Social Choice: Lecture 16
Ram Singh
Course 001
October 31, 2014
Preferences and Choices I
Let
Xbe the set of alternatives
Ri be the ‘weak’ preference relation for individuali, defined overX;
i =1, ...,n
Pi be the strict preference relation for individuali Rbe the set of individual preference relations
Obe the set of individual preference relations that are orderings;O⊂R. (R1, ...,Rn)∈Rnbe a profile of preference relations - one for each individuals. That is,
Rn={(R1, ...,Rn)|Ri ∈Rfor eachi=1, ...,n}
Rbe a ‘weak’ Social preference relation;R∈R
Ram Singh: (DSE) Social Choice October 31, 2014 2 / 11
Preferences and Choices II
Definition
LetS⊆X. A setC(S,R)is the set of ‘best’ elements ofSiff [x ∈C(S,R)]↔(∀y ∈S)[xRy]
The setC(S,R)is also called the choice set generated by the preference relationRdefined over the setS.
Definition
LetS⊆X. A setM(S,R)is the set of Maximual elements ofSiff: For allx ∈S, [x ∈M(S,R)]↔[v(∃y ∈S)(yPx)]
Preferences and Choices III
Proposition
For any giveS⊆Xand preference relation R, C(S,R)⊆M(S,R).
Proposition
IfS⊆Xis finite and and preference relation R is quasi-ordering, then M(S,R) is non-empty.
LetS={x1, ...,xn}. Leta1=x1,
a2=
x2, ifx2Px1
a1, otherwise.
aj+1=
xj+1, ifxj+1Paj aj, otherwise.
You can verify thatanis a maximal element.
Ram Singh: (DSE) Social Choice October 31, 2014 4 / 11
Social Choice Rules (SCR) I
Assumption
Every social preference relationRhas strict preference relationP and indifference preference relationIassociated with it.P andIare such that:
For allx,y ∈X
xPy ↔ xRy andv(yRx) xIy ↔ xRy andyRx
Assumption
We assume individual preferences are orderings, i.e., are reflexive, complete and transitive. That is, for alli =1, ..,n,Ri ∈O.
Social Choice Rules (SCR) II
Definition
A SCR is a function
f :On7→R, such that,
(∀(R1, ...,Rn)∈On)[f(R1, ...,Rn) =R∈R].
Definition
A SCRf is decisive iff∀(R1, ...,Rn)∈On, the social preference relation generated byf is complete, i.e., iff∀(R1, ...,Rn)∈On,f(R1, ...,Rn) =Ris complete.
Definition
A SCR is rational if∀(R1, ...,Rn)∈On, the social preference relation generated byf, i.e.,f(R1, ...,Rn) =R, is an ordering.
Ram Singh: (DSE) Social Choice October 31, 2014 6 / 11
Pareto Criterion as SCR I
Definition
Pareto Criterion: Forx,y ∈X,
xRy¯ ↔ [(∀i ∈N)[xRiy]]
xP¯y ↔ [xRy¯ & v(yRx¯ )]
xI¯y ↔ [xRy¯ &yRx¯ ]
Definition
The SCR is the Pareto Criterion iff
xRy ↔ xRy¯ , i.e., xRy ↔ [(∀i ∈N)[xRiy]]
Pareto Criterion as SCR II
Definition
The SCR is Pareto inclusive, i.e., satisfies the Pareto Criterion if: For all x,y ∈X
(∀i ∈N)[xRiy],i.e.,xRy¯ ⇒ xRy xRy¯ and vyRx¯ ⇒ xPy
Proposition
RelationR¯ is a quasi-ordering. That is, it is reflexive and transitive.
Proposition
If Pareto Criterion is used as a SCR, then for any finiteS⊆Xthe set of maximal elements for is non-empty.
Ram Singh: (DSE) Social Choice October 31, 2014 8 / 11
Pareto Criterion as SCR III
Proposition
Pareto Criterion is a decisive SCR iff
(∀x,y ∈X)[(∃i ∈N)[xPiy]⇒(∀j ∈N)[xRjy]]
Suppose,∃i∈Nsuch thatxPiy, and at the same time∃j ∈Nsuch thatyPjx. In that case, we have
vxRy¯ and v(yRx¯ ),i.e., Therefore, the condition is necessary.
SCRs: Desirable Features I
Definition
A SCR is called a SWF, iff :On7→O. That is,
(∀(R1, ...,Rn)∈On)[f(R1, ...,Rn) =R∈O].
Definition
Condition U: A SCRf satisfies condition of ’unrestricted domain’, if its domain isOn. That is,f generates a social preference relation for every possible profile of individual preferences.
Definition
Condition P: A SCRf satisfies condition of ’weak Pareto principle’, if (∀x,y ∈X)(∀i ∈N)[xPiy ⇒xPy].
Ram Singh: (DSE) Social Choice October 31, 2014 10 / 11
SCRs: Desirable Features II
Definition
Condition I: Take anyS⊆X, and ANY two profiles of individual orderings, say (R1, ...,Rn)and(R10, ...,R0n). Letf(R1, ...,Rn) =Randf(R10, ...,Rn0) =R0. A SCRf satisfies condition of ‘independence of irrelevant alternatives’ if the following holds:
(∀x,y ∈S)(∀i ∈N)[xRiy ↔xRi0y]⇒C(S,R) =C(S,R0)
Definition
Condition D: A SCRf satisfies condition of ’non-dictatorship’, if there is NO individuali ∈Nsuch that
(∀x,y ∈X)[xPiy ⇒xPy].
