Kenneth Arrow’s Proof of the Incompatibility of Certain Voting Systems
Kenneth Arrow was an American mathematician, economist, and social theorist who won the Nobel Prize in Economics in 1972. In 1951, in his paper “Social Choice and Individual Values”, he provided a famous proof demonstrating the incompatibility of certain voting systems with certain criteria. This proof established the foundations for the area of study known today as “social choice theory”. In this article, we will discuss the background and significance of Arrow’s theorem, his proof of incompatibility, and the various implications of its conclusions.
Background of Kenneth Arrow’s Theorem
The theories of voting by majority have existed for centuries and date back to as early as the 17th century. An important early thinker in proposing voting models was the French scholar Jean-Charles de Borda, who introduced the concept of “pareto efficiency” as a criterion of selection. According to this criteria, if any outcome of a voting system is not acceptably efficient, then it should be rejected. Later, in the 19th century, the Swiss mathematician Leonard Jakobsohn proposed the ideas of “preference” and “independence of irrelevant alternatives”, which hold that a voter should be allowed to rank each option in order of preference and those rankings should not be affected by the presence of any other options.
Kenneth Arrow’s Proof of Incompatibility
Arrow’s theorem states that, under certain conditions, it is impossible to find a voting system that satisfies all the criteria of voter independence, preference, and Pareto efficiency. His proof as outlined in his 1951 paper is as follows:
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Assume that a social choice function exists that satisfies all the criteria of Pareto efficiency, preference, and independence of irrelevant alternatives.
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Let there be three voters, labeled X, Y and Z, who each can rank three alternatives A, B, and C.
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Assume that the preferences of each voter can change with the addition or removal of an alternative.
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Consider all possible outcomes of the voting system with respect to these three voters and the three alternatives.
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Suppose that there is an outcome in which the social choice function selects a different alternative from the one favored by the majority.
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Since the voting system satisfies all the criteria of Pareto efficiency, preference, and independence of irrelevant alternatives, it follows that either X or Y must be indifferent between A and B, and yet have C as their least preferred option.
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Therefore, no voting system that satisfies all the criteria of preference, independence of irrelevant alternatives, and Pareto efficiency can exist.
Consequences of Arrow’s Theorem
The proof of Arrow’s theorem demonstrated unambiguously that the concept of “majority rule” is not necessarily the same as “fair decision-making”. The theorem has since highlighted some of the weaknesses of voting systems, and has had far-reaching implications in fields such as economics, politics, and game theory. Arrow’s theorem has been used as evidence in support of proportional representation systems, which aim to provide fairer outcomes in elections. It has also been cited as a reason to be more wary of the validity of opinion polls.
Modern Implications of Arrow’s Theorem
Despite being proposed over 70 years ago, Arrow’s theorem is still highly relevant today. The concept of “cyclical majorities”, whereby different voters in successive elections can side with a different majority each time, is prohibited by the theorem. Arrow’s theorem has also been used to form a mathematical basis for the recent public approval of democratic systems and multiparty governance.
Kenneth Arrow’s 1951 proof of the incompatibility of certain voting systems with certain criteria had immense implications for democracy and decisions making in general. The theorem, which remains relevant today, highlighted the deficiencies of voting systems based solely on majority rule and serves to emphasize the importance of proportional representation and fairness in decision-making. Overall, the proof demonstrated that it is impossible to find a voting system that satisfies all the criteria of preference, independence of irrelevant alternatives, and Pareto efficiency, and it is this impossibility that is still felt today.
