Impossibility Theorems
N. Neil Yu
Stanford University Economics Department Dinner Seminar
Condorcet’s Paradox (Condorcet, 1785)
Definition: In an election, if candidate ai can winaj for every j 6=i in a two-candidate runoff determined bymajority rule, then ai is a Condorcet winner.
Comment: If the social choiceai is not a Condorcet winner, then it loses
toaj for some j6=i. The supporters ofaj would protest.
Condorcet (1785) introduced aCondorcet profilewith no Condorcet winner:
voter 1 voter 2 voter 3 runoff failures
a1 a2 a3 a2loses toa1
a2 a3 a1 a3loses toa2
a3 a1 a2 a1loses toa3
The model of social choice and welfare
Individuals: N ={1,· · · ,N} withN ≥2
Alternatives: A={a1,· · · ,aM} with M ≥3
The collection of all complete, asymmetric, and transitive preference relations: P (strict preferences, for simplicity)
A preference profile: an ordered list ~ = (1, . . . ,N) with n∈ P for every 1≤n≤N
The collection of all preference profiles: PN
SCF Asocial choice function F :PN →Aassigns to each
~
∈ PN a choiceF(~)∈A.
SPF Asocial preference function R :PN → P assigns to each ∈ P~ N a preference relationR(~)∈ P.
Some terminologies
For profile ~,ai dominates aj ifai naj for everyn.
Two profiles~ and ~0 agree on {ai,aj} ifai n aj is necessary and sufficient forai 0n aj for every n. A0 ⊂A isat the top of ~ ifai naj for everyn, every ai ∈A0, and every aj ∈A\A0. If in addition A0={ai,aj}, we call ~ a{i,j}-runoff.
A runoff generating function Tij :PN → PN brings
A Condorcetian Impossibility Theorem
Definition (D):A SCF F is dictatorial if there exists a social choice dictator nsuch that F(~)nai for every ai 6=F(~) and every~.
Definition (O):A SCF F is onto if F(PN) =A.
Definition (WP):A SCF F is weakly Paretian ifai dominating aj impliesF(~)6=aj for everyi,j and every ~.
Definition (CM):A SCF F is Condorcet monotonic if whenever
~
and{i,j}-runoff~0 agree on {ai,aj},F(~) =ai implies F(~0) =ai.
Comment: (CM) requires every social choice to be ageneralized Condorcet winner in that it has to win any runoff it enters that keeps intact its rankings against the opponent. In particular, if F(~) =ai,F(Tij(~)) =ai by (CM). We often apply the contrapositivestatements.
A Condorcetian Impossibility Theorem
Theorem: If a SCF F satisfies (O) and (CM), then it is dictatorial.
Lemma: (O) and (CM) imply (WP).
Proof: Letai dominate aj in ~. (O) ensures that F(~
0
) =ai for
some~0. By (CM), the choice remainsai for Tik(~ 0
), where k∈ {/ i,j}, and further for Tij(Tik(~
0
)). Since aj loses the {i,j}-runoffTij(Tik(~
0
)) that agrees with~ on{ai,aj}, the choice for~ cannot be aj.
Step 1–defining pivotal voters: Pick any ~0 with {ai} and{ai,aj} at the top. Swap the positions of{ai,aj} sequentially from 1 toN. By (WP), the choice is eitherai or aj, starting withai and ending withaj. The (i,j)-pivotal voter nij is the first whose swap makes a
difference. By (CM),nij is independent of which ~ 0
A Condorcetian Impossibility Theorem: Step 2
Step 2–finding F(~00): Consider depicted profiles~00 and ~000 with {ai,aj,ak} at the top, where columns correspond to voters and squares mark possible positions ofak. The definition ofnij informs us thatF(Tij(~
00
)) =ai andF(Tij(~ 000
)) =aj, so F(~ 00
)6=aj and
F(~000)6=ai by (CM). As (WP) rules outak and others, F(~00) =ai.
1 . . . nij−1 nij nij+ 1 . . . N analysis
aj . . . aj ai ai . . . ai ~
00 . . . F(Tij(~
00 )) =ai
ai . . . ai aj aj . . . aj CM⇒ F(~00)6=aj
. . . ak ak . . . ak
WP
⇒ F(~00) =ai
. . .
aj . . . aj aj ai . . . ai F(Tij(~
000 )) =aj ~
000 . . . . . . CM⇒ F(~000)6=ai ai . . . ai ai aj . . . aj
. . . ak . . .
