Social choice theory o More properties of swf: Neutrality between alternatives The swf F(1,2....)is neutral between alternatives if,for every profile of individual preferences (1,a2....), F(1,2,,x1）=-F(-1,-2,-t1） That is,if the social preference is reversed when we reverse the preferences of all agents. This property is often understood as that the swf doesn't a priori distinguish either of the two alternatives. o Example:majority voting satisfies neutrality between alternatives(see MWG pp.792). 4口t16121=至)QC
Social choice theory More properties of swf: Neutrality between alternatives The swf F (α1, α2, ..., αI) is neutral between alternatives if, for every proÖle of individual preferences (α1, α2, ..., αI), F (α1, α2, ..., αI) = F (α1, α2, ..., αI) That is, if the social preference is reversed when we reverse the preferences of all agents. This property is often understood as that the swf doesnít a priori distinguish either of the two alternatives. Example: majority voting satisÖes neutrality between alternatives (see MWG pp. 792)
Social choice theory o More properties of swf: Positive responsiveness Consider a profile of individual preferences (.where alternative x is socially preferred or indifferent to y.i.e.F(a.a)0. Take now a new profile (1,@2....)in which some agents raise their consideration for x.i.e.,(a1,2.....) (a1.a2.,ad)where(c1,2,,）≠(a1,2,,a We say that a swf is positively responsive if the new profile of individual preferences (1,2...)makes alternative x socially preferred,i.e..F(a1.a2.....)=1. o Example:majority voting satisfies neutrality between alternatives(see MWG pp.792) 4口，+6年4三卡4三，三习9C
Social choice theory More properties of swf: Positive responsiveness Consider a proÖle of individual preferences α 0 1 , α 0 2 , ..., α 0 I where alternative x is socially preferred or indi§erent to y, i.e., F α 0 1 , α 0 2 , ..., α 0 I 0. Take now a new proÖle (α1, α2, ..., αI) in which some agents raise their consideration for x, i.e., (α1, α2, ..., αI) α 0 1 , α 0 2 , ..., α 0 I where (α1, α2, ..., αI) 6= α 0 1 , α 0 2 , ..., α 0 I . We say that a swf is positively responsive if the new proÖle of individual preferences (α1, α2, ..., αI) makes alternative x socially preferred, i.e., F (α1, α2, ..., αI) = 1. Example: majority voting satisÖes neutrality between alternatives (see MWG pp. 792)
Social choice theory Positive responsiveness (a,%,a)=(1,0,-1)andF(1,0,-1)=0or1 If (a,,a)=(1,1,-1)then F(a,a,a)=1 1,0,0) 1,0,1) 1,1,1) Any of these (aa,a)satisfy (a)2(a,a,a,) 口1⑤4三1至90C
Social choice theory
Social choice theory o Let us now extend our analysis to non-binary sets of alternatives X.e.g.,X=[a,b.c.... The use of majority voting swf.or weighted voting swf can be subject to non-transitivities in the resulting social preference. That is,the order in which pairs of alternatives are voted can lead to cyclicalities,as shown in Condorcet's paradox (we already talked about it in the first weeks of 501,otherwise see page 270 in JR). o An interesting question is: Can we design voting systems(i.e.,swf that aggregate individual preferences)that are not prone to the Condorcet's paradox and satisfy a minimal set of "desirable"properties? That was the question Arrow asked himself(for his Ph.D. thesis!)obtaining a rather grim result,but a great thesis! 4口，+64三+4三，定990
Social choice theory Let us now extend our analysis to non-binary sets of alternatives X, e.g., X = fa, b, c, ...g The use of majority voting swf, or weighted voting swf can be subject to non-transitivities in the resulting social preference. That is, the order in which pairs of alternatives are voted can lead to cyclicalities, as shown in Condorcetís paradox (we already talked about it in the Örst weeks of 501, otherwise see page 270 in JR). An interesting question is: Can we design voting systems (i.e., swf that aggregate individual preferences) that are not prone to the Condorcetís paradox and satisfy a minimal set of "desirable" properties? That was the question Arrow asked himself (for his Ph.D. thesis!) obtaining a rather grim result, but a great thesis!