There is no ideal voting system. I’ve noticed that many people dream up ‘perfect’ voting systems where everyone’s vote is equal and the results clearly reflect social preferences. What they’re asking for is mathematically impossible. Mathematics always beats utopian fantasies.

Elections are supposed aggregate individual preferences into social preferences. Arrow’s Impossibility Theorem demonstrates that there can be no perfect democratic voting system. The results are inconsistent with the basic concept of democracy. No matter which system you use, some votes will be worth more than others.

Each of Arrow’s axioms are obvious and vital to the idea of democracy. Yet he proved that it is mathematically impossible to be consistent with every single axiom at once. Something has to give. One of the tradeoffs is if a system has no dictatorship, it becomes intransitive. If it is not intransitive, it becomes a dictatorship.

Arrow originally listed 6 axioms for a fair democracy, but later simplified it with a weak pareto principle. Every axiom is vital for fair voting mechanisms but are inconsistent with each other.

The axioms were simplified and restated with a Pareto Principle. This quotes Wiki:

Unrestricted Domain: For each state X and Y, based on the social preference ordering, society prefers either state X to Y or Y to X. i.e. society can compare any pair of candidates. (completeness)
Transitive Property: If society prefers (based on social rule aggregation of individual preferences) state X to Y and prefers Y to Z then society prefers X to Z.
Independence of Irrelevant Alternatives: If for some X, Y, and Z, X is preferred to Y, then changing the position in the ordering of Z does not affect the relative ordering of X and Y i.e. X is still preferred to Y. In other words, changing the position of Z in the preference ordering should not be allowed to “flip” the social choice between X and Y.
Weak Pareto Principle: If all individuals rank X above Y, then society should rank X above Y.
Non-Dictatorship: Societal preferences cannot be based on the preferences of only one person regardless of the preferences of other agents and of that person.

Each axiom is obvious on its own. Every person gets a vote. They can rank their preferences or note their indifference between options. Unanimous consent means that the entire population voted for one choice, so that represents the social preference. Choices should be transitive, so if the preferences rank A>B>C, then A>C. A should win. Irrelevant alternatives should not impact the results. If A>B then adding an irrelevant C in the election should not make B win. There can be no dictatorship, where one voter imposes his preferences on everyone.

Things get unusual when there are greater conflicts of interests and individuals cannot unanimously agree.
Here’s an example. 3 groups (XYZ) rank three preferences (ABC) and find intransitive results.

Here’s a classic example demonstrating how democracy cannot solve one of the simplest social ordering problems
– A B C
X 1 2 3
Y 2 3 1
Z 3 1 2

Transitivity is important. If A>B>C then A>C. This is not the case here.

Break it up into two party elections.
If they have to vote between A and B, the group will choose A.
X prefers A (ranked 1 over 2), Y prefers A (2 over 3) Z prefers B (1 over 3).
If they choose between B and C, they pick B.
X prefers B (2 over 3), Y prefers C (1 over 3), and Z prefers B (1 over 2).

So it seems simple. They prefer A over B, and B over C. But they also prefer C over A.
X prefers A (1 over 3), Y prefers C (1 over 3), Z prefers C (2 over 3).

If you make them vote on all three preferences at once, you get the intransitive, nonsensical result: A>B>C>A>B>C>A etc. There is no social preference in this case, so an election fails to select an aggregate preference.

How do you break these ties? Dictatorship. X imposes its preference for A on the other voters. The election is made null and X wins.

You can also introduce irrelevant alternatives in an election.
Say there are 100 voters and they rank their preferences: 1st choice gets 3 votes, 2nd choice gets 2 votes, 3rd choice gets 1 vote.

In a two party election, A wins over B.There are 52 votes for A, 48 for B. When you introduce an irrelevant and unpopular third choice (C), B wins over A.
Say
49 people vote for A>B>C
48 people vote for B>C>A
3 poeple vote for C>A>B

The result?

This means that politician C can distort the election to help his friend, politician B, win an election B would not normally win.

Simply, only a dictatorship always provides transitive results.

Every electoral system is vulnerable to these flaws. A winner-takes-all system can be distorted just like a ranked preference system.

Some perform better than others. One way to minimize errors is by winnowing the options to two candidates. Basically, elections can be run like sports tournaments, forcing voters to choose between a series of two candidates until they reach the final election. The US political system is most similar to this. The tradeoff is that it limits the options of voters and hides overall social preferences. Other systems, like instant runoffs, are more vulnerable to Arrow’s theorem.

One salvation of democracy is that no one knows who the dictatorial voter group is. There has been much talk about hypothetical “centrist” and “median” voters who can tip the election one way or another, but finding out who these individuals are amongst a massive amount of noise and in limited time is impossible. So democracy suffers from a hidden dictatorship in practice.

There’s another problem with democracy, somewhat related to this impossibility theorem. Arrow proved that individual preferences do not always fairly aggregate into social preferences. That means some voters can be major losers when there are conflicts in interest. What if the losers opt out of the government?

Say 40% strongly prefer A, mildly prefer B, but absolutely oppose C. 60% support C, mildly prefer B, and absolutely oppose A. There will be a problem with these results. Both sides must to compromise on B (which may not be possible), otherwise A may simply walk out of the democratic process rather than tolerate such an extreme conflict of interest. Severe conflicts of interest result in political violence, as each side tests different aspects of their strength.