All Voting Systems are Fundamentally Screwed

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In light of the recent election, there's a lot of talk going on about alternate voting systems like ranked choice, with claims like "Libertarians are stealing our votes!" and "with ranked choice, ____ would have won!" Here's a mathematical approach to why any proposed alternative wouldn't work. This post is based off notes from the first UT math club lecture given by Tom Gannon, you can find them here. Sorry for the clickbait title. Without further ado, let's begin.

How would we precisely define what a voting system is? Informally, we would probably need a list of alternatives to choose from, for each person to make an ordered list of such alternatives, and have a societal preference list as the output. We would also prefer there to be more than two alternatives, since the two party system has been the cause of much complaint. Mathematically, we would define two sets $A=\{\text{set of outcomes}\}$ and $N=\{\text{set of voters}\}$. Then a voting system would be defined as a function \[ F \colon L(A)^N \to L(A), \] where $L(A)$ denotes the set of all total orderings of $A$. Each person's individual ballot would be an $N$-tuple $(R_1,\cdots,R_N)\in L(A)^N$ (also called a preference profile). Let's give some examples of voting systems:

We would probably want our voting system to satisfy some reasonable conditions. For example, if everyone puts in the exact same ballot, then the output should be that ballot that everyone put in right? Also, if everyone ranks candidate $A$ above candidate $B$, then in the final ranking, $A$ should be greater than $B$. These conditions are called the Pareto condition and independence of irrelevant alternatives, respectively. We could also interpret indepedence of irrelevant alternatives as such: if we add a third candidate, it won't change the position of the other two candidates. Let's define these conditions mathematically:

Definition: A voting system satisfies the Pareto condition (unanimity) if given that an alternative $a$ is strictly greater than $b$ for all total orderings $R_1,\cdots,R_N$, then $a$ is strictly greater than $b $ in $F(R_1,\cdots, R_N)$.

Definition: A voting system is said to be independent of irrelevant alternatives (denoted IIA) if for two preference profiles $(R_1,\cdots,R_N)$ and $(S_1,\cdots,S_N)$ such that for all individuals $i$, alternatives $a$ and $b$ have the same order in $R_i$ as in $S_i$, then alternatives $a$ and $b$ have the same order in $F(R_1,\cdots,R_N)$ as in $F(S_1,\cdots,S_N)$.

These conditions seem pretty reasonable right? Let's take a look at what each of our voting systems satisfies.

FPTP Borda LPTP Dictatorship
Pareto? Yes, if everyone chooses the same person he will have the most votes and win Yes, for the same reason as FPTP No, everyone putting the same guy first will result in him losing YES!
IIA? No, introducing a third party may change the results No, same reason as FPTP but more severe since the new candidate will be weighted No, introducing an irrelevant alternative will probably result in him winning YES!

Do you see a pattern? The only voting system that satisfies both Pareto and IIA is a dictatorship! The natural question to ask then, is "are there any other systems that satisfy both Pareto and IIA"? The answer to that is NO! The only system that satisfies both Pareto and IIA is a dictatorship. This is called Arrow's Impossibility Theorem, proving this shows that there doesn't exist a system that satisfies Pareto, is IIA, and isn't a dictatorship.

Theorem (Arrow's Impossibility Theorem): Assume that $V$ is a voting system with more than two alternatives which satisfies both Pareto and is independent of irrelevant alternatives. Then $V$ is a dictatorship.

Corollary: There exists no voting system that

I won't get into the proof: you can find it starting from page $4$ here. But proving this theorem says that any voting system you could possibly come up with will fail at least one of the conditions: so in other words, if we want reasonable voting criterion with more than two candidates, no can do! This may be out of character, but here are some practical reasons why other voting systems haven't been implemented:

Instant Runoff Borda count Dictatorship