I want to consider the voting system against the 4 Arrow's axioms.
So first the voting system simplifies to first-past-the post system where the highest polling candidate wins the election.
How does this fit in with the 4 axioms?
Unrestricted domain
A full ordering is provided given by the number of votes for each candidate
No dictator
There is no individual whose votes decide the final result - everyone's vote matters.
Pareto
If everybody prefers candidate $y$ to $x$ then so will the group, because $y$ will receive more votes
Independence of irrelevant alternatives
What is the effect of removing candidate $X$ on the preference between $Y$ and $Z$? This will have no effect because if $Y$ has more votes than $Z$ then this will remain the case even if $X$ is removed.
By my working, all $4$ axioms are fulfilled but this contradicts Arrow's Impossibility Theorem: there is no social welfare function meeting all $4$ above conditions.
Where have I gone wrong?
Perhaps Pareto needs to be reconsidered. This is "If each individual has $x>_iy$ then the group has $x>_gy$"
But I suppose the group preference is there is just one candidate who is better than them all. So say if candidate $A$ wins, $B$ has the second most votes and $C$ has the third most votes. Every individual has $B$ is preferred to $C$ but this is not reflected by the group.
Would really appreciate some input on this.
Edit: My interpretation of the 4th condition was incorrect, please see the below answer
Independence of irrelevant alternatives fails. Your conclusion that removing $X$ will have no effect on the preference between $Y$ and $Z$ is wrong. If one person has preference $Z\gt Y\gt X$ and everyone else has preference $X\gt Y\gt Z$, then $Z$ is socially preferred to $Y$ if $X$ is in the race, whereas otherwise $Y$ is socially preferred to $Z$. (I'm assuming that by "votes" you meant "first preferences".)