arrow's impossiblity theorem or "mathematics you might not believe existed"
DESCRIPTION
Illustrations to the proof provided in http://en.wikipedia.org/wiki/Arrow's_impossibility_theorem#Informal_proofTRANSCRIPT
Mathematics you might not believe existed
Margus Niitsoo
“Social choice theory”
“Social choice theory”
M > OO > MO > M
“Social choice theory”
M > OO > MO > M
O First, M second
“Social choice theory”
“Social choice theory”
M > C > OO > M > CC > O > M
“Social choice theory”
M > C > OO > M > CC > O > M
But we still need a fair ordering!
What ordering is fair?● If everyone thinks A > B, then in
the final order, A > B ● i.e. if everyone thinks A should win,
he should actually win!
● “Unanimity”
What ordering is fair?● If you move C around in one
vote, the final relative standing of A and B should not change.● i.e. if a vote A > B > C is changed to A > C >
B or C > A > B, then if A>B before in the final result, it should remain so
● “Ind. Irrel. Alt.”
And now to the weird math● Theorem (K.Arrow,1950)
The only fair voting scheme is a Dictatorship.
i.e. the final ordering is based solely on the preferences of one fixed voter – the dictator.
Proof in 4 parts
●Pivotal voter for B exists●Pivot is instantaneous●That voter dictates A/C●That voter is a total dictator
Proof adapted from John Geanakoplos via Wikipedia.
Part 1: Pivotal voter for B
B > A > CB > C > AB > A > C
B > X > Y
A > C > BC > A > BA > C > B
X > Y > B
Part 1: Pivotal voter for B
B > A > CB > C > AB > A > C
B > X > Y
Part 1: Pivotal voter for B
B > A > CB > C > AA > C > B
??
Part 1: Pivotal voter for B
B > A > CC > A > BA > C > B
??
Part 1: Pivotal voter for B
A > C > BC > A > BA > C > B
X > Y > B
● Change in the final result has to be in one of the 3 steps
● Call that voter “Pivotal”● In the following examples,
2. voter is pivotal
Part 1: Pivotal voter for B
Proof in 4 parts
●Pivotal voter for B exists●Pivot is instantaneous●That voter dictates A/C●That voter is a total dictator
Assume pivot is gradual:
B > A > CB > C > AA > C > B
B > X > Y
B > A > CC > A > BA > C > B
X > B > Y
Take the inbetween step
B > A > CC > A > BA > C > B
X > B > Y
B > A > CC > A > BA > C > B
X > B > Y
And play with A and C
B > A > CC > A > BA > C > B
X > B > Y
B > A > CA > C > BA > C > B
X > B > Y
And play with A and C
B > A > CC > A > BA > C > B
X > B > Y
B > A > CA > C > BA > C > B
A > B > C(Unanimity)
Which gives an ordering...
B > A > CC > A > BA > C > B
A > B > (Y)
B > A > CA > C > BA > C > B
A > B > C(I.I.A – look only at A and B )
… that is contradictory
B > A > CC > A > BA > C > B
A > B > (Y)
B > C > AC > A > BC > A > B
C > B > A(I.I.A now implies C > B )
Therefore pivot : top->bot
B > A > CB > C > AA > C > B
B > X > Y
B > A > CC > A > BA > C > B
X > Y > B
Proof in 4 parts
●Pivotal voter for B exists●Pivot is instantaneous●That voter dictates A/C●That voter is a total dictator
Take any ordering ...
A > C > BB > A > CC > B > A
??
… and play with B
A > C > BB > A > CC > B > A
??
B > A > CA > B > CC > A > B
??(I.I.A implies that A and C are ordered the same in both)
However, A>B
B > A > CA > C > BC > A > B
X > Y > B
B > A > CA > B > CC > A > B
?? A>B ??
(I.I.A just looking at A and B)
And B>C
B > A > CB > A > CC > A > B
B > X > Y
B > A > CA > B > CC > A > B
A > B > C
(I.I.A just looking at B and C)
So we get A > C
A > C > BB > A > CC > B > A
??A > C??
B > A > CA > B > CC > A > B
A > B > C(I.I.A implies that A and C are ordered the same in both)
Or C > A, as 2. dictates
A > C > BB > C > AC > B > A
??C > A??
B > A > CC > B > AC > A > B
C > B > A(I.I.A implies that A and C are ordered the same in both)
Proof in 4 parts
●Pivotal voter for B exists●Pivot is instantaneous●That voter dictates A/C●That voter is a total dictator
● We showed that:● If you pivot for B, you dictate A/C
● Equivalently:● If you pivot for A,
you dictate B/C● If you pivot for C,
you dictate A/B
Different pivots
Assume different pivots● Maybe voter 1. dictates A/B
and voter 3. dictates B/C?● Problem if:
1: A>B, 3: B>C, 2: C>A● i.e. cannot happen!
– 1 dictates A/B and B/C but not A/C also can't!
Therefore nr 2. is a dictator!
Arrow's impossibility theorem:
● Theorem (K.Arrow,1950)
No fair voting scheme exists (unless you like dictators)
Gibbard–Satterthwaite● Theorem:
Even if you want just the winner, you have to accept:● A dictator, or,● One candidate who cannot win,
even in theory, or● Tactical voting is possible
– i.e. lying in your vote
More negative results● No system of axioms where you
could prove all true statements that you can form inside it. (Gödel, 1931)
More negative results● No system of axioms where you
could prove all true statements that you can form inside it. (Gödel, 1931)
● No computer program that checks whether other programs go into an infinite loop. (Turing, 1936)
More negative results● No system of axioms where you
could prove all true statements that you can form inside it. (Gödel, 1931)
● No computer program that checks whether other programs go into an infinite loop. (Turing, 1936)
● Also, no free lunch!(Wolpert, Macready 1997)
Thank you!