Illustrations to the proof provided in http://en.wikipedia.org/wiki/Arrow's_impossibility_theorem#Informal_proof

1. Mathematics you might
not believe existed
Margus Niitsoo

2. “Social choice theory”

3. “Social choice theory”
M>O
O>M
O>M

4. “Social choice theory”
M>O
O>M
O>M
O First, M second

5. “Social choice theory”

6. “Social choice theory”
M>C>O
O>M>C
C>O>M

7. “Social choice theory”
M>C>O
O>M>C
C>O>M
But we still
need a fair
ordering!

8. 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”

9. 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.”

10. 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.

11. 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.

12. Part 1: Pivotal voter for B
B>A>C A>C>B
B>C>A C>A>B
B>A>C A>C>B
B>X>Y X>Y>B

13. Part 1: Pivotal voter for B
B>A>C
B>C>A
B>A>C
B>X>Y

14. Part 1: Pivotal voter for B
B>A>C
B>C>A
A>C>B
??

15. Part 1: Pivotal voter for B
B>A>C
C>A>B
A>C>B
??

16. Part 1: Pivotal voter for B
A>C>B
C>A>B
A>C>B
X>Y>B

17. Part 1: Pivotal voter for 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

18. Proof in 4 parts
● Pivotal voter for B exists
● Pivot is instantaneous
● That voter dictates A/C
● That voter is
a total dictator

19. Assume pivot is gradual:
B>A>C B>A>C
B>C>A C>A>B
A>C>B A>C>B
B>X>Y X>B>Y

20. Take the inbetween step
B>A>C B>A>C
C>A>B C>A>B
A>C>B A>C>B
X>B>Y X>B>Y

21. And play with A and C
B>A>C B>A>C
A>C>B C>A>B
A>C>B A>C>B
X>B>Y X>B>Y

22. And play with A and C
B>A>C B>A>C
A>C>B C>A>B
A>C>B A>C>B
A>B>C X>B>Y
(Unanimity)

23. Which gives an ordering...
B>A>C B>A>C
A>C>B C>A>B
A>C>B A>C>B
A>B>C A > B > (Y)
(I.I.A – look only at A and B )

24. … that is contradictory
B>C>A B>A>C
C>A>B C>A>B
C>A>B A>C>B
C>B>A A > B > (Y)
(I.I.A now implies C > B )

25. Therefore pivot : top->bot
B>A>C B>A>C
B>C>A C>A>B
A>C>B A>C>B
B>X>Y X>Y>B

26. Proof in 4 parts
● Pivotal voter for B exists
● Pivot is instantaneous
● That voter dictates A/C
● That voter is
a total dictator

27. Take any ordering ...
A>C>B
B>A>C
C>B>A
??

28. … and play with B
A>C>B B>A>C
B>A>C A>B>C
C>B>A C>A>B
?? ??
(I.I.A implies that A and C are
ordered the same in both)

29. However, A>B
B>A>C B>A>C
A>C>B A>B>C
C>A>B C>A>B
X>Y>B ?? A>B ??
(I.I.A just looking at A and B)

30. And B>C
B>A>C B>A>C
B>A>C A>B>C
C>A>B C>A>B
B>X>Y A>B>C
(I.I.A just looking at B and C)

31. So we get A > C
A>C>B B>A>C
B>A>C A>B>C
C>B>A C>A>B
??A > C?? A>B>C
(I.I.A implies that A and C are
ordered the same in both)

32. Or C > A, as 2. dictates
A>C>B B>A>C
B>C>A C>B>A
C>B>A C>A>B
??C > A?? C>B>A
(I.I.A implies that A and C are
ordered the same in both)

33. Proof in 4 parts
● Pivotal voter for B exists
● Pivot is instantaneous
● That voter dictates A/C
● That voter is
a total dictator

34. Different pivots
● 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

35. 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!

36. Therefore nr 2. is a dictator!

37. Arrow's impossibility theorem:
● Theorem (K.Arrow,1950)
No fair voting scheme
exists (unless you like
dictators)

38. 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

39. More negative results
● No system of axioms where you
could prove all true statements that
you can form inside it. (Gödel, 1931)

40. 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)

41. 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)

42. Thank you!