1. Social Choice Session 2 Carmen Pasca and John Hey

2. Arrow’s Impossibility Theorem In social choice theory, Arrow’s impossibility theorem... states that, when voters have three or more discrete alternatives (options), no voting system can convert the ranked preferences of individuals into a community-wide ranking while also meeting a certain set of criteria. These criteria are called unrestricted domain, non-dictatorship, Pareto efficiency, and independence of irrelevant alternatives. (Wikipedia) Is this obvious or not? This lecture we give a proof, borrowed from John Bone at the University of York. We have also included another proof borrowed from Kevin Feasel. Use whichever you prefer.

3. Arrow Arrow asks the question: “can we aggregate individual preferences into social preferences?” He invokes the following conditions: Universal Domain: is applicable to any profile. Consistency: produces a complete, transitive ordering of available alternatives. Pareto: if everyone prefers x to y then so should society. Independence of Irrelevant Alternatives: orders x and y on the basis only of individual preferences on x and y.

4. The result Arrow shows that if one wants these conditions then the only way to get them is through Dictatorship. On the one hand this seems trivial (“how can you aggregate different preferences?”)...... but one should look at the result as telling us that we need to drop something if we want something that in some way represent the preferences of society – whatever that means. We need to drop some of the assumptions...... or change our way of looking at the problem.

5. The Story and Two of the Axioms We consider a very simple society with two members, Jen and Ken, and three things to order: a, b and c. Think of these as ways to organise society. Universal Domain: is applicable to any profile. This means that whatever the preferences are we should be able to aggregate them. Pareto: if everyone prefers x to y then so should society. This seems unquestionable.

6. 6 a b ca c bb a cb c ac a bc b a a b c a c b b a c b c a c a b c b a 111213141516 212223242526 313233343536 414243444546 515253545556 616263646566 Universal Domain& Pareto Ken Jen aPbaPb bPcbPc aPcaPc

7. 7 a b ca c bb a cb c ac a bc b a a b c a c b b a c b c a c a b c b a 111213141516 212223242526 313233343536 414243444546 515253545556 616263646566 Universal Domain& Pareto Ken Jen aPbaPb bPcbPc aPcaPc aPbaPb aPcaPc

8. 8 a b ca c bb a cb c ac a bc b a a b c a c b b a c b c a c a b c b a 111213141516 212223242526 313233343536 414243444546 515253545556 616263646566 Universal Domain& Pareto Ken Jen aPbaPb bPcbPc aPcaPc aPbaPb bPcbPc aPcaPcaPcaPc

9. 9 a b ca c bb a cb c ac a bc b a a b c a c b b a c b c a c a b c b a 111213141516 212223242526 313233343536 414243444546 515253545556 616263646566 Universal Domain& Pareto aPbaPb Ken Jen aPbaPb bPcbPc aPcaPc aPbaPb bPcbPc aPcaPcaPcaPc bPcbPc

10. 10 a b ca c bb a cb c ac a bc b a a b c a c b b a c b c a c a b c b a 111213141516 212223242526 313233343536 414243444546 515253545556 616263646566 Universal Domain& Pareto aPbaPb aPbaPb bPcbPc aPcaPc aPbaPb cPbcPb aPcaPc bPabPa bPcbPc aPcaPc bPabPa bPcbPc cPacPa aPbaPb cPbcPb cPacPa bPabPa cPbcPb cPacPa aPbaPb bPcbPcbPcbPc aPcaPcaPcaPc aPbaPbaPbaPb cPbcPbcPbcPb aPcaPcaPcaPc bPcbPcbPcbPc bPabPabPabPa aPcaPcaPcaPc bPabPabPabPa bPcbPcbPcbPc cPacPacPacPa cPacPacPacPa cPbcPbcPbcPb aPbaPbaPbaPb bPabPabPabPa cPbcPbcPbcPb cPacPacPacPa Ken Jen

11. Now the other two axioms Consistency: produces a complete, transitive ordering of available alternatives. We have a complete transitive ordering for society. Independence of Irrelevant Alternatives: orders x and y on the basis only of individual preferences on x and y. This is particularly important – we are going to use it frequently. In the slides that follow we first impose consistency and then repeatedly we invoke the independence of irrelevant alternatives.

