Social choice theory o Weighted voting swf: Needless to say,simple majority is a special case of weighted majority,whereby the weights satisfy B;=1 for every individual i. The vote of every individual receives the same weight. o Intuitively,if the number of individuals who prefer alternative x to y is larger than the number of individuals prefering y to x. then F(c1,2,,）=1. .This swf is Paretian,given that F(1.1.....1)=1.sincejai=N>0;and F(-1,-1,-1)=-1 since,a;=-N<0. 4口，+年4三卡4三卡三习9C
Social choice theory Weighted voting swf: Needless to say, simple majority is a special case of weighted majority, whereby the weights satisfy βi = 1 for every individual i. The vote of every individual receives the same weight. Intuitively, if the number of individuals who prefer alternative x to y is larger than the number of individuals prefering y to x, then F (α1, α2, ..., αI) = 1. This swf is Paretian, given that F (1, 1, ..., 1) = 1, since ∑ i βi αi = N > 0; and F (1, 1, ..., 1) = 1 since ∑ i βi αi = N < 0
Social choice theory Dictatorial swf: The property of Paretian in swf is so lax that even Dictatorial swf satisfy it. o Let's first define a dictatorial swf: We say that a swf is dictatorial if there exists an agent h, called the dictator,such that,for any profile of individual preferences (a1,a2....), h=1 implies F(@1.a2....)=1,and h=-1 implies F(a1,a2.....)=-1, That is,the strict preference of the dictator prevails as the social preference. We can understand the dictatorial swf as a extreme case of weighted voting... where >0 for the dictator and B;=0 for all other individuals in the society ih. 4口11G1三1=1至)9C
Social choice theory Dictatorial swf: The property of Paretian in swf is so lax that even Dictatorial swf satisfy it. Letís Örst deÖne a dictatorial swf: We say that a swf is dictatorial if there exists an agent h, called the dictator, such that, for any proÖle of individual preferences (α1, α2, ..., αI), αh = 1 implies F (α1, α2, ..., αI) = 1, and αh = 1 implies F (α1, α2, ..., αI) = 1, That is, the strict preference of the dictator prevails as the social preference. We can understand the dictatorial swf as a extreme case of weighted voting... where βh > 0 for the dictator and βi = 0 for all other individuals in the society i 6= h
Social choice theory Dictatorial swf: Since weighted voting swf is Paretian,then the dictatorial swf (as a special case of weighted voting)must also be Paretian. Extra confirmation: F(1.1.....1)=1,since=n>0:and F(-1,-1,,-1)=-1 since,i=-ph<0 4口，+6年4三卡4三，三习9C
Social choice theory Dictatorial swf: Since weighted voting swf is Paretian, then the dictatorial swf (as a special case of weighted voting) must also be Paretian. Extra conÖrmation: F (1, 1, ..., 1) = 1, since ∑ i βi αi = βh > 0; and F (1, 1, ..., 1) = 1 since ∑ i βi αi = βh < 0
Social choice theory o More properties of swf: Symmetry among agents (or anonymity): The swf F(a1,@2....)is symmetric among agents (or anonymous)if the names of the agents do not matter. That is,if a permutation of preferences across agents does not alter the social preference.Precisely,let π：{1,2.}→{1,2.1} be an onto function(i.e.,a function that,for every indvidual i. there is a j such that ()=i).Then,for every profile of individual preferences (1.2...),we have F(a1,2,a1）=F(al.2x0) o Example:majority voting satisfies anonymity
Social choice theory More properties of swf: Symmetry among agents (or anonymity): The swf F (α1, α2, ..., αI) is symmetric among agents (or anonymous) if the names of the agents do not matter. That is, if a permutation of preferences across agents does not alter the social preference. Precisely, let π : f1, 2, ..., I g ! f1, 2, ..., I g be an onto function (i.e., a function that, for every indvidual i, there is a j such that π(j) = i). Then, for every proÖle of individual preferences (α1, α2, ..., αI), we have F (α1, α2, ..., αI) = F απ(1) , απ(2) , ..., απ(I) Example: majority voting satisÖes anonymity
Social choice theory Anonymity holds in simple majority (%4,a)-(1-11) m(1)-3.red arrow 2a=111 Social preference coincides despite (2)=1,green arrow changing individual (3)=2,purple arrow ∑a=(-+11-1 identities Weighted voting does not neeessarily satisfy anonymity Using the same (a,,)and m)function as above ∑6a=月1+6(1)+月1=月+月-月 Social preference does not necessarily coincide ∑月4=月1+月(1)+月1=月+月-月 4口，+6年4三卡4三，三习9C
Social choice theory