Social Choice
Setup
Individual preferences form a profile, ≿, strict preferences denoted ≻i
SCR takes profile and produces complete and transitive ordering of social states: if society
prefers x to y , denote x ⊵ y
Arrow’s SCR: UPID
Unrestricted domain: admit pattern of individual preferences ie ⊵ for any ≿
Pareto principle: if x ≻ i y ∀i ⇒ x ⊵ y
Independence of irrelevant alternatives: if individual preference orderings over x and y do
not change, SCR should produce same social ordering over x and y even if preferences over
other alternatives change
(Non)-Dictatorship: no agent i s.t. irrespective of preferences of others, ∀ x , y , if
x ≻i y ⇒ x ⊵ y
Arrow’s impossibility frontier: when choosing between more than two outcomes, there is no
way of aggregating social choice as preferences must be consistent and UPID must be
satisfied – we must sacrifice one
Consistency: a shorthand way of saying complete, reflexive, and transitive
Majority Voting
x ⊵ y iff ¿ [x ≿ i y ]>¿ [ y ≿i x ]
Satisfies UPID but may not be transitive as x ⊵ y , y ⊵ z , z ⊵ x
Getting out of Impossibility
Relax at least 3 social states – majority voting works for two
Relax D: appoint queen
Relax P: write down any transitive social ranking
Relax IIA: Borda Count
Replace transitivity with quasi-transitivity ie only for strict relations: liberum veto oligarchy
Relax unrestricted domain
Single-Peaked Preferences
If alternatives one-dimensional and all (odd number) agents have
single peaked preferences, majority voting produces a SCR and
majority of votes select median alternative over overs
o Why? Take ( x , y , z ) where z > y > x . In a referendum
between z and y, C would vote for z, but A and B choose y.
Now consider a referendum between x and y, A chooses x,
but B and C choose y. Therefore y is the best compromise.
Can also observe that B and C would choose z over x,
while A would choose x
Here y ⊵ z ⊵ x
Principle of min differentiation: if two parties choose platforms on which to stand, both will
choose the same (preferred by median voter)
This is not a counter-example to Arrow’s UPID, because we have restricted the domain to
single-peaked preferences ie agents are not allowed to have whatever preferences they want –
they must be single-peaked
Liberalism
For each i there is at least one pair of alternatives x and y s.t. if x ≿ i y then x ⊵ y and if
y ≿ x then y ⊵ x eg which side of bed you sleep on
Sen Theorem: if at least 3 social states, no SCR satisfies U, P, and Liberalism, even if
liberalism only applies to two individuals
Incentives
Instead of SCR, use social choice function, F, which selects one best social state and doesn’t
need to produce a complete and transitive ordering
Agents have preferences over outcomes of SCF
Setup
Individual preferences form a profile, ≿, strict preferences denoted ≻i
SCR takes profile and produces complete and transitive ordering of social states: if society
prefers x to y , denote x ⊵ y
Arrow’s SCR: UPID
Unrestricted domain: admit pattern of individual preferences ie ⊵ for any ≿
Pareto principle: if x ≻ i y ∀i ⇒ x ⊵ y
Independence of irrelevant alternatives: if individual preference orderings over x and y do
not change, SCR should produce same social ordering over x and y even if preferences over
other alternatives change
(Non)-Dictatorship: no agent i s.t. irrespective of preferences of others, ∀ x , y , if
x ≻i y ⇒ x ⊵ y
Arrow’s impossibility frontier: when choosing between more than two outcomes, there is no
way of aggregating social choice as preferences must be consistent and UPID must be
satisfied – we must sacrifice one
Consistency: a shorthand way of saying complete, reflexive, and transitive
Majority Voting
x ⊵ y iff ¿ [x ≿ i y ]>¿ [ y ≿i x ]
Satisfies UPID but may not be transitive as x ⊵ y , y ⊵ z , z ⊵ x
Getting out of Impossibility
Relax at least 3 social states – majority voting works for two
Relax D: appoint queen
Relax P: write down any transitive social ranking
Relax IIA: Borda Count
Replace transitivity with quasi-transitivity ie only for strict relations: liberum veto oligarchy
Relax unrestricted domain
Single-Peaked Preferences
If alternatives one-dimensional and all (odd number) agents have
single peaked preferences, majority voting produces a SCR and
majority of votes select median alternative over overs
o Why? Take ( x , y , z ) where z > y > x . In a referendum
between z and y, C would vote for z, but A and B choose y.
Now consider a referendum between x and y, A chooses x,
but B and C choose y. Therefore y is the best compromise.
Can also observe that B and C would choose z over x,
while A would choose x
Here y ⊵ z ⊵ x
Principle of min differentiation: if two parties choose platforms on which to stand, both will
choose the same (preferred by median voter)
This is not a counter-example to Arrow’s UPID, because we have restricted the domain to
single-peaked preferences ie agents are not allowed to have whatever preferences they want –
they must be single-peaked
Liberalism
For each i there is at least one pair of alternatives x and y s.t. if x ≿ i y then x ⊵ y and if
y ≿ x then y ⊵ x eg which side of bed you sleep on
Sen Theorem: if at least 3 social states, no SCR satisfies U, P, and Liberalism, even if
liberalism only applies to two individuals
Incentives
Instead of SCR, use social choice function, F, which selects one best social state and doesn’t
need to produce a complete and transitive ordering
Agents have preferences over outcomes of SCF
