Lecture41 Wed April 29 -...

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CS 4850 Lecture 41 –scribes: Heqing Li & Sujith Vidanapathirana April 29, 2009 I. Continuation of Arrow Theorem Proof Arrow Theorem There’s no way to satisfy all of the 3 axioms: 1) Unanimity 2) Independency of irrelevant alternatives 3) Non dictator We have been trying to prove 1) + 2) => No 3). If everyone ranks b as the least preferred, the global ranking would have b at the bottom. Starting from this preference list, if we move b to the top one by one, somewhere in the process of moving b up to the 1 st place, b should become the 1 st choice in the global ranking. Individual rankings Global ranking Assume, the non b ranking entries outside vb are arbitrary. When we move b down in vb, b would move the same way in the global ranking. Vb is a dictator. For any other candidate being ranked, a, the only time that b can switch order with a in global ranking is when b and a switch in vb. Thus, b must go through the global ranking the same way as in vb. bbbbbbb b bbb bbbb vb b
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CS 4850 Lecture 41 –scribes: Heqing Li & Sujith Vidanapathirana April 29, 2009 All the rankings outside vb are arbitrary, vb is almost a dictator, dictator of anything but b.
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Lecture41 Wed April 29 -...

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