Workshop on Computer Science and Decision Theory
نویسندگان
چکیده
Arrow’s theorem has fostered a lot of work on the problem to aggregate individual preferences into a collective preference or more generally into a collective choice function. Still more generally the ”russian school” (see e.g. Aizerman and Aleskerov 1995, Aizerman and Malishevski 1981 or Aleskerov 1999) has considered the problem to aggregate individual choice functions into a collective choice function. In their work they emphasize the role of three axioms on choice functions, namely the heredity axiom (H), the concordance axiom (C) and the Outcast axiom (O). Indeed, the combination of these axioms gives significant classes of choice functions. For instance, a choice function is ”classically” rational (respectively rationalizable by a partial order, or pathindependent) if and only if it satisfies axioms (H) and (C) (respectively axioms (H),(C) and (O), or axioms (H) and (O)). Moreover, one can describe rational -in an extended sensechoice mechanisms, inducing choice functions statisfying each of these three axioms. We present the order structure of the sets of choice functions satisfying each of these axioms. Indeed, these sets are always partially ordered by the point-wise order between functions. Moreover, we shall see that they are always lattices. The lattice of choice functions satisfying axiom (H) has a nice structure since it is a distributive lattice with intersection and union as meet and join operations. The lattice of choice functions satisying axiom (C) is atomistic and the study of the dependence relation δ of this lattice allows us to prove that it is lower bounded. Since it is also atomistic, it has many other properties;
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