12. 12 a b ca c bb a cb c ac a bc b a Ken a b c a c b b a c b c a c a b c b a Jen 111213141516 212223242526 313233343536 414243444546 515253545556 616263646566 aPbaPb aPbaPb bPcbPc aPcaPc aPbaPb cPbcPb aPcaPc bPabPa bPcbPc aPcaPc bPabPa bPcbPc cPacPa aPbaPb cPbcPb cPacPa bPabPa cPbcPb cPacPa aPbaPb bPcbPcbPcbPc aPcaPcaPcaPc aPbaPbaPbaPb cPbcPbcPbcPb aPcaPcaPcaPc bPcbPcbPcbPc bPabPabPabPa aPcaPcaPcaPc bPabPabPabPa bPcbPcbPcbPc cPacPacPacPa cPacPacPacPa cPbcPbcPbcPb aPbaPbaPbaPb bPabPabPabPa cPbcPbcPbcPb cPacPacPacPa cPbcPbcPbcPbcPbcPb cPbcPbcPbcPbcPbcPb cPbcPbcPbcPbcPbcPb aPcaPc Consistency& IIA requires (at least) one of these

13. Now what? We can now use an implication of what we have found. Note in cell 36 that bPa and cPb; and hence we can deduce that cPa. Let us insert that in cell 36 and explore the implications. This allows us to use the Independence of Irrelevant Alternatives Axiom again.

14. 14 a b ca c bb a cb c ac a bc b a a b c a c b b a c b c a c a b c b a 111213141516 212223242526 313233343536 414243444546 515253545556 616263646566 aPbaPb aPbaPb bPcbPc aPcaPc aPbaPb cPbcPb aPcaPc bPabPa bPcbPc aPcaPc bPabPa bPcbPc cPacPa aPbaPb cPbcPb cPacPa bPabPa cPbcPb cPacPa aPbaPb bPcbPcbPcbPc aPcaPcaPcaPc aPbaPbaPbaPb cPbcPbcPbcPb aPcaPcaPcaPc bPcbPcbPcbPc bPabPabPabPa aPcaPcaPcaPc bPabPabPabPa bPcbPcbPcbPc cPacPacPacPa cPacPacPacPa cPbcPbcPbcPb aPbaPbaPbaPb bPabPabPabPa cPbcPbcPbcPb cPacPacPacPa cPbcPbcPbcPbcPbcPb cPbcPbcPbcPbcPbcPb cPbcPbcPbcPbcPbcPb cPacPacPacPacPacPa cPacPacPacPacPacPa cPacPacPacPacPacPa Consistency& IIA Ken Jen

15. Now what? Once again we can now use an implication of what we have found. Note in cell 14 that bPc and cPa; and hence we can deduce that bPa. Let us insert that in cell 14 and explore the implications. This allows us to use the Independence of Irrelevant Alternatives Axiom again.

16. 16 a b ca c bb a cb c ac a bc b a a b c a c b b a c b c a c a b c b a 111213141516 212223242526 313233343536 414243444546 515253545556 616263646566 aPbaPb aPbaPb bPcbPc aPcaPc aPbaPb cPbcPb aPcaPc bPabPa bPcbPc aPcaPc bPabPa bPcbPc cPacPa aPbaPb cPbcPb cPacPa bPabPa cPbcPb cPacPa aPbaPb bPcbPcbPcbPc aPcaPcaPcaPc aPbaPbaPbaPb cPbcPbcPbcPb aPcaPcaPcaPc bPcbPcbPcbPc bPabPabPabPa aPcaPcaPcaPc bPabPabPabPa bPcbPcbPcbPc cPacPacPacPa cPacPacPacPa cPbcPbcPbcPb aPbaPbaPbaPb bPabPabPabPa cPbcPbcPbcPb cPacPacPacPa cPbcPb cPbcPb cPbcPbcPbcPb cPbcPbcPbcPb cPbcPbcPbcPbcPbcPb cPacPacPacPacPacPa cPacPacPacPacPacPa cPacPacPacPacPacPa bPabPabPabPabPabPa bPabPabPabPabPabPa bPabPabPabPabPabPa Ken Jen Consistency& IIA

17. Now what? Once again we can now use an implication of what we have found. Note in cell 23 that bPa and aPc; and hence we can deduce that bPc. Let us insert that in cell 23 and explore the implications. This allows us to use the Independence of Irrelevant Alternatives Axiom again.

