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Rubinstein2005-page61

Rubinstein2005-page61 - b are preferable to d Convexity is...

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October 21, 2005 12:18 master Sheet number 59 Page number 43 Consumer Preferences 43 M = ( max k { x k } , ... , max k { x k } ) is at least as good as x . Both 0 and M are on the main diagonal. By continuity, there is a bundle on the main diagonal that is indifferent to x (see the problem set). By mono- tonicity this bundle is unique; we will denote it by ( t ( x ) , ... , t ( x )) . Let u ( x ) = t ( x ) . To see that the function u represents the preferences, note that by transitivity of the preferences x % y iff ( t ( x ) , ... , t ( x )) % ( t ( y ) , ... , t ( y )) , and by monotonicity this is true iff t ( x ) t ( y ) . Convexity Consider, for example, a scenario in which the alternatives are can- didates for some position and are ranked in a left-right array as fol- lows: —– a b —– c —– d —— e —. In normal circumstances, if we know that a voter prefers b to d , then: We tend to conclude that c is preferred to d , but not necessarily that a is preferred to d (the candidate a may be too extreme). We tend to conclude that d is preferred to e (namely, we do not Fnd it plausible that both e and
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Unformatted text preview: b are preferable to d ). Convexity is meant to capture related intuitions that rely on the existence of “geography” in the sense that we can talk about an alternative being between two other alternatives. The convexity assumption is appropriate for a situation in which the argument “if a move from d to b is an improvement then so is a move part of the way to c ” is legitimate, while the argument “if a move from b to d is harmful then so is a move part of the way to c ” is not. ±ollowing are two formalizations of these two intuitions (Fg. 4.2). We will see that they are equivalent. Convexity 1: The preference relation % satisFes convexity 1 if x % y and α ∈ ( 0, 1 ) implies that α x + ( 1 − α) y % y . This captures the intuition that if x is preferred to y , then “going a part of the way from y to x ” is also an improvement....
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