A Condorcetian Impossibility Theorem: Step 3
Step 3–finding F(~000): Hence, F(Tik(~ 00
)) =ai, implying that in
definingnik, no swap beforenij makes a difference, i.e.,nik ≥nij.
Butj andk are arbitrary, sonij =nik, i.e., ni− refers to the same individual. Moreover,Tik(~
00
) and~000 agree on {ai,ak}, so F(~000)6=ak due toak’s loss to ai in {i,k}-runoffTik(~
00 ). We are left withF(~000) =aj.
1 . . . nij−1 nij nij+ 1 . . . N analysis
aj . . . aj ai ai . . . ai ~
00 . . .
ai . . . ai aj aj . . . aj
. . . ak ak . . . ak F(~00) =ai . . . CM⇒F(~000)6=ak aj . . . aj aj ai . . . ai
~
000 . . . . . . F(~000)6=ai
ai . . . ai ai aj . . . aj ∴F(~
000 ) =aj
A Condorcetian Impossibility Theorem: Step 4
Step 4: Hence, F(Tjk(~ 000
)) =aj, demanding by (CM) that
aj nij ak implies F(~)6=ak. (∗)
1 . . . nij−1 nij nij+ 1 . . . N analysis . . .
aj . . . aj aj ai . . . ai
~
000 . . . . . .
ai . . . ai ai aj . . . aj F(~
000 ) =aj . . . ak . . .
In definingnjk, (∗) says that no swap before nij makes a difference, sonjk ≥nij or nj−≥ni−. Buti andj are arbitrary, confirming nj− =ni−. The single pivotal voter eliminates any alternative
except her favorite by (∗).
Comment: The proof builds upon “pivotal voter” proofs of Arrow’s
theorem (Barbera, 1980; Geanakoplos, 2005; Reny, 2001; Yu, 2012).
The Theorem of Gibbard (1973) and Satterthwaite (1975)
Notation: (0
n, ~−n)comes from replacingn’s preferences in~ with
0 n∈ P.
Definition (SP--misreporting leads to worse choices): A SCFF is strategy-proof if F(0n, ~−n)6=F(~) implies
F(~)nF(0
n, ~−n) for every n, every ~, and every 0n.
Theorem: If a SCFF satisfies (O) and (SP), then it is dictatorial.
Lemma: (SP) imply (CM).
Proof: Let~ and{i,j}-runoff~0 agree on{ai,aj}. Suppose that
F(~) =ai andF(01, ~−1) =ak fork 6=i. In one case,ai1aj. When
others report~−1, individual 1 endowed with01is better off reporting
1, contradicting (SP). In the other case ofaj 1ai, given~−1, the
same misreport is tempting ifk6=j. But ifk =j, individual 1 endowed
with1is better off reporting01, violating (SP). SoF(01, ~−1) has to
remainai. Likewise, the process of updating~ to~
0
Arrow’s General Impossibility Theorem (Arrow, 1963)
Definition (AD):A SPFR isArrow dictatorial if there exists asocial
preference dictator nsuch that ainaj impliesaiR(~)aj for everyi,j
and every~.
Definition (U):A SPFR is unanimousif ainaj for everynimplies
aiR(~)aj for everyi,j and every~.
Definition (IIA):A SPFRis independent of irrelevant alternatives if
whenever~ and~0 agree on{ai,aj},aiR(~)aj impliesaiR(~ 0
)aj.
Theorem: If a SPFR satisfies (U) and (IIA), then it is Arrow dictatorial.
Proof: GivenR, we can define a SCFFR that selects the alternative
ranked highest byR. FR obviously satisfies(O), because by (U),
FR(~) =a
i if only{ai} is at the top. To see(CM), let~ and
{i,j}-runoff~0 agree on{ai,aj}. WhenFR(~) =ai,aiR(~)aj, so by (IIA),aiR(~
0
)aj. ButaiR(~ 0
)ak fork ∈ {/ i,j}by (U), so FR(~ 0
) has to
beai. Therefore,FR presents asocial choice dictatorn. She is a social
preference dictator too. Ifai naj, individualndictates
FR(Tij(~)) =ai, soaiR(Tij(~))aj and by (IIA)aiR(~)aj.