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Voting Theory

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Overview Voting Paradoxes Condorcet Criterion Arrow’s Impossibility Theorem

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Voting Paradox Recall, democratic theory predicated on the idea that somehow the vote reveals “the will of the people” That means we need to be able to move from individual preferences to something like a “social preference” The winner of the election is in some meaningful sense reflective of what “the people” want

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Voting Paradox Yet as we examine the various voting systems put forth in the world we need to keep in mind some conceptual problems with voting theory It may not be possible to move from individual to group preferences smoothly or meaningfully

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Voting Paradox Voter 1X, Y, Z Voter 2Y, Z, X Voter 3Z, X, Y In this population, what do “the people” want?

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Condorcet Candidate One way to determine what the people prefer to is consider the choice to be the one which defeats all others in a pair-wise comparison We call that the Condorcet candidate after the Marquis de Condorcet ( )

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Voting Paradox Voter 1X, Y, Z Voter 2Y, Z, X Voter 3Z, X, Y In this population, what do “the people” want?

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Voting Paradox Voter 1X, Y, Z Voter 2Y, Z, X Voter 3Z, X, Y Note: X>Y, Y>Z, Z>X

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Voting Paradox Is there a way around this problem? Condorcet first discovered the problem, but his solution isn’t always going to work Raises potentially troubling issue for democratic theory Can any voting system reveal aggregate meaningfully from individual to group preferences?

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Arrow’s Impossibility Theorem 1.Universal Admissibility of Individual Preferences –All possible orderings by indiviudlars are admissiable –No institutions (e.g., parties) can restrict the orderings so that certain preferences scales cannot be expressed

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Arrow’s Impossibility Theorem 2.Positive Association of individual and social values Given that X>Y is the social ordering, if individuals either raise or do not change the ranking of X in their preference scales and the ranking of Y remains unchanged, then It is still the case that X>Y This restriction ensures that the method of adding individuals’ preference scales reflects, in a nonperverse way, these preferences: the social ranking of X does not respond negatively to changes in rankings by individuals

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Arrow’s Impossibility Theorem 3.Independence of Irrelevant Alternatives –If “S” is a subset of the set of available alernatives and the preference scales of individuals change with respect to alternatives not in S –Then the social ordering for alternatives in S does not change

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Arrow’s Impossibility Theorem 4.Citizen’s Sovereignty –For any two alternatives X and Y, there exist individual preference scales such that X is preferred to Y in the social ordering In other words, the social outcome is not imposed At the extreme, if all individuals should prefer “X to Y,” then X cannot be prohibited by the social outcome Outlaws the possibility that the social outcome is unrelated to the preference scales of the society’s members

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Arrow’s Impossibility Theorem 5.Nondictatorship –For any two alternatives X and Y, there is no individual such that whenver he or she prefers X to Y, X is always preferred to Y in the social ordering There is no individual who can dictate the social ordering of alternatives

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Arrow’s Impossibility Theorem The intent of the assumptions is to link society’s ordering of alternatives to individuals’ preference scales in a nonarbitrary way We want the social outcome responsive to the preference scales of individuals

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Arrow’s Impossibility Theorem Arrow then demonstrates that given these basic assumptions, no socialordering is possible that doesn’t violate one or the other of the assumptions There is no method of summing individuals preferences that satisfies all 5 assumptions If 1 through 3, then either the 4 or 5 is being violated (that is, order is imposed from without or from within)

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