Arrow's Theorem: How to Make America More Democratic

Democracy, rule by the people and for the people, is the ideal on which this country was founded. The ability to vote for our leaders is prized by all Americans, but it can come at a cost. Donald Trump is the current president of the United States, even though he is disliked by the majority of registered voters. This article isn’t, as one might think, about the electoral college. Rather, it’s about our voting system as a whole. Take, for example, this election’s Republican primaries. There were 17 candidates who ran, and, as we all know, Donald Trump won. Why? Well, for one thing, all the “mainstream” republican votes were split over numerous candidates. So, the most radical stood out, and managed to win the primary. It’s a well known effect, and actually very common: most candidates run on a radical platform in the primaries, and tone it down for the general election. So it is perfectly possible that the majority of Republicans were against Trump during the primary, but with so many options to choose from, their votes got split up. So if in fact the majority of voters preferred, say, Ted Cruz to Donald Trump, then should Trump be the one to win? Is it really a democracy if the candidate preferred by the majority loses?

These are the questions Kenneth Arrow asked himself as he tried to devise the perfect voting system. His plan was simple: to come up with a set of parameters for this perfect system, and then design the system itself.

Here are the parameters he decided on: first, the system must be ordinal, as opposed to cardinal. Thus, each candidate is ranked such that a voter can compare any 2 candidates and say which one is preferred. An ordinal system has voters rating candidates 1st, 2nd, 3rd, while a cardinal one uses 1, 2, 3 instead.

Second, the system must not be a dictatorship: the social ordering cannot simply copy the ordering of a single voter. Or, to put it differently, no voter is a dictator: the system must take into account all the voters’ preferences.

The third requirement is an unrestricted domain: the output of the system should have each candidate ranked, and every time the same set of voter preferences is input, the output must be the same. The first part ensures that the ordering is complete and any 2 candidates can be compared, while the second eliminates any potential randomness.

The next condition is the one most systems fail: independence of irrelevant alternatives. The addition of a candidate to an election should have no effect on the final result unless that candidate wins. The US system of each voter picking a candidate lacks this condition. If a voter prefers candidate A to B, and C is added, the voter’s switching to C will have an effect on the relative positions of A and B.

The final requirement is non-imposition: every possible ordering should be achievable. This means, for example, given an election with candidates A, B and C, the rankings ABC, ACB, BAC, BCA, CAB, CBA should all be attainable through some set of voter preferences.

So, those were the parameters Arrow set for the system he was trying to make. What was the result? Nothing. He determined that, with at least 3 candidates and 2 voters, it is impossible to make a system that satisfies those requirements. That’s right: there is no perfect voting system. In his 1951 book Social Choice and Individual Values, Arrow provides a very thorough proof of that impossibility, now known as Arrow’s Impossibility Theorem.

So what does this mean for democracy? If there is no perfect system, what should we use? Well we can find the next best thing. We can relax parameters until a system is found. But which ones to relax?

Non-dictatorship is certainly out, as is non-imposition. The unrestricted domain is necessary as well. Independence of irrelevant alternatives is a necessity as well, for a true democracy. The voters’ opinions of two candidates relative to each other don’t change if another candidate is considered, so the system should reflect that. So that leaves just the first: the ordinal ranking system. Can we change that and create a system that works with all the other requirements?

As it turns out, the answer is yes. We can create a cardinal system that will satisfy all the remaining criteria. Take the simplest cardinal system: approval voting. Each voter assigns a candidate either a 1 or a 0. The candidate with the highest total score wins. Does that work with the other parameters? Let’s see. All the voters’ preferences are taken into account, so it’s not a dictatorship. All the candidates are ranked the same way by total score, so unrestricted domain is satisfied. Independence of irrelevant alternatives is also present: whatever you vote for one candidate has no effect on your votes for the others. And finally, non-imposition is satisfied: there is no arrangement of candidates that can’t be achieved. So approval voting does work, as do many other similar systems, such as each voter scoring each candidate out of 10, and the candidates being ranked based on their totals.

Cardinal voting systems are far more fair than ordinal ones. They allow two candidates fighting for one group of voters to still have a chance against one candidate fighting for a different, equally sized group. Cardinal systems for the most part also get more information from the voter: with our current system only the voter’s favorite candidate is learned. So far, my arguments have extended only to primaries. But this idea is also important for the general election. Currently, only candidates from the two major parties stand a chance at winning, even if a third-party candidate is generally well liked. Showing support in the form of voting for a third party candidate is seen as “throwing one’s vote away,” but with a cardinal system that doesn’t have to be the case. All ordinal systems run the risk of splitting a voter base, and fear of such a split is what kept Bernie Sanders from running as an independent. Bernie would take votes from Clinton, which would have helped Trump win. A system which allows such a situation is entirely undemocratic, and doesn’t reflect the will of the people in the least. Our country should look towards completely changing how we vote, so that our leaders represent the country as a whole, and not just the radicals.