of voter preferences, but also how individuals choose to vote is related to game theory, and my conjecture on why a Condorcet winner almost always exists is related to graphs.