Collective Choices and Voting

What do a French Revolutionist from the 18th century and a Nobel Laureate from the 20th century have to do with the US election in 2016? Why is German Chancellor Merkel such a great Social Democrat? And how did Iran's Guardian Council make Rouhani President?

The Congress of the Game Theory Society

Once in four years there is an event that brings game theorists from around the globe together: the Congress of the Game Theory Society. The last one took place in 2016 in Maastricht. For game theorists, who are interested in individual decision making, in designing mechanisms that improve societies, and in finding socially optimal outcomes, this Congress was extraordinary. Not only in a good way though: Turkey had just seen a failed coup d'etat and many of our Turkish colleagues (who hosted the congress in 2012) where not allowed to leave their country and join this event. Also, Donald J. Trump was just nominated the presidential candidate of the Republican Party. An unexpected development which inspired 2007 Nobel Laureate Eric Maskin to open his plenary talk with the sentences:

Donald Trump is the Republican nominee for U.S. Presidency. How could that have happened?
Now, with two general elections in Europe behind us and one to come, it seems to be a good time to think a little what can go wrong if a voting mechanism is not carefully designed.

What is Voting Trying to Achieve?

From our Western viewpoint this seems a silly question: we vote for parliaments, presidents, bills, and call that a democratic process. And we call it democratic because we are allowed to vote and our vote counts. Nevertheless, a voting process is not self-sufficient. It is implied by the idea that different people have different opinions — or to be more precise: different preferences — and that as a society we should somehow find a compromise, an outcome that is supported by a majority. If there are only two candidates that can be elected, these preferences are easily determined from our vote: if we vote for candidate A it means we prefer candidate A over candidate B. One of the candidates will (very likely) get a majority, and the larger part of the population is happy. But what happens if there are more than two candidates? In this case a vote does not reveal our entire preference order, but only reveals the top of our list. We must expect that in this case the "optimal" outcome might not be achieved, simply because not all available information is revealed. (I shall come back to this point later.) But: suppose we could not only vote for one candidate but rather make an ordered list of candidates in accordance with our preference: would that make it easier to find a good outcome? 1972 Nobel Laureate Kenneth Arrow (who died just two months ago) answered this question in his dissertation "Social Choice and Individual Values" in 1951, which happened to become the basis of what is known as Social Choice Theory.

Welfare Economics and Arrow's Impossibility Theorem

Abram Bergson introduced the Social Welfare Function in 1938. It produces a ranking over potential economic states of a society (that specify economic output, distribution, etc.). Paul Samuelson (Nobel Laureate of 1970) provided in his 1947 dissertation a complete mathematical description of how to evaluate a social state based on different ethical assumptions (e.g. the weight one should place on efficiency or equal distribution of wealth) that would specify a Social Welfare Function. So, given all the utility functions of all individuals, and all production constraints of all companies, the Bergson-Samuelson Social Welfare Function would provide a ranking, specifying what outputs are better than others. Arrow reformulated the problem in his PhD thesis: abstractly speaking every member of a society likes some states more than others, that is every member has a ranking over potential states. If one asks everyone for their ranking, should one not be able to find the best outcome? And should that not hold for any potential answers one can get (that is all preference people might have)? Arrow showed that if there are three or more different states the only way to aggregate arbitrary rankings in a somehow consistent way is to select one member of society and do whatever she says. In other words: one would need a dictator.

Single Peaked Preferences, and the Condorcet Winner

Arrow formulated his result for "the big picture", but his idea remains valid for arbitrary sets of alternatives. In particular, it holds if the set of alternatives is simply the set of candidates in an election. So, the aggregation of voters' preferences into one optimal candidate is, in general, impossible... The difficulty to find the optimal candidate in an election had already been recognised by Marie Jean Antoine Nicolas Caritat, Marquis de Condorcet in 1785, and his finding was nicely illustrated during the last US presidential elections: Trump had a majority against Clinton, Clinton had a majority against Sanders, but Sanders would have had a majority against Trump (according to many polls). If everyone would submit their preferences over all candidates it would be very difficult to justify the election of any of the three. The decision in this case was made by the order in which voting took place.
Although Arrow's Impossibility Theorem is quite a negative result — it basically says that it is impossible to aggregate arbitrary preferences in a consistent way — economists have tried to circumvent it in many ways. In the context of political elections one could argue as follows: Typically, parties have a position somewhere on a spectrum from the far-left to the far-right. And a voter who prefers a candidate on the left will typically prefer a centrist candidate over a far-right candidate, and vice versa. Also, a voter who prefers the center will become more and more unhappy the further away from the center the elected candidate is. So, within this spectrum from the far-left to the far-right, for all voters there is a top candidate and the further away a candidate is (on that spectrum) the less she is liked by the voter. Maybe, when we designing a rule to find the best outcome we do not need this rule to work for all preferences, but only for those preferences that have such a structure. Such preferences are called "single-peaked" for obvious reasons. (You can order Trump, Clinton, and Sanders on the political spectrum from left to right, but given the election results there must have been some people who preferred Sanders over Trump over Clinton, that is the two outer candidates were both preferred over the centrist. So their preferece order would have two "peaks".) If we assume that all people have single peaked preferences, then there will always be a candidate who has a majority against all other candidates in a pairwise voting. This candidate is the so called Condorcet Winner. Many economists agree that if there is a Condorcet Winner, she should be selected, simply because she would beat all other candidates in a direct comparison.


