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Unformatted text preview: Today in Comparative Politics • 2010 Rutgers/Ritsumeikan Spring and Summer Exchange Program • Majority rule • Single peakedness • Median voter theorem • Chaos in multidimensional problems • Arrow impossibility theorem Intransitive majority preferences • A preference relation R is transitive if XRy and yRz imply xRz. • When majority preferences cycle, • The majority preference relation is intransitive. • The outcome under a binary procedure is completely determined by the agenda. • Whoever controls the agenda has great power here. • “Anonymous” majority rule doesn’t look so egalitarian now. • Opportunities for strategic voting are numerous. Does majority rule always cycle? • Majority voting is very commonly used in distributivepolitics settings. • Saw last time how easy it is for cycles to occur in that case. • But there is an important circumstance in which majority rule is well behaved. Single peakedness • Assume that all individual preferences are strict. • Assume that there is an odd number of voters. • Suppose that in every triple of alternatives {x, y, z} there exists one alternative, say x, that every individual agrees is not worst . • That is, for all i either xP i y or xP i z • Then preferences over {x, y, z} are said to be single peaked . • Why “single peaked”? A singlepeaked preference profile y x z 1 2 3 4 If majority preferences cycle, single peakedness doesn’t hold 1 2 3 x y z y z x z x y Nonsingle peaked preference profile x y z 1 2 3 x z y 1 3 2 y x z 2 1 3 y z x 2 3 1 z x y 3 1 2 z y x 3 2 1 Black ’s theorem • If the number of voters is odd and preferences are singlepeaked, the majority preference relation is transitive. • Proof: pretty easy algebra • I will leave the argument in the Sakai slide deck marked again in yellow. Suppose x P maj y P maj z P maj x xPyPz xPzPy yPxPz yPzPx zPxPy zPyPx Suppose x P maj y P maj z P maj x xPyPz n 1 xPzPy n 2 yPxPz n 3 yPzPx n 4 zPxPy n 5 zPyPx n 6 Suppose x P maj y P maj z P maj x If x is not worst, n 4 = n 6 = 0 xPyPz n 1 xPzPy n 2 yPxPz n 3 yPzPx n 4 zPxPy n 5 zPyPx n 6 Suppose x P maj y P maj z P maj x If x is not worst, n 4 = n 6 = 0 If y is not worst, n 2 = n 5 = 0 xPyPz n 1 xPzPy n 2 yPxPz n 3 yPzPx n 4 zPxPy n 5 zPyPx n 6 Suppose x P maj y P maj z P maj x If x is not worst, n 4 = n 6 = 0 If y is not worst, n 2 = n 5 = 0 If z is not worst, n 1 = n 3 = 0 xPyPz n 1 xPzPy n 2 yPxPz n 3 yPzPx n 4 zPxPy n 5 zPyPx n 6 Suppose x P maj y P maj z P maj x If x is not worst, n 4 = n 6 = 0 If y is not worst, n 2 = n 5 = 0 If z is not worst, n 1 = n 3 = 0 xPyPz n 1 xPzPy n 2 yPxPz n 3 yPzPx n 4 zPxPy n 5 zPyPx n 6 xP maj y: n 1 + n 2 + n 5 > n 3 + n 4 + n 6 (1) Suppose x P maj y P maj z P maj x If x is not worst, n 4 = n 6 = 0 If y is not worst, n 2 = n 5 = 0 If z is not worst, n 1 = n 3 = 0 xPyPz n 1 xPzPy n 2 yPxPz n 3 yPzPx n 4 zPxPy n 5 zPyPx n 6 xP maj y: n 1 + n 2 + n 5 > n 3 + n 4 + n 6 (1) yP maj z: n 1 + n 3 + n 4 > n 2 + n 5 + n 6 (2) Suppose x P...
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 Fall '09
 BLAIR
 Comparative Politics, Media, Voting system, CCRS, xPyPz n1, yPzPx n4, xPzPy n2

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