3Voting ParadoxRecall, 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
4Voting ParadoxYet as we examine the various voting systems put forth in the world we need to keep in mind some conceptual problems with voting theoryIt may not be possible to move from individual to group preferences smoothly or meaningfully
5Voting Paradox Voter 1 X, Y, Z Voter 2 Y, Z, X Voter 3 Z, X, Y In this population, what do “the people” want?
6Condorcet CandidateOne way to determine what the people prefer to is consider the choice to be the one which defeats all others in a pair-wise comparisonWe call that the Condorcet candidate after the Marquis de Condorcet ( )
7Voting Paradox Voter 1 X, Y, Z Voter 2 Y, Z, X Voter 3 Z, X, Y In this population, what do “the people” want?
8Voting Paradox Voter 1 X, Y, Z Voter 2 Y, Z, X Voter 3 Z, X, Y Note: X>Y, Y>Z, Z>X
9Voting Paradox Is there a way around this problem? Condorcet first discovered the problem, but his solution isn’t always going to workRaises potentially troubling issue for democratic theoryCan any voting system reveal aggregate meaningfully from individual to group preferences?
10Arrow’s Impossibility Theorem Universal Admissibility of Individual PreferencesAll possible orderings by indiviudlars are admissiableNo institutions (e.g., parties) can restrict the orderings so that certain preferences scales cannot be expressed
11Arrow’s Impossibility Theorem Positive Association of individual and social valuesGiven 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, thenIt is still the case that X>YThis 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
12Arrow’s Impossibility Theorem Independence of Irrelevant AlternativesIf “S” is a subset of the set of available alernatives and the preference scales of individuals change with respect to alternatives not in SThen the social ordering for alternatives in S does not change
13Arrow’s Impossibility Theorem Citizen’s SovereigntyFor any two alternatives X and Y, there exist individual preference scales such that X is preferred to Y in the social orderingIn other words, the social outcome is not imposedAt the extreme, if all individuals should prefer “X to Y,” then X cannot be prohibited by the social outcomeOutlaws the possibility that the social outcome is unrelated to the preference scales of the society’s members
14Arrow’s Impossibility Theorem NondictatorshipFor 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 orderingThere is no individual who can dictate the social ordering of alternatives
15Arrow’s Impossibility Theorem The intent of the assumptions is to link society’s ordering of alternatives to individuals’ preference scales in a nonarbitrary wayWe want the social outcome responsive to the preference scales of individuals
16Arrow’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 assumptionsThere is no method of summing individuals preferences that satisfies all 5 assumptionsIf 1 through 3, then either the 4 or 5 is being violated (that is, order is imposed from without or from within)