Group decision Rules Majority rule Condorcet paradox(voting cycle Borda rule
Group Decision Rules • Majority rule , • Condorcet paradox (voting cycle) • Borda rule
Mathematical model A set of voters v=v1,v2,V3,.,ng a set of alternatives or outcomes S=s1, S2, S3,... Sm, with s=m; and A set of preference relation P=R1, R2, R3.Rn, called a preference profile, the preference relation Ri for each voter i is a permutation(order) of elements in S
• A set of voters V={v1,v2,v3,…,Vn} • A set of alternatives or outcomes S={s1,s2,s3,…Sm}, with |S|=m; and • A set of preference relation P={R1,R2,R3…Rn}, called a preference profile, – the preference relation Ri for each voter i is a permutation (order) of elements in S. Mathematical model
Example 1 Majority Rule 3 rational people have rational preferences over 2 alternatives x, y! Person 123 1st x Y X 1:X>Y L.e. Person 2. y>X 2nd Y X Y 3:X>Y How to Aggregate their preferences? How to choose?
Example 1 Majority Rule • 3 rational people have rational preferences over 2 alternatives {x,y} Person 1 2 3 1 st X Y X 1 : X>Y Pref. i.e.Person 2 : Y>X 2 nd Y X Y 3 : X>Y How to Aggregate their preferences? How to choose?
Using majority rule Since more than y2 people(two out of three) prefer x to y Then the group prefers x to y
• Using majority rule. • Since more than ½ people (two out of three) prefer x to y. • Then the group prefers x to y
Example 2 Condorcet Paradox 3 rational people have rational preferences over 3 alternatives x, y, z) Person 23 1st X Y Z 1.X>Y>Z Pref. 2nd Y Xie Person 2:y>z>X 3rd Z 3:2>X>Y
Example 2 Condorcet Paradox • 3 rational people have rational preferences over 3 alternatives {x,y,z} Person 1 2 3 1 st X Y Z 1 : X>Y>Z Pref. 2 nd Y Z X i.e. Person 2 : Y>Z>X 3 rd Z X Y 3 : Z>X>Y