Mathematics to understand and improve democracy

Democracy is a fundamental pillar of many modern societies and its proper functioning depends to a large extent on fair and efficient voting systems. Mathematics allows us to design and understand these systems since the works of the Marquis de Condorcet in the eighteenth century, which gave rise to the theory of social choice. This branch of mathematical economics focuses on understanding the processes of aggregation of preferences and information. It studies, for example, how different voting systems move from voter preferences to election results.

One of the relevant issues to consider in choice theory is the property of transitivity, which deals with how the preferences of an individual or collective relate to each other and are organized. For example, let's imagine that we are choosing between three types of ice cream: chocolate, vanilla and strawberry (A, B and C). The transitivity property states that, if we prefer chocolate ice cream to vanilla ice cream, and we prefer vanilla ice cream to strawberry ice cream, then we should also prefer chocolate ice cream to strawberry ice cream. This is what is expected in a set of rational preferences.

Read moreThe Condorcet paradox: do votes guarantee the victory of the candidate preferred by voters? | The Science Game

Violations of this property lead to "irrational" results, such as that of the Dutch book (and in Spanish). Let's imagine that we have other, now intransitive, preferences about ice cream: A>B, B>C, C>A. If, from the start, we have chocolate ice cream, as we prefer strawberry ice cream, we are willing to exchange ours, along with a small amount of δ money (say, € 1) for strawberry ice cream, therefore, C > A + δ. Next, we would sell C plus a small amount of money δ' to get B, since we prefer B over C. Finally, we would sell B plus a small amount of money δ'' to get A again, since we prefer A over B. At the end of the process, we would have A again, but we would have lost the amounts of money δ, δ' and δ'' in the process.

Mathematics allows us to design fair and efficient voting systems, from the work of Condorcet in the eighteenth century, which gave rise to the theory of social choice: the processes of aggregation of voter preferences

However, Condorcet showed that preferences at the social level can be intransitive, in general. For example, imagine an election with three candidates (A, B, and C) and three voters with transitive preferences: the former prefers A over B and B over C. The second, to B on C and C on A. The third, to C on A and A on B.

Therefore, if we were to vote by majority for the candidates in pairs, A would win B (two votes to one), B would win C (two votes to one) and C wins A (two votes to one). This preference cycle (A>B, B>C, C>A) is intransitive, even though none of the voters had intransitive preferences. As a result of this intransitivity, it would be possible to design a two-round system that gives the winner to the candidate we want. For example, if we want C to win, it is enough to establish a first round between A and B and the winner (A) faces C. Similarly, if we want A to win, we would set up a first round between B and C, and the winner (B) would face A.

Caricature of a humorous weekly that ironizes about the electoral farce during the regime of the Restoration (nineteenth century) in Spain.Tomás Padró Pedret (La Flaca)

Arrow's impossibility theorem delves into this idea. Theoretical economist Kenneth Arrow showed that, under certain reasonable conditions, it is impossible to design a voting system that always generates "consistent" results. In other words, there is no voting system that simultaneously satisfies all desirable conditions in a democratic system, such as non-dictatorship — there should not be a voter whose preference always prevails; the Pareto condition—if all individuals prefer one option over another, the collective result must also reflect that unanimous preference; and independence from irrelevant alternatives — the outcome between two candidates should not depend on the preferences of other candidates.

Despite these limitations, another classic Condorcet result (the jury's theorem) offers, a priori, a more optimistic view of decision processes. Suppose we are faced with a decision between two options and, unlike the previous case, there is a right choice. For example, a jury that must make a decision on the innocence or guilt of a defendant. This theorem suggests that, if each voter has an equal greater than 50% probability of making the correct decision, then a group of independent voters is more likely to get a majority decision right, compared to an individual alone. In addition, as the number of voters increases, the probability of making the right decision approaches 100%.

The economist Arrow showed that it is impossible to design a 'coherent' voting system. In other words, there is no model that simultaneously satisfies all desirable conditions.

For example, suppose a jury where each member has a 60% chance of making the right decision. However, as we see in the figure, a jury of 25 people would have a probability of almost 85%.

However, it is unrealistic to think that all voters are equally likely to get it right and this is important for applying the theorem. Depending on the probabilities of each voter, the probability of the jury getting it right tends to one or not, as the theorem thesis posits.

Recently, following a Bayesian approach, the a priori probability that the thesis predicted by the theorem is fulfilled has been estimated. That is, if we choose an arbitrary sequence of voters with different probabilities of making the correct decision, will it be true that the probability of the jury getting it right tends to 100% as we increase the number of members? The answer is no. More precisely, if a random sequence of probabilities is taken following any "symmetric" distribution, the thesis predicted by the theorem will not be fulfilled in "almost" all cases, that is, it will be fulfilled with zero probability.

These mathematical results remind us of the importance of reflecting on the voting structures and systems we use in our societies. By understanding the strengths and weaknesses of these systems, we can work on improving them and ensuring that our democracies are as epistemically efficient as possible.

Álvaro Romaniega holds a PhD in Mathematics from the Institute of Mathematical Sciences (ICMAT) and is currently a researcher in stochastic finance.

Coffee and Theoremsis a section dedicated to mathematics and the environment in which they are 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 its 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."

Editing and coordination: Ágata A. Timón G Longoria (ICMAT).

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