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Arrow Paradox

The Arrow Impossibility Theorem, also called the Arrow Paradox and with little precision, the Impossibility Theorem of democracy, shows that it is possible to design rules for making social or political decisions that obey a certain set of criteria "reasonable."

was enunciated and demonstrated for the first time by the Nobel laureate Kenneth Arrow, as part of his doctoral thesis Social choice and individual values, and popularized in his book of the same name released in 1951. The original article, A Difficulty in the Concept of Social Welfare, was published in The Journal of Political Economy, Vol 58 (4), pp. 328-346, August 1950.

simplified Statement of Theorem

Impossibility Theorem Arrow provides that a company needs to agree an order of preference among different options. Each individual in society has its own order of personal preference. The problem is finding a general mechanism (a social choice function) that transforms the set of individual preference orders in order of preference for the society, which must satisfy several desirable properties:

unrestricted or domain
• universality: the social choice function should create a complete order for every possible set of individual preference orders (the result of the vote should be able to sort each other all the preferences and the voting mechanism should be able to process all possible sets of voter preferences)
• Non-imposition or citizen sovereignty, every order of preference of society should be achievable by some set of individual preference orders. (Each result should be achievable in some way).
• Absence of dictatorship: the social choice function should not simply follow the order of preference of a single individual while ignoring others.
• positive association between individual and social values \u200b\u200bor monotony: if an individual modifies its preference order by promoting a certain option, the preference order of society must respond by promoting that same option or, at most, without changing it, but not degrading it. (An individual should not affect a candidate to promote.)
• Independence of irrelevant alternatives: if we restrict our attention to a subset of options and we apply the social choice function by themselves, then the result should be compatible with the corresponding full set of options. The changes in the way that an individual ordering the alternatives "irrelevant" (ie outside the subset) should not impact the system to make society the subset "relevant."
  

Arrow's theorem says that if the decision-making body has at least two members and at least three options to decide, then it is impossible to design a social choice function that simultaneously satisfies all these conditions.

Obtained another version of the theorem by replacing the criterion of monotonicity with the criterion of unanimity:

• Unanimity or Pareto efficiency: if every individual prefers a certain option to another, so must the resulting societal preference order . This statement is stronger, since both assume monotheism as the independence of irrelevant alternatives implies Pareto efficiency. Interpretations

Arrow's Theorem

Arrow's Theorem is usually expressed in mathematical language does the phrase "No voting system is fair." However, this statement is incorrect or, at best, inaccurate, and that would be needed to clarify what is meant by a fair voting mechanism. Although Arrow himself uses the term "fair" to refer to their criteria, not at all clear that way.

The most discussed is the criterion of independence of irrelevant alternatives as it seems too "strong." And so, with a narrower definition of "irrelevant alternatives" to exclude those candidates from the set of Smith, Condorcet methods satisfy some all criteria.

In any case, Arrow's Theorem is a significant result with profound implications for the field of decision theory.