Although the assumptions of single-peaked preferences is rather strong (for instance there are many similarities in the programmes of the far-left and far-right candidates in the current French election, Mélenchon and Le Pen), let's assume for now that they were accurate. Then, if everybody submits a list of candidates in the order of his preference, there will be a Condorcet winner. That is a candidate who, if compared to any other candidate, would always have the support of a majority of the population. However, in practice voters do not submit a complete ordered list of candidates but only their top choice. How does this affect the outcome? Unfortunately, it destroys the nice finding that the Condorcet Winner will be elected. So, the loss of information that is revealed (we only learn about people's top choices, and not about their entire rankings) can lead to different outcomes. Nevertheless, by definition none of the other candidates will obtain an absolute majority. There are voting mechanisms that are run in several rounds, and in each round one or more candidates are removed from the list until there are only two candidates left. In the last round one of them will have an absolute majority. In this case in each round a little more information about people's preferences is revealed: Whenever someone's top choice is removed, he will reveal his second-best choice and so on.
One Example: In the French presidential election a candidate will win if (s)he has an absolute majority. If no candidate has an absolute majority in the first round, there will be a second round of voting between the top two candidates. In this case, we learn only a little more about people's preferences, namely their first choice (in the first round), and their top choice between the two candidates in the second round. But is this enough to save the Condorcet Winner? Unfortunately not: The Condorcet Winner might be the second choice of everybody and thereby not get a single vote in the first round. So (s)he might not reach the second round at all.
Another Example: When the Torries decided about the new prime minister after the Brexit Referendum in 2016, they used the following mechanism: Everybody votes for one candidate on the list, the candidate with the fewest votes is removed from the list, and a new election with the shorter list is held. This is repeated until there is only one candidate left. In this mechanism we learn even more about people's preferences than in the French system. Nevertheless, it might happen that there is a Condorcet winner who does not win the election: Suppose (for instance) there are three candidates A, B, C, and five voters. Two voters prefer A over B over C, two voters prefer C over B over A, the last one prefers B over A over C. Then there are three voters who prefer B over C, and there are three voters who prefer B over A. Nevertheless, when we only count top votes, B will not reach the next round, as A and B will receive more votes.

Political Profiles and the Hinterland

A political candidate needs to be the top choice of voters in order to get votes. Being the second choice often does not pay off. When we stick to the above model, where the political spectrum is a line from the far-left to the far-right and all voters have single-peaked preferences, any candidate must choose a political agenda, that is (abstractly speaking) a point on this line. And this point must be chosen wisely as it needs to maximise the number of voters whose top choice is close to that point. In a way, candidates play a game: they can "steal" each other's voters by strategically choosing their agenda. (It may be a bold claim that politicians are not idealists with a fixed political position, but rather opportunists who try to maximise the number of voters. On the other hand, isn't this exactly what democracy is about? Representation of the maximal number of voters by the elected?) If there are only two candidates, one left one right, all voters left of the left candidate's profile will vote for left, all voters right of the right candidate's profile will vote right. Economists refer to those votes as the "Hinterland". Candidates mainly compete for votes in the middle by taking more central positions and increasing their Hinterland. But then new parties at the extremes might emerge. The development of the political landscape in Germany over the last 15 years illustrates this:
From 2003 to 2005 the German administration under the Social Democrats (SPD) and the Green Party (Bündnis 90/Die Grünen) with Chancellor Gerhard Schröder (SPD) implemented the "Agenda 2010", an extensive restructuring of the German social system. This was basically a shift from the left towards the center. For many left-wing SPD members this programme was too far away from their core values. In 2005 they formed, together with some union members, a new, more socialist, party (WASG) that was later joint by the socialists (PDS) to become Die Linke. This new party left of the SPD took many votes from the SPD's former Hinterland — one factor that vastly contributed to the election of the Christ Democrats (CDU) with Chancellor Merkel in 2005. While SPD tried to gain back their voters from the left, CDU under Merkel shifted towards the left as well incorporating topics such as Environment Protection (e.g. the moratorium on nuclear power in 2011) and European Integration into their core programme, and allowing hundreds of thousands of refugees to enter the country in 2016. This made room on the right of CDU for those conservatives that did not feel represented by CDU anymore, and lead to the foundation of a new party (AfD). (While they started off as a new conservative party, they have recently attracted attention mainly by the right-extreme statements of some of their members.)

Divide and Conquer

The previous story illustrated that the choice of a candidate's position on the political spectrum is not only an ideological question, but is also affected by the behaviour of her rivals. But it also suggests that more parties on one side of the spectrum will positively affect those on the other side. This is particularly true if simple majorities are relevant such as in the first round of the French presidential elections. An extreme case of this phenomenon happened during the 2013 presidential elections in Iran. While 40 candidates registered for the election, only eight of them were approved by the Guardian Council. Six of them were considered political hardliners, and Hassan Rouhani and Mohammad Reza Aref being the most moderate. When Aref withdrew and endorsed Rouhani, the latter was the only moderate candidate (with a huge Hinterland), and he won the election with just over 50% of the votes in the first round.

Bottom Line

Designing a voting rule is complicated and there are many pitfalls. A Condorcet Winner, who is typically a good compromise as she is liked by many (even if she is not the top choice of a majority), is in many cases not elected because of the design of voting mechanisms. This is because these mechanisms do not reveal enough information to actually identify a Condorcet Winner. Many claim that Western societies become more and more divided. But maybe, this is a problem that could be handled with a better voting design: Maybe, voting mechanisms that remove candidates who do not get a large majority in the first round make it very difficult for moderate candidates to get elected. They might be very popular, but simply not be the top choice of sufficiently many.