The Limits of Democracy: What Arrow’s Impossibility Theorem Reveals about Fair Elections
How do we arrive at group choices based on individual preferences?
What does it mean for our political systems to be ‘democratic’?
How do we vote, and why do we do it that way? What is the process to then elect out representatives after voting? What are our voting systems trying to achieve?
Today we see how some incarnations of democracy are mathematically impossible.
Today in a nutshell
Democracy has two main components: a way to vote and a way to choose a winner (based on the votes).
Depending on how we vote and choose, one gets several incarnations of democracy (‘electoral systems’). Each of these systems has its limitations.
For a ranked based voting, any way to choose a winner is undemocratic!
What is democracy?
According to Cambridge dictionary, democracy is “a system of government in which power is held by elected representatives who are freely voted for by the people, or held directly by the people themselves”.
This leaves a lot of freedom on how to implement democracy:
A voting system. How do we ‘vote’ for these candidates? Do we just select our favourite representative, our top three, give a rating to each…?
A decision system. How do we choose the ‘elected representatives’ (after voting)? That is, how do we put together the choices of each individual?
The two types of democracy
A key property that distinguishes democratic systems is whether individuals vote directly or through some previously elected representatives. The former is known as direct democracy, and the latter as representative democracy.
Note that in a representative democracy, our representatives then take decisions by voting directly.
Direct democracy
This is a form of democracy in which individuals vote directly, without relying on intermediate representatives. Think about it as raising your hand in a room filled with the whole population of the country (or the ones eligible to vote).
It was common in Ancient times (such as Athenian democracy) when the population was lower and affairs could be resolved this way. But it still exists nowadays! The Swiss cantons of Appenzell Innerrhoden and Glarus still use this system.
Representative democracy
This is the most common form of democracy, where individuals vote for some representatives that will then make the political decisions.
This is the type of democracy we see in most (Western) countries, where we elect some sort of Parliament and/or President that later make decisions.
How to vote and decide
Whether you are trying to vote something through a direct method, or trying to elect some representative who will later decide for you, there is a fundamental question of how do we vote. We also have to figure out how we arrive to a common decision after everyone has voted.
Say you and your friends are thinking about which movie to watch: Pulp Fiction, The Godfather or Forrest Gump.
How do you choose which to watch?
First past the post
This is the most common method in our daily choices. In the movie example, each of you will choose the movie you’d like to watch the most (voting system). The movie that has the most votes will be the winner (deciding system).
This method is very common in politics too, and is still used nowadays in the UK, the US, India…
First past the post and winner-takes-all have a very fundamental problem: it is often the case that most of the people did not vote for the winner.
For instance, say there are 3 candidates in an election and 1 million people voting.
300,000 people vote for Candidate 1
300,000 people vote for Candidate 2
400,000 people vote for Candidate 3
Candidate 3 will be elected according to this system. But 400,000 people voted for him, while 600,000 didn’t!
It also encourages strategic vote. If you know beforehand that Candidate 1 is a weak unpopular candidate, you might feel voting for him will be useless—he will certainly not have the most votes to be elected, so why vote for him? You might therefore vote for another candidate, even if that is not the one you like the most.
Ranked choice voting
To avoid the problems above, we could instead rank all the candidates from our most to our least favourite (voting system).
For instance, we could do a ranked choice voting and then elect a winner as follows:
If some candidate has been voted as the first preference by at least 50% of the population, they are elected as president.
If not, we take the candidate that has received the least number first preference votes. We eliminate them from the polls and redistribute the votes he received according to their voters’ second preference.
(For instance, say my ranking is (1) Candidate B (2) Candidate A (3) Candidate C. If Candidate B had the least number of first preference votes, he will be eliminated and my vote now goes to Candidate A (my second preference).)Now, with one less candidate, we look again whether any of them has been voted by at least 50% of the population.
We repeat this until one candidate has at least 50% of the first-preference votes.
This deciding system is known as Instant-Runoff Voting (IRV), and it is used in countries such as Australia’s House of Representatives, some states in the US or Fiji.
This way of choosing a winner also has its limitations, the most important being that it is very unstable:
Imagine an election where Candidate A and Candidate C are very opposite to each other (e.g. far left and far right in politics), while Candidate B lies in the middle.
People vote and the results are 40% of the people having Candidate A as their first preference, 20% of people having Candidate B and 30% having Candidate C. Candidate A is preferred to Candidate C.
No candidate has at least 50% of the votes, so we eliminate Candidate B and redistribute their votes according to their voters’ second preference.
If the people who voted for Candidate B all (or most) have Candidate C as their second preference, redistributing the votes might make Candidate C preferred to Candidate A.
We see the results of the election are unstable. If Candidate B is in the election, Candidate C will win; if they are not, Candidate A will win. We certainly do not want a random person popping into an election to alter the results!
