The Arrow Possibility Theorem says that it is not generally possible to find a social choicemechanism fulfilling (1) non-dictatorship, (2) independence of irrelevant alternatives, (3)Pareto efficiency. But simple majority voting on pairwise alternatives seems to do well inProfile II in Question 1, and indeed, for all single peaked preference arrays. Is this a coun-terexample to the Arrow Possibility Theorem? If not, which of the four properties is (are)not fulfilled. Explain.
Question
The Arrow Possibility Theorem says that it is not generally possible to find a social choicemechanism fulfilling (1) non-dictatorship, (2) independence of irrelevant alternatives, (3)Pareto efficiency. But simple majority voting on pairwise alternatives seems to do well inProfile II in Question 1, and indeed, for all single peaked preference arrays. Is this a coun-terexample to the Arrow Possibility Theorem? If not, which of the four properties is (are)not fulfilled. Explain.
Solution
No, this is not a counterexample to the Arrow Possibility Theorem. The simple majority voting on pairwise alternatives does seem to fulfill the conditions of non-dictatorship, independence of irrelevant alternatives, and Pareto efficiency in certain cases, such as single-peaked preference arrays. However, it fails to meet these conditions in all cases, which is what the Arrow Possibility Theorem states.
The property that is not fulfilled is the "independence of irrelevant alternatives". This property states that the social preference between alternatives x and y should depend only on the individual preferences between x and y, and not on individual preferences regarding other alternatives.
In the case of simple majority voting, the outcome can be influenced by the presence or absence of irrelevant alternatives. For example, consider three alternatives A, B, and C, and three voters with the following preferences:
Voter 1: A > B > C Voter 2: B > C > A Voter 3: C > A > B
If we only consider alternatives A and B, A wins by 2 votes to 1. But if we introduce the irrelevant alternative C, B wins by 2 votes to 1. Thus, the introduction of an irrelevant alternative has changed the outcome, violating the independence of irrelevant alternatives.
Similar Questions
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