As an abstract social choice function, relative utilitarianism has been analyzed by Cao ( 1982 ), Dhillon ( 1998 ), Karni ( 1998 ), Dhillon and Mertens ( 1999 ), Segal ( 2000 ), Sobel ( 2001 ) and Pivato ( 2008 ) . ( Cao ( 1982 ) refers to it as the ` modified Thomson solution'. ) When interpreted as a ` voting rule', it is equivalent to Range voting.
42.
Each choice function on a collection " X " of nonempty sets is an element of the family of sets, where a given set can occur more than once as a factor; however, one can focus on elements of such a product that select the same element every time a given set appears as factor, and such elements correspond to an element of the Cartesian product of all " distinct " sets in the family.
43.
Given a finite colored directed bipartite graph with " n " vertices V = V _ 0 \ cup V _ 1, and " V " colored with colors from " 1 " to " m ", is there a choice function selecting a single out-going edge from each vertex of V _ 0, such that the resulting subgraph has the property that in each cycle the largest occurring color is even.
44.
A social-choice function is called "'manipulable by player i "'if there is a scenario in which player i can gain by reporting untrue preferences ( i . e ., if the player reports the true preferences then \ operatorname { Soc } ( P ) = a, if the player reports untrue preferences then \ operatorname { Soc } ( P') = a', and player i prefers a'to a ).
45.
The result is an explicit choice function : a function that takes the first box to the first element we chose, the second box to the second element we chose, and so on . ( A formal proof for all finite sets would use the principle of mathematical induction to prove " for every natural number " k ", every family of " k " nonempty sets has a choice function . " ) This method cannot, however, be used to show that every countable family of nonempty sets has a choice function, as is asserted by the axiom of countable choice.
46.
The result is an explicit choice function : a function that takes the first box to the first element we chose, the second box to the second element we chose, and so on . ( A formal proof for all finite sets would use the principle of mathematical induction to prove " for every natural number " k ", every family of " k " nonempty sets has a choice function . " ) This method cannot, however, be used to show that every countable family of nonempty sets has a choice function, as is asserted by the axiom of countable choice.
47.
The result is an explicit choice function : a function that takes the first box to the first element we chose, the second box to the second element we chose, and so on . ( A formal proof for all finite sets would use the principle of mathematical induction to prove " for every natural number " k ", every family of " k " nonempty sets has a choice function . " ) This method cannot, however, be used to show that every countable family of nonempty sets has a choice function, as is asserted by the axiom of countable choice.
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