Transcript Διαφάνεια 1

How “impossible” is it to
design a Voting rule?
Angelina Vidali
University of Athens
The setting
N = f 1; : : : ; ng
individuals (voters)
A = f a1 ; : : : ; am g
alternatives
ui (a) > ui (b)
Voter i prefers a to b
ui
a
b
c
voter i
u = (u1 ; : : : ; un )
a preference profile
Voting rule vs Social welfare
function
• A Voting rule
is a mapping
f :U!
set of all (strict)
preference
profiles u
A
set of
alternatives
• it chooses the “winning
alternative”
Gibbard-Satterthwaite ‘70s
• A Social welfare
function is a
mapping
f : U ! U¤
set of all (weak)
preference
profiles u
• it chooses the
“social ranking”
Arrow ‘50s
Condorcet rule
• a beats b in a pairwise election if the majority
of voters prefers a to b
• a is a Condorcet winner if a beats any other
candidate in a pairwise election
4
the Condorcet paradox
c
a
b
c
a
b
voter 1
a
voter 2
bb
a
c
b
c
voter 3
5
Some voting rules
• Plurality vote: a single winner is chosen by having more
votes than any other individual representative.
• Borda vote: The Kth-ranked alternative gets a score of
N-K. All scores are summed and the candidate with the
highest total score wins.
• Instant Runoff Voting: If no candidate receives a majority
of first preference rankings, the candidate with the
fewest number of votes is eliminated and that
candidate's votes redistributed to the voters' next
preferences among the remaining candidates. This
process is repeated until one candidate has a majority of
votes.
A manipulable voting rule
• If individual i reports a false preference profile
instead of his true preference the outcome of
the elections
is 0better
for>him.
u (f (u
; u ))
u (f (u))
i
KKE
i
¡ i
i
If I vote ΚΚΕ the outcome f(u) will be either ΝΔ or ΠΑΣΟΚ
if I vote ΠΑΣΟΚ the outcome will be ΠΑΣΟΚ.
It is better for me to report ΠΑΣΟΚ.
ΠΑΣΟΚ
ΝΔ
voter i
non-manipulable=strategyproof
The voter who lies determines the
winner in a tie!
• Ties is the most tricky part…
KKE
ΠΑΣΟΚ
ΠΑΣΟΚ
ΝΔ
ΝΔ
ΠΑΣΟΚ
ΝΔ
KKE
ΠΑΣΟΚ
ΠΑΣΟΚ
ΝΔ
KKE
ΝΔ
KKE
KKE
voter 1
voter 2
voter 3
voter 4
voter 5
A dictator
The dictators top alternative is
the outcome of the elections.
ui (f (u)) ¸ ui (a) for all a 2 A and u 2 U
Neutral
• The names of the candidates don’t matter.
i.e. f commutes with permutations of [m]
f (¼(u)) = ¼(f (u))
Example:
• b is the winner
if all voters a b in their preference profiles
• then a is the winner.
Monotonicity
“If our preference for the government increases
it is reelected”
• Let f be a strategyproof voting rule, f(u)=a.
• As long as, for all voters, the alternatives that
were worse than a in u, remain worse in v the
allocation remains the same.
0
a1
B
Ba
B 2
u= B a
B
@a
3
a4
1
¢¢¢
.. C
.C
C
C
C
A
0
a1
B
B a
B
Ba
B 3
@a
2
a4
1
¢¢¢
.. C
.C
C
C
C
A
as long as the red
elements stay below a
the outcome remains a
…even if one of the black
elements moves below a
Pareto Optimality
1
• “If everybody prefers a0to b . . .
.. .. .. a
a
then b is not elected.” B
C
B ..
C
..
B . a . a bC
C
u= B
B
C
.
.
B b .. a b .. C
@
A
..
.. ..
. b b . .
Pareto Optimality follows from Monotonicity
Gibbard (‘73)-Satterthwaite (‘75) theorem
n¸ 3
• If the number of alternatives
• then a voting rule that is strategyproof and onto
is dictatorial.
• follows from Arrow’s impossibility theorem
(1951)
using the correspondence between:
Independence of Irrelevant Alternatives and
strategyproofness
Independence of Irrelevant Alternatives
The social relative ranking of two alternatives
a, b depends only on their relative ranking.
