Section-3-Social-Welfare-Functions-for-Preferences-slides

Section-3-Social-Welfare-Functions-for-Preferences-slides -...

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Unformatted text preview: Social Welfare Functions that Aggregate Preferences Todd Sarver Northwestern University Econ 310-2 Fall 2011 Todd Sarver (Northwestern University) Social Welfare Functions for Preferences Econ 310-2 Fall 2011 1 / 23 General Setting In this lecture, we are interested in aggregating individual preferences into a single social preference in order to make a social decision. Environment Individuals: 1 , 2 , 3 ,...,n Alternatives: Denoted by capital letters such as A,B,C,... Preferences: Each individual i has a complete and transitive preference i over the set of alternatives. Preference Profile: A list of the preferences of all individuals, ( 1 , 2 ,..., n ) . Todd Sarver (Northwestern University) Social Welfare Functions for Preferences Econ 310-2 Fall 2011 2 / 23 Social Preferences Based on the individuals preferences 1 , 2 ,..., n we would like to make a choice from the set of feasible social alternatives. Todd Sarver (Northwestern University) Social Welfare Functions for Preferences Econ 310-2 Fall 2011 3 / 23 Social Preferences Based on the individuals preferences 1 , 2 ,..., n we would like to make a choice from the set of feasible social alternatives. We can do so by aggregating the individuals preferences into a single social preference. Definition A social preference is a single preference * representing societys preferences over the feasible alternatives. Todd Sarver (Northwestern University) Social Welfare Functions for Preferences Econ 310-2 Fall 2011 3 / 23 Social Preferences Based on the individuals preferences 1 , 2 ,..., n we would like to make a choice from the set of feasible social alternatives. We can do so by aggregating the individuals preferences into a single social preference. Definition A social preference is a single preference * representing societys preferences over the feasible alternatives. Once we have a social preference, selecting the socially optimal alternative is analogous to a standard individuals maximization problem simply select the alternative that is best according to this social ranking. But how do we decide on the social preference? Todd Sarver (Northwestern University) Social Welfare Functions for Preferences Econ 310-2 Fall 2011 3 / 23 Social Welfare Functions for Preferences Definition 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 preference as output. Preference Profile ( 1 , 2 , . . . , n )- Social Welfare Function for Preferences- Social Preference * Todd Sarver (Northwestern University) Social Welfare Functions for Preferences Econ 310-2 Fall 2011 4 / 23 Example Suppose there are two alternatives, A and B , and two individuals, Will and Sydney. The SWFP needs to give a social preference over A and B for every possible profile of preferences for Will and Sydney....
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