Between June 6 and 9, elections are held to renew the European Parliament. To select their representatives, citizens have a voting system, which reflects, in a certain way, your preferences. However, like any other electoral system, the method used has its limitations; It is not perfect – this statement is based on a theorem, Arrow’s impossibility theorem. This and other mathematical tools allow us to analyze and understand voting systems; For example, they assure that there is no minimally democratic voting system in which the so-called useful vote cannot occur.
The useful vote, that is, opting for an option other than the preferred one with the aim of maximizing satisfaction with the electoral result, can present certain dilemmas. On the one hand, it is a more complicated procedure than simply voting for the preferred option, since it is impossible to determine which alternative should be chosen without knowing the preferences of the rest of the voters, which we cannot know for sure. On the other hand, this strategy means not reflecting our true political preferences with the vote.
But could we define a voting system in which all participants would be guaranteed that, if they vote for their favorite candidate, the electoral result that will be obtained will be at least as good for them as the one that would be decided if they opted for another voting strategy? Mathematics can help us see if this ideal voting system exists, in which a useful vote would not make sense.
To carry out the analysis, we are going to focus on the simplest type of elections, ones in which only one candidate – for example, a president – is elected. In this context, which offers the branch of mathematics called social choice theory, we can see a voting system as a “game”, whose development depends on the voting strategy followed by each voter and whose result is the candidate who wins the elections. If a voter has a voting strategy that ensures satisfaction with the result equal to or better than that obtained with any other strategy, that voter is said to have a dominant strategy. If every voter, regardless of her preferences, always has a dominant strategy at her disposal, it is said to be a simple game. In that case, players do not need to rack their brains to find the strategy that will be most beneficial to them, since they can always carry out the dominant strategy, which guarantees them a return equal to or greater than any other.
The ideal voting system we are looking for is, then, a simple game in which the strategy of honest voting – each person votes for their preferred candidate – is dominant for all players. However, as the philosopher Allan Gibbard demonstrated in 1973, it is not easy to construct simple games with good properties. In fact, the Gibbard’s theorem states that if a game is simple and admits at least three different outcomes, then it is, necessarily, dictatorial. That is, there is a player – called dictator– who has an infallible strategy that allows him to obtain any result; Thus, if the dictator wants the result of the game to be a specific one, he always has a strategy to achieve it, independent of the strategies chosen by the other players.
Therefore, Gibbard’s theorem indicates that the ideal voting system that we are looking for, in which honest voting is the optimal strategy, cannot exist, since, if it did, it would be a dictatorial system. In conclusion, regardless of the voting system we use, there will always be situations in which the useful voting strategy makes sense and, consequently, the dilemmas derived from it are unavoidable. Life is not that easy.
Andrés Laín Sanclemente He is a predoctoral researcher at the Institute of Mathematical Sciences (ICMAT).
Editing and coordination: Ágata A. Timón G Longoria (ICMAT).
Coffee and Theorems is a section dedicated to mathematics and the environment in which it is created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest advances in this discipline, share meeting points between mathematics and other social and cultural expressions and remember those who marked their development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: “A mathematician is a machine that transforms coffee into theorems.”
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