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Can a preference relation be complete but not transitive?

  1. Feb 5, 2014 #1
    1. The problem statement, all variables and given/known data
    This is not a homework problem, but a topic in a microeconomics book that I am unclear about.

    My book argues that the set X = {a, b, c, d} of preferences can be (i) transitive but (ii) incomplete.

    Is it possible for a similar set of preferences to be (i) complete but (ii) intransitive, depending on how we define the relations between the elements? My book does not go into this, but I am curious, and any explanation would be much appreciated.

    2. Relevant equations

    (i) Completeness axiom: For every pair x, y [itex]\in[/itex] X, either x is weakly preferred to y, y is weakly preferred to x, or both (that is, we are indifferent between x and y).

    (ii) Transitivity axiom: For every triple x, y, z [itex]\in[/itex] X, if x is weakly preferred to y and y is weakly preferred to z, then x is weakly preferred to z.


    3. The attempt at a solution

    I understand that the set X = {a, b, c, d} can be transitive but incomplete. My book explains this with a table (picture attached; my book says that this is only one way of proving that the set X can be transitive but incomplete, and that there are many other ways); the incompleteness arises because we have not defined a relationship between c and d, making them incomparable.

    However, the set is transitive because (i) a is weakly preferred to b, and b is weakly preferred to c; (ii) a is weakly preferred to b, and b is weakly preferred to d; (iii) a is weakly preferred to c and a is also weakly preferred to d.

    I don't understand how it is possible for a set to be complete but intransitive. If we have defined relationships between all the elements such that any two pairs are comparable, doesn't transitivity automatically follow? Is my understanding correct?
     

    Attached Files:

  2. jcsd
  3. Feb 5, 2014 #2
  4. Feb 9, 2014 #3
    No worries, I have figured this problem out. Please close this thread.
     
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