# Can mathematics unlock the political situation?

Politics is the discipline that studies power. To analyze this field, a good piece of wood to hold on to so as not to swell is mathematics. A first tool for quantifying power is the Banzhaf Index (IB) which records the number of winning coalitions in which a political subject is essential. Let's give a fictional example. At one mayor's office, a minimum of eight councillors are required to approve a measure. Suppose there are three matches A, B and C, and A has six councillors, B has three and C has two. The winning coalitions are AB, AC and ABC. Party A is crucial in all three coalitions, while B and C are only essential in AB and AC, respectively. Then, IB(A)-3/5, while IB(B)-IB(C)-1/5. We see that the power of a party is not determined by its number of representatives by the number of seats, but by its IB. The paradox of new members states that by extending the distribution of power to new members, the IB of the former does not decrease. This result is generalizable at any consistent power index

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Information and power are intimately related concepts. The media broadcasts news, supporting or decreasing auctoritas a ruler and a system. On the other hand, the origin of the term "statistic", which comes from "state", is not coincidental, since ancient times, statesmen ordered to produce censuses, cadasatheers, sampling and establish correlations between social phenomena. And the more information a politician has, the better he can develop his strategy and an action plan. In the age of computing, big data and machine learning are essential tools in information processing and decision-making.

With these technological advances, would direct representation be possible, where every citizen votes their preferences from their smartphone? The Condorcetparadox, Arrow's paradox and the teorem of Kirman-Sondermann's hidden dictatorship are mathematical results that negate this possibility, without a dictator or oligarchy appearing whose judgment would coincide always with collective preferences. These conclusions support the iron law of the oligarchsby Robert Michels, who states that every social organization is always dominated by a small core of people, due to reasons of efficiency, initiative and tactics.

In today's democratic system, political representation is needed to respect the interests of civil society that is not present in decision-making or law-making. The minimum unit of power is the district and the electoral system used to elect deputies varies by country. In the Anglo-Saxon world, a single-member system is used, electing a single representative acting on an imperative mandate, responding to his district's grievances. In Spain, the D'Hondt system is in force, in which parties obtain deputies corresponding to an algorithm with successive sorts depending on the votes and divisions.

Voting is a fundamental political fact because in addition to electing representatives, it measures the legitimacy of the system. But if it is so important, one might wonder about the role of referendums in political life. In this regard, from a mathematical point of view, it should be clarified that a proposal to be put to the vote must meet two requirements: it cannot be a truth or a logical falsehood (paradox of Wollheim) and cannot deny the system in which it has (paradox of democracy).

To understand the political negotiations of the investiture in Spain, it is useful to know some strategies. From an economic point of view, Anthony Downs, author of The economic theory of democracy explains the ideological adaptations of the different formations to occupy the maximum spectrum, such as hotels to occupy the maximum length of coastline. A famous model of game theory, the prisoner's dilemma, teaches us that two bosses do not usually cooperate by taking common risks, but rather that the dominant strategy is to look for their particular interest. The game of the hen consists of braking a car as close as possible to a cliff without falling; and the winner is the one closest to the precipice. This game explains numerous political phenomena, such as the current conflicts between parties when accepting demands to make pactar.