Social Choice_Arrow's Theorem

# Social Choice_Arrow's Theorem - Arrows Impossibility...

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Arrow°s Impossibility Theorem 1 Matt Van Essen University of Alabama 1 The proof presented is based on Reny (2000). Van Essen (U of A) Impossibility Theorem 1 / 44

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Arrow°s Impossibility Theorem Theorem If the number of possible alternatives is greater than or equal to three, then there is no social ranking function F that satis°es the Pareto e¢ ciency, Independence of Irrelevant Alternatives, and Non-dictatorship. Van Essen (U of A) Impossibility Theorem 2 / 44
Arrow°s Impossibility Theorem The proof of Arrow°s Theorem that we present is from a great paper by economist P. Reny. In particular, we shall see that the proof mirrors the proof that we presented for the Muller-Satterthwaite Theorem. Van Essen (U of A) Impossibility Theorem 3 / 44

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Arrow°s Impossibility Theorem Step 1: A Good Place to Start We begin the proof by picking a special "extreme" pro±le where we know the top and bottom alternatives in social ranking. In particular, choose any a , b 2 A and consider a set of rankings where each voter i °s ranking R i places alternative a at the top of the ranking and b at the bottom of their ranking. This pro±le is illustrated in in Figure 1. Van Essen (U of A) Impossibility Theorem 4 / 44
Arrow°s Impossibility Theorem R 1 ° ° ° R n ± 1 R n R n + 1 ° ° ° R N Social Ranking a ° ° ° a a a ° ° ° a a ° ° ° ° ° ° ° ° ° ° ° ! ° ° ° ° ° ° ° b . . . b b b ° ° ° b b Figure 1 Van Essen (U of A) Impossibility Theorem 5 / 44

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Arrow°s Impossibility Theorem Since F satis±es the Pareto property and all voter°s are unanimous in their preference for a the social ranking a is alone at the top of the social ranking. In addition, since b is at the bottom of everyone°s ranking, by unanimity, b must be at the bottom of the social ranking. Van Essen (U of A) Impossibility Theorem 6 / 44
Arrow°s Impossibility Theorem Step 2: Smoking Out a Pivotal Voter It turns out that one voter, a ²pivotal³voter, is going to have a lot of sway over when the relative social ranking between a and b changes. We would like to ±nd this particular voter. Therefore in this step of the proof we are going to make a series of small changes designed to identify this pivotal voter. Van Essen (U of A) Impossibility Theorem 7 / 44

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Arrow°s Impossibility Theorem (a) Moving b on Voter 1°s Ranking without Changing the Social Ranking of a and b : We are going to change the pro±le illustrated in Figure 1 by moving alternative b up one position in voter 1°s ranking. In particular, alternative b is now ranked second to last in 1°s ranking. This is illustrated in Figure 2. Van Essen (U of A) Impossibility Theorem 8 / 44
Arrow°s Impossibility Theorem R 1 ° ° ° R n ± 1 R n R n + 1 ° ° ° R N Social Ranking a ° ° ° a a a ° ° ° a a ° ° ° ° ° ° ° ° ° ° ° ! ° b ° ° ° ° b ° ° ° ° b b b ° ° ° b Figure 2 Van Essen (U of A) Impossibility Theorem 9 / 44

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Arrow°s Impossibility Theorem What happens to the social ranking of a and b ?
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• Spring '12
• Vanessen
• Social Choice and Individual Values, Voting system, van Essen, impossibility theorem, Social choice theory, Arrow's impossibility theorem

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