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310-2-08-Lecture02-Social-Choice - Lecture 2 Social Choice...

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Lecture 2 — Social Choice Todd Sarver Northwestern University Spring 2008 1 Social Welfare Functions that Aggregate Preferences In this lecture, we are interested in aggregating individual preferences intro a single social preference . The environment is described below: Individuals: 1 , 2 , 3 ,... ,n Alternatives: Let X denote the set of feasible alternatives, and let A,B,C,... denote elements of X . Preferences: Each individual i has a complete and transitive preference followsequal i over the alter- natives in X . Preference Profile: A list of the preferences of all individuals, ( followsequal 1 , followsequal 2 ,..., followsequal n ). Social Ranking: A single preference followsequal * representing “society’s preferences.” Our goal is to aggregate the individuals’ preferences to get a social ranking. This will enable us not only to determine the socially best alternative in X , but also to implement a social choice from any subset of X . That is, if a choice must be made from a set ˜ X that contains some (but perhaps not all) of the alternatives in X , the social preference followsequal * indicates which of the alternatives in ˜ X should be chosen. We refer to the mechanism for aggregating the individuals’ preferences as a social welfare function for preferences (SWFP) . The SWFP takes a preference profile as input and gives a single social ranking as output: Preference Profile ( followsequal 1 , followsequal 2 ,..., followsequal n ) -→ Social Welfare Function for Preferences -→ Social Ranking followsequal * We begin by considering a few examples. First, consider the case of two alternatives. Example 1 (Majority Rule with 2 Alternatives) Suppose there are two alternatives, A and B . Majority rule chooses the alternative the “gets the most votes”:
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Lecture 2 — Social Choice 2 A follows * B if A follows i B for a majority of individuals i B follows * A if B follows i A for a majority of individuals i What about a tie? If there is a tie, we could use an arbitrary tie-breaking rule, such as A follows * B . The above example illustrates that majority rule is a type of a social welfare function. However, there are also other social welfare functions, such as letting followsequal * = followsequal 3 . That is, the SWFP could make society’s preference the same as that of individual 3, so A followsequal * B if and only if A followsequal 3 B . A social welfare function that assigns the social ranking to be the same as that of some fixed individual is referred to as a dictatorship . Of course, many people find this type of SWFP unsatisfactory from a normative perspective. We now move on to the case of three or more alternatives. Example 2 (Pairwise Majority Rule) Suppose there are three alternatives, A , B , and C , and three individuals, Erin, Jeff, and Liz. Suppose the rankings of these individuals are the following: Erin Jeff Liz A B C B C A C A B In this table, the most preferred alternative is at the top and the least preferred is at the bottom.
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