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Unformatted text preview: Pareto Efficiency (also called Pareto Optimality) 1 Definitions and notation Recall some of our definitions and notation for preference orderings. Let X be a set (the set of alternatives ); we have the following definitions: 1. A relation R on X is a subset of X × X . We often write xRy instead of ( x,y ) ∈ R and we say “ x is R-related to y ”. 2. If R is a relation on X , we denote the complement of R by R / (instead of ∼ R , because ∼ will be given another meaning). Thus, xR /y means that x is not R-related to y . 3. A strict preordering of X is a transitive and irreflexive relation P on X . We sometimes write x y for xPy , and also y ≺ x . We say that x is “preferred” to y . A complete preordering of X is a transitive and complete relation % on X . 4. If P is a strict preordering, we denote the corresponding indifference relation by I , defined by xIy ⇐⇒ [ xP/y & yP/x ]. We also write x ∼ y for xIy , and x % y (also y- x ) for [ xPy or xIy ]. Note that ∼ is both reflexive and symmetric, but it need not be transitive; and that % is complete, but it need not be transitive. (Can you provide a counterexample to show that transitivity may fail?) If ∼ is transitive , then % is a complete preordering. 5. If % is a complete preordering, then ∼ is transitive, and [ x % y & y z ] implies x z for any x,y,z ∈ X . 2 Aggregation of rankings into a single ranking Let X be a set of alternatives , generically denoted by x ; let N be a set of n individuals , gener- ically denoted by i ; and let P be a set of admissible preorderings (“rankings,” or “preferences”) over X , generically denoted by P . We want to have a rule we can use to aggregate a list P = ( P 1 ,...,P n ) of individual rankings into a single “aggregate” ranking, P . In other words, we want to have an aggregation function or rule a : P n-→ P , i.e., ( P 1 ,...,P n ) a 7-→ P . (1) 1 (Note the similarity with the notation for the sample mean of a list of n numbers: x = 1 n ∑ i x i . The sample mean is a way of aggregating a list of numbers into a single “representative” number – i.e., it’s a function that maps a list of numbers into a single number). Instead of framing the problem as one of aggregating a list of rankings into a single ranking, we could alternatively frame the problem as one of aggregating a list of utility functions into a single utility function. We will return to this idea in Section 7. 2.1 Examples Here are several examples of sets X of alternatives for which we might wish to aggregate a list of individual rankings into a “representative” ranking: 1. X is a set of allocations x = ( x 1 ,..., x n ) ∈ R nl + . 2. X is a set of candidates for a job, or for a political position....
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- Spring '11
- Economics, Social Choice and Individual Values, pareto, Utility Functions