But, at least the voting method is much more powerful than above: we have our favourite candidate but also a ranking for the rest of the candidates.
We will now see something striking: any ranked choice voting system is doomed to fail!
Our model for democracy
Consider the following electoral system.
The voting system is a ranked choice voting system: to exercise their vote, individuals will rank all the candidates, stating who is their first, second and up to last favourite candidate.
For instance, you could have Alice, Bob and Charles as candidates to presidency. Your vote will look something like: (1) Charles (2) Alice (3) Bob.
Once everyone has voted, we need to have a way to choose the winner (our deciding system). Even better, we would like to make a final ranking based on the rankings submitted by each individual (then we could select the winner of the election as the first candidate of the final ranking).
That is, if there are 100 individuals in our population, we want an assignment:
This assignment must satisfy some rules to call it democratic. For instance, if we just take the ranking of individual 1 as the final ranking, then our system is actually a dictatorship (with individual 1 as the dictator)—we do not want this. Similarly, we want some common-sense properties: if every person in the population has Candidate A as their first choice, they should certainly be the first choice in the final ranking.
These leads us to the following axioms: any democratic way of choosing a winner should satisfy them.
Axioms of democracy
Any assignment that outputs a final ranking based on each of the individual rankings must satisfy the following properties:
Unanimity: if every individual prefers candidate A to candidate B, then the final ranking also prefers candidate A to candidate B.
Transitivity: this means that the deciding system of our democracy produces an ordered list of candidates.
Non-dictatorial: no individual dictates the final ranking.
Independence of irrelevant alternatives. Say the population prefers Candidate A to Candidate B. If a new Candidate C appears in the game, Candidate A should still be preferred to Candidate B.1
If you think about it, all these are very common assumptions for a democratic way of choosing a winner. (1) says something like ‘if the preference is common for everyone, just take it’, (2) says that we should not choose candidates that everyone dislikes; (3) is the most common definition of a democracy; and (4) says that adding irrelevant candidates shouldn’t affect the outcome of the others (this was the problem with IRV).
But there is a problem:
No democracy exists
In his PhD thesis, Nobel Prize economist Kenneth Arrow showed the following striking result:
ARROW’S IMPOSSIBILITY THEOREM: in a society with at least three people, no deciding system satisfying the “Axioms of democracy” exists.
This is not an opinion or some observation on history—it is a mathematical theorem. Democracy can’t work!
What about real life?
Remember to always think what information we have.
Arrow’s impossibility theorem is a very striking result—and believe me that it is actually true. But it concerns a very specific type of voting system (ranked choice voting system) with some concrete (although very reasonable) axioms on how to make a social choice.
The assumptions we made might not hold in our democratic systems.
Myself, and most of you reading, are Spanish. We do not rank candidates when voting. Instead, we just choose our preferred one, and then our system selects a winner based on absolute majority.2
This system has its limitations: we have seen several times in the last decade how the results of an election are not able to choose a President, and a second election must be performed. It also encourages the ‘strategic voting’ that we mentioned above.
This is certainly not optimal.
On the other hand, it escapes the assumptions of Arrow’s impossibility theorem: there’s hope for democracy!
But there are many countries that do use ranked based voting. Australia, Ireland or some states in the US are such examples.
Don’t panic! These countries are not dictatorships.
Arrow’s impossibility theorem says that no system can satisfy all axioms. Thus there might be some axiom that these systems don’t satisfy—but they can (and do) satisfy the non-dictatorial axiom. They are not democracies as defined by our ‘Axioms of democracy’, but they are not dictatorships.
(As we explained above, the axiom they do not satisfy is the “Independence of irrelevant alternatives”.)
Other alternatives
Luckily for us, there are other voting systems that are not ranked based or just selecting a favourite candidate:
You could vote for all the candidates you support, i.e. vote YES! or NO!, to each candidate (this is implemented in the Spanish Senate).
You could give a score to each candidate, say from 0 (dislike) to 10 (love). This is different than simply ranking all the candidates first to last, and it is the system used to elect the Secretary General of the United Nations.
These systems do not fall under Arrow’s impossibility theorem:
There is hope for democracy!
Reference
If you like this topic, I highly recommend Veritasium’s great video (also available in Spanish). I learned a lot watching it, and it helped me understand Arrow’s thesis better.
Candidate C might be preferred to Candidate A and B, to Candidate B but not to Candidate A or to neither of both—the point is that the relative position of Candidate A and B should remain the same.
Technically, we don’t even choose a preferred candidate, but rather a political party. These choices then lead to a certain number of deputies for each party (using the famous D’Hont method), based on which a president is nominated by the king. There is then a “investiture vote” in which all deputies vote: if the nominated president gets at least have of the deputies support, he is elected as president. If not, another vote is casted by the deputies, and more ‘yes’ than ‘no’’s are enough for the president to be elected.