ARROW AND THE IMPOSSIBILITY THEOREM

Amartya Sen ARROW AND THE IMPOSSIBILITY THEOREM PDF

• Arrow’s impossibility theorem
• Amartya Sen.

Amartya Sen’s lecture on Kenneth Arrow’s “general impossibility theorem” can be summarized as follows - The Arrow’s theorem illustrates the problems in deriving social decisions from individual preferences, based on the basic values of democracy.

• The proof of the theorem shows that the contradiction follows from several axioms (unrestricted domain of definition, independence from irrelevant alternatives, Pareto principle, non-dictatorship).
• The proof uses a complement that shows the expansion and contraction of a definite set of individuals.
• The combination of axioms makes it impossible to take into account the nature of the choices and personal circumstances, and limits the voting rules.
• Allowing for inter-individual comparisons could address the issue of welfare economics.
• Political decision-making must also address issues of individual liberties and rights.
• Arrow’s theorem is a landmark achievement that revolutionized the intellectual world of social thought by showing that serious problems can arise from seemingly reasonable combinations of axioms.

What is the expansion and contraction of a definitive set of individuals?

The “expansion and contraction of the definite set of individuals” refers to the two complements (intermediate results) in the proof of Arrow’s theorem.

• Spread of Decisiveness: if a set G is decisive for a pair {x, y} of two alternatives, then G is (overall) decisive for all pairs. In other words, determinism for one pair of alternatives spills over to all other pairs.
• Contraction of Decisive Sets: If a set G is decisive and contains more than one individual, then the true subset (smaller set) of G is also decisive. In other words, if you reduce individuals from a decisive set, the set remains decisive. These corollaries play an important role in the proof of Arrow’s Theorem. The “deterministic extension” shows that determinacy for a pair of alternatives spills over to all other pairs, and the “deterministic set reduction” shows the process that ultimately leads to the existence of the dictator. Combining these corollaries leads to Arrow’s general impossibility theorem.

Explain “Allowing interindividual comparisons can address the problem of welfare economics.”

This part of the paper proposes allowing utility comparisons between individuals as one way to overcome the impossibility of social choice presented by Arrow’s theorem.

Arrow’s theorem states that individual preferences are independently ordinalized for each individual and does not allow for utility comparisons between individuals. In other words, it is not possible to determine how much more utility one individual has than another. Under this restriction, social welfare functions that simultaneously satisfy the Pareto principle, independence from irrelevant alternatives, and the axiom of unrestricted domain of definition are limited to dictatorial ones.

However, Sen argues that allowing for inter-individual utility comparisons avoids this impossibility. For example, a proposal to distribute a portion of income from the poorest to the rest would be a Pareto improvement in a selfish society, but an interindividual comparison would determine that this proposal would undermine social welfare.

• Quadratic Votingによって個人間の効用比較を実現する手があるわけだ

By incorporating interindividual comparisons, a broader information base can be used in determining social welfare and assessing equity. This allows us to better address the issues that welfare economics needs to address.

However, as Sen also points out, it is not easy to apply interindividual comparisons to political decision making. Therefore, a different approach to the problem of political decision making is needed than interindividual comparisons.

“Political decision-making must also address issues of individual liberty and rights,” he explained.

This part of the article emphasizes the importance of individual freedom and rights in political decision-making, apart from the problems of social choice presented by Arrow’s theorem.

While Arrow’s theorem illustrates the difficulties in deriving social decisions from individual preferences, the issue referred to here is that individual liberties and rights should be protected, even if a large number of people disagree.

For example, following the argument of John Stuart Mill., minority rights and individual liberties should be protected even though the majority may oppose the minority’s freedom to choose its own lifestyle or attempt to restrict the exercise of individual liberties. The principle of majority rule alone does not adequately address such issues of individual liberty and rights.

Even the combination of axioms presented in Arrow’s theorem does not take this issue into account. This means that mechanisms are needed in political decision-making to protect individual liberties and rights, rather than relying solely on the principle of majority rule.

From this perspective, political decision making must address the issue of how to protect individual liberties and rights in addition to the issue of aggregating social preferences. This suggests a challenge for political decision making from a different angle than Arrow’s theorem.

Why not introduce Quadratic Voting as a strategic process for individuals to properly express their utility?

Quadratic Voting (QV) is proposed as a voting method that takes into account the intensity of individual preferences and discourages strategic behavior; with QV, voters can purchase multiple votes, with the price of each vote proportional to the square of the number of votes. This provides an incentive to express one’s preferences more honestly.

Considering QV in the context of Arrow’s theorem, there are several possible advantages:

• QV can capture information about the intensity of an individual’s preferences. This allows for richer, more informed social decisions.
• Because strategic behavior can be discouraged, individuals will be more honest in expressing their true preferences. This may help satisfy the “independence from irrelevant alternatives” condition in Arrow’s theorem.
• QV also allows minority views to be taken into account. Minorities with strong preferences can assert their views by buying more votes.

However, QV also has its challenges:

• Income and wealth inequality may influence voting outcomes. Wealthy individuals or groups may have undue influence.
  • This is “$1 of money” in QV with money.subjective value が貧乏人には高く金持ちには低いことに起因している
    • There is no reason why the credit used to buy votes in QV must be money, so just introduce some kind of non-purchaseable credit.
• QV assumes a cardinal comparison of individual preferences. However, utility comparisons between individuals still present difficulties.
• The complex nature of the voting system requires voter understanding and acceptance.

Thus, QV can be considered as one potential solution to the problem of social choice presented by Arrow’s theorem, but at the same time, there are challenges to its implementation: while QV may allow for more accurate aggregation of individual preferences, it should be carefully considered in terms of equity and feasibility. However, it will need to be carefully considered in terms of fairness and feasibility.

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