If one candidate dies the choice to be made
among the set S of surviving candidates should be
independent of the the preferences of individuals
for candidates not in S.
[See: Social Choice and Individual Values, K.J. Arrow p.26]
see: www.scorevoting.net
a voting method that violates IIA
• The alternative with the highest
At first
weighted sum of votes wins.
weight
4
3
2
1
a
b
c
d
a
b
c
d
voter 1
voter 2
c
d
a
b
voter
a is chosen
a:4+4+2=10
b:7
c:8
d:6
but if b leaves:
a:10
c:10
d:7
3 we get a tie
between a and c
The proof of G-S theorem for
n=2 voters
So a wins in both u and v.
1 a dictator for a.
voter 0
1 becomes
a
u = @b
c
0
a
v = @b
c
b
aA
c
1
b
cA
a
c cannot win (Pareto Optimality)
Assume a wins (w.l.o.g.)
Then b wins (monotonicity) cdc
c cannot win (Pareto Optimal.)
Suppose b wins
The proof of G-S theorem for
n=2 voters (2)
•
•
•
•
Repeat for every pair of alternatives {a,b}
A1={x| player 1 is a dictator for x}
AjA
2)jis·a 1
dictator for y}
2={y|
n (Aplayer
\
A
1
2
because: if it had two
distinct elements then one of them should
belong to A1 or A;2.
• Finally some Ai= all the element belong to
Aj and j is the dictator.
Towards a Quantitative version
Gibbard-Satherwaite theorem:
“Every non-trivial (=non dictatorial) voting
rule is strategically vulnerable.”
How often?
For what fraction of profiles does such a
manipulation exist?
Impartial culture assumption
• Voters vote independently and randomly
• We draw independently and uniformly a
random ranking for each voter
• possible rankings for voter i: m!
• P(each ranking)= 1/m!
Manipulation power of a voter: Mi(f)
The manipulation power Mi( f ), of voter i on the
social choice function
u0 f, is the probability that :
if voter i reports a i chosen uniformly at random
this is a profitable manipulation of f for voter i .
$
i
$
What is the probability I can gain
something by just drawing one of
the m alternatives randomly and
reporting this instead of my true
preference?
individual i
ε-strategyproof
P [f (u0; u¡ i ) > f (u)] · ²
i
=
The manipulation power Mi( f ), of voter i on
the social choice function f, satisfies
M i (f ) < ²
δ-far from dictarorship
• The distance between two functions f,g is
¢ (f ; g) = Pu 2 U [f (u) 6
= g(u)]
• f is δ-far from dictatorship, if for any
dictatorship g
¢ (f ; g) > ±
Quantitative version of G-S theorem
[Friedgut-Kalai-Nisan FOCS’08]
²
For every >0 if f is a voting rule for n voters is
• neutral
• ²
among 3 alternatives
•
-far from dictatorship,
1)
then one of the voters has a(non-negligible
n
manipulation power of
.
Quantitative version of G-S theorem
[Xia-Conitzer EC’08]
-
+
+
a list of assumptions…
• homogeneity
• anonymity
• non-imposition
• a canceling out condition
• there exists a stable profile that is still stable after
one given alternative is uniformly moved to
different positions
(they argue that many known voting rules satisfy them)
for arbitrarily many alternatives and players
Quantitative version of G-S theorem
2 voters
[Dobzinski-Proccacia WINE’08]
²<
1
32m 9
For
if a voting rule f for 2 voters is
• Pareto optimal (annoying condition!)
• among at least 3 alternatives
²
• with manipulation
power <
16m 8 ²
then f is
-far from dictatorship.
no neutrality assumption here!
m can also be greater than 3
Some open problems
• Quantitative version of G-S theorem for
more than 2 voters (with less conditions!)
• What about the impartial culture
assumption? is it plausible?
• Find quantitative versions of known
mechanism design results:
straptegyproof
ε-strategyproof
Endnote
"Most systems are not going to work badly
all of the time, all I proved is that all can
work badly at times."
K. J. Arrow
…or do they work badly most of the time???