Social Choice Theory Yuyi Wang ETH Zurich 0口11日1三1=至)9C
Social Choice Theory Yuyi Wang ETH Zurich
Social choice theory o Consider a group of />2 individuals must choose an alternative from a set X. We will first consider that set X is binary X={x,y} These two alternatives could represent the set of candidates competing for office,the policies to be implemented,etc. o Preferences: Every individual i's preference over x and y can be defined as a number: x;={1,0,-1} indicating that he prefers x to y,is indifferent between them, or he prefers alternative y to x,respectively. 4口，+8卡4三三，三为Q0
Social choice theory Consider a group of I 2 individuals must choose an alternative from a set X. We will Örst consider that set X is binary X = fx, yg These two alternatives could represent the set of candidates competing for o¢ ce, the policies to be implemented, etc. Preferences: Every individual iís preference over x and y can be deÖned as a number: αi indicating that he prefers or he prefers alternative y = f1, 0, 1g x to y, is indi§erent between them, to x, respectively
Social choice theory o We now seek to aggregate individual preferences with the use of a social welfare functional (or social welfare aggregator). o Social welfare functional: A social welfare functional(swf)is a rule F(1,2,x1）∈{1.0，-1} which,for every profile of individual preferences (c1,2,,d1）∈{1,0，-l',assigns a social preference F(a1,2,,1）∈{1,0，-1} Example: For individual preferences (@1.42,3)=(1,0,1),the swf F(1,0,1)=1,thus prefering alternative x over y. 4口11G4三1=1至)9C
Social choice theory We now seek to aggregate individual preferences with the use of a social welfare functional (or social welfare aggregator). Social welfare functional: A social welfare functional (swf) is a rule F (α1, α2, ..., αI) 2 f1, 0, 1g which, for every proÖle of individual preferences (α1, α2, ..., αI) 2 f1, 0, 1g I , assigns a social preference F (α1, α2, ..., αI) 2 f1, 0, 1g. Example: For individual preferences (α1, α2, α3) = (1, 0, 1), the swf F (1, 0, 1) = 1, thus prefering alternative x over y
Social choice theory Properties of swf: A swf is Paretian if it respects unanimity of strict preference: That is,if it strictly prefers alternative x when all individuals strictly prefer x.i.e..F(1.1.....1)=1, but strictly prefers alternative y when all individuals strictly prefer y,i.e.,F(-1,-1,,-1)=-1, o Note: This property is satisfied by many swf. Weighted voting and Dictatorship are two examples(let's show that). 4口，+6年4三卡4三，三习9C
Social choice theory Properties of swf: A swf is Paretian if it respects unanimity of strict preference; That is, if it strictly prefers alternative x when all individuals strictly prefer x, i.e., F (1, 1, ..., 1) = 1, but strictly prefers alternative y when all individuals strictly prefer y, i.e., F (1, 1, ..., 1) = 1, Note: This property is satisÖed by many swf. Weighted voting and Dictatorship are two examples (letís show that)
Social choice theory o Weighted voting swf: .We first add individual preferences,assigning a weight B;>0 to every individual,where(β1，β2，，f,)≠0，as follows ∑iBk;∈R. .We then apply the sign operator,which yields 1 when iBii>0,0 when iBiaj=0.and -1 when iBji<0. Hence, F(a1.a2..)=sign Bjai o In order to check if this swf is Paretian,we only need to confirm that F(L,1l)=1,since∑f,a=∑B,>0and F(-1,-1-=-1s5nceB=-A,<0
Social choice theory Weighted voting swf: We Örst add individual preferences, assigning a weight βi 0 to every individual, where (β1 , β2 , ..., βI ) 6= 0, as follows ∑i βi αi 2 R. We then apply the sign operator, which yields 1 when ∑i βi αi > 0, 0 when ∑i βi αi = 0, and 1 when ∑i βi αi < 0. Hence, F (α1, α2, ..., αI) = sign ∑ i βi αi In order to check if this swf is Paretian, we only need to conÖrm that F (1, 1, ..., 1) = 1, since ∑ i βi αi = ∑ i βi > 0; and F (1, 1, ..., 1) = 1 since ∑ i βi αi = ∑ i βi < 0