Rubinstein2005-page23

# Rubinstein2005-page23 - f ( x , z ) = x ² z . However,...

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October 21, 2005 12:18 master Sheet number 21 Page number 5 Preferences 5 2. The use of similarities as an obstacle to transitivity . In some cases, the decision maker expresses indifference in a comparison be- tween two elements that are too “close” to be distinguishable. For example, let X (the set of real numbers). Consider an indi- vidual whose attitude is “the more the better”; however, he ﬁnds it impossible to determine whether a is greater than b unless the difference is at least 1. He will assign f ( x , y ) = x ² y if x y 1 and f ( x , y ) = I if | x y | < 1. This is not a preference relation since 1 . 5 0 . 8 and 0 . 8 0 . 3, but it is not true that 1 . 5 0 . 3. Did we require too little? Another potential criticism of our deﬁni- tion is that our assumptions might have been too weak and that we did not impose some reasonable further restrictions on the concept of preferences. That is, there are other similar consistency require- ments we may impose on a legal response to qualify it as a descrip- tion of preferences. For example, if f ( x , y ) = x ² y and f ( y , z ) = I , we would naturally expect that
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Unformatted text preview: f ( x , z ) = x ² z . However, this ad-ditional consistency condition was not included in the above def-inition since it follows from the other conditions: If f ( x , z ) = I , then by the assumption that f ( y , z ) = I and by the no order effect, f ( z , y ) = I , and thus by transitivity f ( x , y ) = I (a contra-diction). Alternatively, if f ( x , z ) = z ² x , then by no order effect f ( z , x ) = z ² x , and by f ( x , y ) = x ² y and transitivity f ( z , y ) = z ² y (a contradiction). Similarly, note that for any preferences f , we have if f ( x , y ) = I and f ( y , z ) = y ² z , then f ( x , z ) = x ² z . The Questionnaire R A second way to think about preferences is through an imaginary questionnaire R consisting of all questions of the type: R(x,y) (for all x , y ∈ X , not necessarily distinct). “Is x at least as preferred as y ?” Tick one and only one of the following two options: ± Yes ± No...
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## This note was uploaded on 12/29/2011 for the course ECO 443 taught by Professor Aswa during the Fall '10 term at SUNY Stony Brook.

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