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### 24_Desirable_properties

Course: MATH 125, Fall 2009
School: St. Xavier
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Word Count: 781

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properties Desirable of social choice procedures We now outline a number of properties that are desirable for these social choice procedures: 1. Pareto [named for noted economist Vilfredo Pareto (18481923)] If alternative a is preferred to alternative b on every voter's ballot, then b is never a winner. 2. Condorcet [named for another economist, Marie-Jean-AntoineNicolas de Caritat, the Marquis de Condorcet...

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properties Desirable of social choice procedures We now outline a number of properties that are desirable for these social choice procedures: 1. Pareto [named for noted economist Vilfredo Pareto (18481923)] If alternative a is preferred to alternative b on every voter's ballot, then b is never a winner. 2. Condorcet [named for another economist, Marie-Jean-AntoineNicolas de Caritat, the Marquis de Condorcet (17431794)] If in a pairwise comparison with each of the other alternatives, alternative a emerges the winner (whereby it is referred to as the Condorcet winner), then a is the only winner of the election. 3. Monotonicity (or non-perversity) Suppose alternative a is one of the winners. If one of the voters then modifies his ballot by moving a higher up in his preference ordering, and the election is redecided, a is still declared a winner. 4. Independence of irrelevant alternatives Suppose a is a winner and b is not. If some of the voters then modify their ballots in such a way that their prior preference of a over b (or of b over a) is not altered in the process, then when the election is redecided, b is still not declared a winner. Which of the five social choice procedures satisfy these desirable properties? The following table gives the answers: Pareto Plurality Borda Hare Seq. Pairs Dictator Yes Yes Yes No Yes Condorcet No No No Yes No Monotonicity I.I.A. Yes Yes No Yes Yes No No No No Yes For arguments that show why the cells marked Yes are true, see Taylor, pp. 108113. For examples that illustrate why the cells marked No are ture, see Taylor, pp. 113120. In particular, it is important to note that none of the social choice procedures satisfy all four of the desirable properties. This is not because of some flaw in each one of these procedures; indeed, it is a flaw in the inherent nature of social choice. Theorem. No social choice procedure for dealing with more than three alternatives can satisfy both the Condorcet criterion and the Independence of irrelevant alternatives criterion. Before we prove this theorem, let us first consider a phenomenon known as the Condorcet voting paradox (attributed, of course, to the Marquis de Condorcet). It illustrates some of the difficulties that any social choice must function deal with. Suppose that a public with three voters faces an election amongst three alternatives a, b, c. If the three ballots are p1 a b c p2 c a b p3 b c a then which alternative should a "good" social choice procedure choose for a winner? If a is chosen the winner, then p2 and p3 could argue that they form a majority of the public who prefer c to a. If b is chosen the winner, then p1 and p2 could argue that they form majority of the a public who prefer a to b. And if c is chosen the winner, then p1 and p3 could argue that they form a majority of the public who prefer b to c. Thus no social procedure function can settle on a unique winner that isn't inferior to some other alternative to a majority of voters! A proof of the theorem we stated earlier proceeds as follows: Proof. Suppose there were some social choice procedure that did satisfy both the Condorcet criterion and the Independence of irrelevant alternatives criterion. How then would this procedure deal with the set of ballots p1 a b c p2 c a b p3 b c a which we looked at earlier? Consider what happens if the third voter switches the positions of alternatives b and c. In this event, c would defeat both a and b in a pairwise comparison, hence for that set of ballots, c would have to be the unique winner, as we are assuming that ...

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