Social Choice: Lecture 19
Ram Singh
Course 001
November 7, 2014
Impossibility Result I
Theorem
There is no SWF that satisfies conditions U, P, I and ND simultaneously.
Proof: Take anyx,y ∈X. Then,∃V ⊆Nsuch thatV isD(x,y).
How many conditions we need to prove:
∃V ⊆Nsuch that V isD(x,y)?
There exist several profiles(R1, ...,Rn),(R01, ...,R0n), etc. such that (∀i ∈N)[xPiy,xPi0y, ...]etc.
(∀(R1, ...,Rn),(R10, ...,Rn0)∈On)[(∀i ∈N)(xPiy)⇒xPy]
Impossibility Result II
Therefore,
V =NisD(x,y).
Therefore,
V={V|V isD(u,v) for some u,v ∈X}.
is non-empty. Recall,]V1=1, and isD(x,y). Also, Proposition
Suppose a SWF satisfies conditions U, P, and I
V is D(x,y)for some x,y ∈X]⇒[V isD(u,¯ v)for all u,v ∈X. So,]V1=1 isD(u,¯ v)for allu,v ∈X.
Impossibility Result III
Theorem
If a SWF satisfies conditions U, P and I, then∃i ∈Nsuch that (∀x,y ∈X)(∀(R1, ...,Rn)∈On)[xPiy ⇒xPy].
Theorem
If a SWF satisfies conditions U, I, and ND then∃x,y ∈Xand
∀(R1, ...,Rn)∈Onsuch that
(∀i ∈N)[xPiy]but ∼xPy.
SWF: Examples I
Proposition
There exists a SWF f :D7→Othat satisfies conditions U, P, and ND, but does not satisfy condition I.
If preferences are as follows
1 2 3
x y z
y z x
z x y
, the usual rank-score of each
alternative is 6 Proposition
MMR satisfies conditions U, P, I, and ND but is not a SWF.
Let
N(xPy)number of individuals who strictly preferx overy N(xRy)number of individuals who weakly preferx overy
SWF: Examples II
Definition
A Method of Majority Rule is a SWF such that:
(∀x,y ∈X)[xRy ⇔[N(xPy)≥N(yPx)], or (∀x,y ∈X)[xRy ⇔[N(xRy)≥N(yRx)].
If preferences are as follows
1 2 3
x y z
y z x
z x y
, the MMR gives us
xPy,yPz, andzPx That is,
MMR satisfies all conditions butR6∈O.
Rinduced by MMR is neither transitive nor quasi-transitive.
Possibility Results I
Proposition
There exists a SCR that satisfies conditions U, P, I, and ND.
Condition O*: The social preference relationRinduces a non-empty choice set.
Will the choice set be non-empty ifR∈O?
IsR∈Oa sufficient condition for the choice set to be non-empty?
IsR∈Oa necessary condition for the choice set to be non-empty?
Does there exist a SCR that satisfies conditions U, P, I, and ND, and O* ?
Possibility Results II
Proposition
Let R be any preference relation. IfSis finite and R is reflexive, complete and quasi-transitive, then the Choice set is non-empty.
Proposition
IfSis finite, there exists a SCR that satisfies conditions U, P, I, and ND, and O*.
Define
xRy ⇔∼[(∀i ∈N)(yRix)and(∃i ∈N)(yPix)]
That is, Clearly, thisRis reflexive and complete. Note that xPy ⇔ [(∀i∈N)(xRiy)and(∃i ∈N)(xPiy)]
yPx ⇔ [(∀i∈N)(yRix)and(∃i ∈N)(yPix)]
Possibility Results III
By definitionRsatisfies conditions U and I. Moreover, (∀i ∈N)(xPiy)⇒[∼yRx andxRy] Therefore,Rsatisfies condition P.
Note that
xPy ⇒ [(∀i∈N)(xRiy)and(∃i ∈N)(xPiy)]
yPz ⇒ [(∀i∈N)(yRiz)and(∃i∈N)(yPiz)]
Therefore,
[xPy andyPz]⇒[(∀i ∈N)(xRiz)and(∃i ∈N)(xPiz)]
That is,
[xPy andyPz]⇒xPz SoRis quasi transitive.
Majority Rule
Proposition
There exists a SWF f :D7→Othat satisfies conditions P, I, and ND, but D⊂⊂On
Definition
Single Peakedness.Ris single peaked if there exists a re-arrangement of alternatives inX, say{y1,y2, ...,ym}, and somey∗, sayy∗=yk, such that
j <j0 ≤k ⇒ yj0Pyj
l0>l ≥k ⇒ xlPxl0
Proposition
If preferences are single-peaked and number of individuals is odd, there exists a SWF f :D7→Othat satisfies conditions P, I, and ND.
Answer is : MMR
Liberal Paradox I
Definition
Liberalism L: For everyi ∈N, there is a pair of distinct alternatives (x,y)∈X×Xsuch that
xPiy ⇒xPy andyPix ⇒yPx
Definition
Minimal Liberalism L*: For at least two individualsLiberalismholds.
Proposition
No SWF can satisfy conditions U, P and L*
Suppose conditions U, P and L* hold. Let
Liberal Paradox II
j be decisive for(x,y) k be decisive for(z,w)
xPjy,zPkw and(∀i)[wPix &yPiz] This gives us,
xPy, zPw, wPx andyPz,i.e., xPz, zPw, andwPx, a contradiction.
The preferences are as follows
i j k
. w y
. x z
. y w
. z x
,
Summing Up I
Implications of relaxing conditionR∈O Implications of relaxing/chanding condition I Implications of relaxing/chanding condition P Implications of relaxing/chanding condition ND Implications of relaxing/chanding condition U There are trade-offs among
Rationality of society Individual liberty Democracy
