Rubinstein2005-page21 - ( x , y ) of distinct elements in X...

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October 21, 2005 12:18 master Sheet number 19 Page number 3 Preferences 3 f ( x , y ) = 1 means that x is better than y , f ( x , y ) = 2 means that y is better than x and f ( x , y ) means indifference.) Not all legal answers to the questionnaire Q qualify as preferences over the set X . We will adopt two “consistency” restrictions: First, the answer to Q ( x , y ) must be identical to the answer to Q ( y , x ) . In other words, we want to exclude the common “framing effect” by which people who are asked to compare two alternatives tend to prefer the “±rst” one. Second, we require that the answers exhibit “transitivity.” In other words, the answers to Q ( x , y ) and Q ( y , z ) must be consistent with the answer to Q ( x , z ) in the following sense: If “ x is preferred to y ” and y is preferred to z ” then “ x is preferred to z ,” and if the answers to the two questions Q ( x , y ) and Q ( y , z ) are “indifference” then so is the answer to Q ( x , z ) . To summarize, here is my favorite formalization of the notion of preferences: Defnition 1 Preferences on a set X are a function f that assigns to any pair
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Unformatted text preview: ( x , y ) of distinct elements in X exactly one of the three values x y , y x or I so that for any three different elements x , y and z in X , the following two properties hold: No order effect : f ( x , y ) = f ( y , x ). Transitivity : if f ( x , y ) = x y and f ( y , z ) = y z then f ( x , z ) = x z and if f ( x , y ) = I and f ( y , z ) = I then f ( x , z ) = I . Note again that I , x y , and y x are merely symbols representing verbal answers. Needless to say, the choice of symbols is not an arbitrary one. (Why do I use the notation I and not x y ?) A Discussion of Transitivity The transitivity property is an appealing property of preferences. How would you react if somebody told you he prefers x to y , y to z and z to x ? You would probably feel that his answers are confused. Furthermore, it seems that, when confronted with an intransitivity...
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