Desirable Voting Rule Properties
Kenneth Arrow’s Impossibility Theorem: The only voting rule with all of properties 1, 2 and 3 is dictatorial.
Desirable Voting Rule Properties
B ill x y z B ertha B ob y z x z x y
Majority Vote Results x beats y No y beats z socially z beats x best alternative! Majority voting does not always aggregate transitive individual preferences into a transitive social preference.
B ill x(1) y(2) z(3)
Rank-order vote results B ertha B ob (low score wins). x-score = 6
y(1) z(2) x(3)
z(1) x(2) y(3)
Aggregating Preferences
B ill x(1) y(2) z(3)
Rank-order vote results B ertha B ob (low score wins). x-score = 6 y-score = 6 y(1) z(1)
z(2) x(3)
x(2) y(3)
Aggregating Preferences
B ill x(1) y(2) z(3)
These are truthful preferences. Bob introduces a new alternative
Manipulating Preferences
B ill x(1) y(2) z(3) α (4) B ertha B ob y(1) z(2) α (3) x(4) z(1) x(2) y(3) α (4)
/ Chapter Thirty-Two Welfare
What do we do in this chapter?
We at the end of this course We are exploring a few very important topics of microeconomics Welfare economics is an extremely important topic
These are truthful preferences. Bob introduces a new alternative and then lies. Rank-order vote results. x-score = 8
Manipulating Preferences
B ill x(1) y(2) z(3) α (4) B ertha B oby(4)
Aggregating Preferences
B ill x y z B ertha B ob y z x z x y
Majority Vote Results x beats y No y beats z socially z beats x best alternative!
Aggregating Preferences
Social Choice
Different economic states will be preferred by different individuals. How can individual preferences be “aggregated” into a social preference over all possible economic states?
Desirable Voting Rule Properties
1. If all individuals’ preferences are complete, reflexive and transitive, then so should be the social preference created by the voting rule. 2. If all individuals rank x before y then so should the voting rule. 3. Social preference between x and y should depend on individuals’ preferences between x and y only.
Manipulating Preferences
B ill x(1) y(2) z(3) B ertha B ob y(1) z(2) x(3) z(1) x(2) y(3)
These are truthful preferences.
Manipulating Preferences
B ill x(1) y(2) z(3) B ertha B ob y(1) z(2) x(3) z(1) x(2) y(3)
These are truthful preferences. Bob introduces a new alternative
Manipulating Preferences
B ill x(1) y(2) z(3) α (4) B ertha B ob y(1) z(2) α (3) x(4) z(1) x(2) y(3) α (4)
Social Welfare Functions
1. If all individuals’ preferences are complete, reflexive and transitive, then so should be the social preference created by the voting rule. 2. If all individuals rank x before y then so should the voting rule. 3. Social preference between x and y should depend on individuals’ preferences between x and y only.
Social Welfare Functions
1. If all individuals’ preferences are complete, reflexive and transitive, then so should be the social preference created by the voting rule. 2. If all individuals rank x before y then so should the voting rule. 3. Social preference between x and y should depend on individuals’ preferences between x and y only.
Aggregating Preferences
x, y, z denote different economic states. 3 agents; Bill, Bertha and Bob. Use simple majority voting to decide a state?
Aggregating Preferences
Manipulating Preferences
B ill x(1) y(2) z(3) α (4) B ertha B ob y(1) z(2) α (3) x(4) z(1) α(2) x(3) y(4)
These are truthful preferences. Bob introduces a new alternative and then lies. Rank-order vote results. x-score = 8 z wins!! y-score = 7 z-score = 6 α-score = 9
Aggregating Preferences
B ill x(1) y(2) z(3)
Rank-order vote results B ertha B ob (low score wins).
y(1) z(2) x(3)
z(1) x(2) y(3)
Aggregating Preferences
x(2) y(3)
Rank-order voting is indecisive in this case.
Manipulating Preferences
As well, most voting schemes are manipulable. I.e. one individual can cast an “untruthful” vote to improve the social outcome for himself. Again consider rank-order voting.
B ill x(1) y(2) z(3) α (4) B ertha B ob y(1) z(2) α (3) x(4) z(1) α(2) x(3) y(4)
These are truthful preferences. Bob introduces a new alternative and then lies. Rank-order vote results. x-score = 8 y-score = 7 z-score = 6
These are truthful preferences. Bob introduces a new alternative and then lies. Rank-order vote results. x-score = 8 y-score = 7