Rubinstein2005-page45 - A which is the-maximal from the...

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October 21, 2005 12:18 master Sheet number 43 Page number 27 Choice 27 Figure 3.1 Violation of condition . We will now identify a condition under which a choice function can indeed be presented as if derived from some preference relation (i.e., can be rationalized). Condition : We say that C satisFes condition if for any two problems A , B D , if A B and C ( B ) A then C ( A ) = C ( B ) . (See Fg. 3.1.) Note that if % is a preference relation on X , then C % (deFned on a set of subsets of X that have a single most preferred element) satisFes . Alternatively, consider the “second-best procedure” in which the decision maker has in mind an ordering % of X and for any given choice problem set A chooses the element from
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Unformatted text preview: A , which is the %-maximal from the nonoptimal alternatives. The second-best proce-dure does not satisfy condition ∗ : If A contains all the elements in B besides the %-maximal, then C ( B ) ∈ A ⊂ B but C ( A ) 6= C ( B ) . We will now show that condition ∗ is a sufFcient condition for a choice function to be formulated as if the decision maker is maxi-mizing some preference relation. Proposition: Assume that C is a choice function with a domain containing at least all subsets of X of size no greater than 3. If C satisFes ∗ , then there is a preference % on X so that C = C % ....
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