18. 18 a b ca c bb a cb c ac a bc b a a b c a c b b a c b c a c a b c b a 111213141516 212223242526 313233343536 414243444546 515253545556 616263646566 aPbaPb aPbaPb bPcbPc aPcaPc aPbaPb cPbcPb aPcaPc bPabPa bPcbPc aPcaPc bPabPa bPcbPc cPacPa aPbaPb cPbcPb cPacPa bPabPa cPbcPb cPacPa aPbaPb bPcbPcbPcbPc aPcaPcaPcaPc aPbaPbaPbaPb cPbcPbcPbcPb aPcaPcaPcaPc bPcbPcbPcbPc bPabPabPabPa aPcaPcaPcaPc bPabPabPabPa bPcbPcbPcbPc cPacPacPacPa cPacPacPacPa cPbcPbcPbcPb aPbaPbaPbaPb bPabPabPabPa cPbcPbcPbcPb cPacPacPacPa cPbcPb cPbcPb cPbcPbcPbcPb cPbcPbcPbcPb cPbcPbcPbcPbcPbcPb cPacPacPacPacPacPa cPacPacPacPacPacPa cPacPacPacPacPacPa bPabPabPabPabPabPa bPabPabPabPabPabPa bPabPabPabPabPabPa bPcbPcbPcbPcbPcbPc bPcbPcbPcbPcbPcbPc bPcbPcbPcbPcbPcbPc Ken Jen Consistency& IIA

19. Now what? Once again can now use an implication of what we have found. Note in cell 51 that aPb and bPc; and hence we can deduce that aPc. Let us insert that in cell 51 and explore the implications. This allows us to use the Independence of Irrelevant Alternatives Axiom again.

20. 20 a b ca c bb a cb c ac a bc b a a b c a c b b a c b c a c a b c b a 111213141516 212223242526 313233343536 414243444546 515253545556 616263646566 aPbaPb aPbaPb bPcbPc aPcaPc aPbaPb cPbcPb aPcaPc bPabPa bPcbPc aPcaPc bPabPa bPcbPc cPacPa aPbaPb cPbcPb cPacPa bPabPa cPbcPb cPacPa aPbaPb bPcbPcbPcbPc aPcaPcaPcaPc aPbaPbaPbaPb cPbcPbcPbcPb aPcaPcaPcaPc bPcbPcbPcbPc bPabPabPabPa aPcaPcaPcaPc bPabPabPabPa bPcbPcbPcbPc cPacPacPacPa cPacPacPacPa cPbcPbcPbcPb aPbaPbaPbaPb bPabPabPabPa cPbcPbcPbcPb cPacPacPacPa cPbcPb cPbcPb cPbcPbcPbcPb cPbcPbcPbcPb cPbcPbcPbcPbcPbcPb cPacPacPacPacPacPa cPacPacPacPacPacPa cPacPacPacPacPacPa bPabPabPabPabPabPa bPabPabPabPabPabPa bPabPabPabPabPabPa bPcbPcbPcbPcbPcbPc bPcbPcbPcbPcbPcbPc bPcbPcbPcbPcbPcbPc aPcaPcaPcaPcaPcaPc aPcaPcaPcaPcaPcaPc aPcaPcaPcaPcaPcaPc Ken Jen Consistency& IIA

21. Finally For the final time can now use an implication of what we have found. Note in cell 62 that cPb and aPc; and hence we can deduce that aPb. Let us insert that in cell 62 and explore the implications. This allows us to use the Independence of Irrelevant Alternatives Axiom again.

22. 22 a b ca c bb a cb c ac a bc b a a b c a c b b a c b c a c a b c b a 111213141516 212223242526 313233343536 414243444546 515253545556 616263646566 aPbaPb aPbaPb bPcbPc aPcaPc aPbaPb cPbcPb aPcaPc bPabPa bPcbPc aPcaPc bPabPa bPcbPc cPacPa aPbaPb cPbcPb cPacPa bPabPa cPbcPb cPacPa aPbaPb bPcbPcbPcbPc aPcaPcaPcaPc aPbaPbaPbaPb cPbcPbcPbcPb aPcaPcaPcaPc bPcbPcbPcbPc bPabPabPabPa aPcaPcaPcaPc bPabPabPabPa bPcbPcbPcbPc cPacPacPacPa cPacPacPacPa cPbcPbcPbcPb aPbaPbaPbaPb bPabPabPabPa cPbcPbcPbcPb cPacPacPacPa cPbcPb cPbcPb cPbcPbcPbcPb cPbcPbcPbcPb cPbcPbcPbcPbcPbcPb cPacPacPacPacPacPa cPacPacPacPacPacPa cPacPacPacPacPacPa bPabPabPabPabPabPa bPabPabPabPabPabPa bPabPabPabPabPabPa bPcbPcbPcbPcbPcbPc bPcbPcbPcbPcbPcbPc bPcbPcbPcbPcbPcbPc aPcaPcaPcaPcaPcaPc aPcaPcaPcaPcaPcaPc aPcaPcaPcaPcaPcaPc aPbaPbaPbaPbaPbaPb aPbaPbaPbaPbaPbaPb aPbaPbaPbaPbaPbaPb The Dictator Ken Jen Consistency& IIA

23. Two questions for the break (1) You may recall that we started this line of logic with one of two possible ways of getting consistency. What do you think happens if we take the second possible way? (2) Why did we not pick up any intransitivities on the way? That is, for example, why did we not find something like aPbbPccPa in one of the cells? Magic?

24. 24 a b ca c bb a cb c ac a bc b a a b c a c b b a c b c a c a b c b a 111213141516 212223242526 313233343536 414243444546 515253545556 616263646566 aPbaPb aPbaPb bPcbPc aPcaPc aPbaPb cPbcPb aPcaPc bPabPa bPcbPc aPcaPc bPabPa bPcbPc cPacPa aPbaPb cPbcPb cPacPa bPabPa cPbcPb cPacPa aPbaPb bPcbPcbPcbPc aPcaPcaPcaPc aPbaPbaPbaPb cPbcPbcPbcPb aPcaPcaPcaPc bPcbPcbPcbPc bPabPabPabPa aPcaPcaPcaPc bPabPabPabPa bPcbPcbPcbPc cPacPacPacPa cPacPacPacPa cPbcPbcPbcPb aPbaPbaPbaPb bPabPabPabPa cPbcPbcPbcPb cPacPacPacPa aPcaPcaPcaPcaPcaPc aPcaPcaPcaPcaPcaPc aPcaPcaPcaPcaPcaPc cPbcPb Ken Jen Consistency& IIA

25. 25 a b ca c bb a cb c ac a bc b a a b c a c b b a c b c a c a b c b a 111213141516 212223242526 313233343536 414243444546 515253545556 616263646566 aPbaPb aPbaPb bPcbPc aPcaPc aPbaPb cPbcPb aPcaPc bPabPa bPcbPc aPcaPc bPabPa bPcbPc cPacPa aPbaPb cPbcPb cPacPa bPabPa cPbcPb cPacPa aPbaPb bPcbPcbPcbPc aPcaPcaPcaPc aPbaPbaPbaPb cPbcPbcPbcPb aPcaPcaPcaPc bPcbPcbPcbPc bPabPabPabPa aPcaPcaPcaPc bPabPabPabPa bPcbPcbPcbPc cPacPacPacPa cPacPacPacPa cPbcPbcPbcPb aPbaPbaPbaPb bPabPabPabPa cPbcPbcPbcPb cPacPacPacPa aPbaPbaPbaPbaPbaPb aPbaPbaPbaPbaPbaPb aPbaPbaPbaPbaPbaPb aPcaPcaPcaPcaPcaPc aPcaPcaPcaPcaPcaPc aPcaPcaPcaPcaPcaPc Ken Jen Consistency& IIA

26. 26 a b ca c bb a cb c ac a bc b a a b c a c b b a c b c a c a b c b a 111213141516 212223242526 313233343536 414243444546 515253545556 616263646566 aPbaPb aPbaPb bPcbPc aPcaPc aPbaPb cPbcPb aPcaPc bPabPa bPcbPc aPcaPc bPabPa bPcbPc cPacPa aPbaPb cPbcPb cPacPa bPabPa cPbcPb cPacPa aPbaPb bPcbPcbPcbPc aPcaPcaPcaPc aPbaPbaPbaPb cPbcPbcPbcPb aPcaPcaPcaPc bPcbPcbPcbPc bPabPabPabPa aPcaPcaPcaPc bPabPabPabPa bPcbPcbPcbPc cPacPacPacPa cPacPacPacPa cPbcPbcPbcPb aPbaPbaPbaPb bPabPabPabPa cPbcPbcPbcPb cPacPacPacPa aPbaPbaPbaPbaPbaPb aPbaPbaPbaPbaPbaPb aPbaPbaPbaPbaPbaPb cPbcPbcPbcPbcPbcPb cPbcPbcPbcPbcPbcPb cPbcPbcPbcPbcPbcPb aPcaPcaPcaPcaPcaPc aPcaPcaPcaPcaPcaPc aPcaPcaPcaPcaPcaPc Ken Jen Consistency& IIA

27. 27 a b ca c bb a cb c ac a bc b a a b c a c b b a c b c a c a b c b a 111213141516 212223242526 313233343536 414243444546 515253545556 616263646566 aPbaPb aPbaPb bPcbPc aPcaPc aPbaPb cPbcPb aPcaPc bPabPa bPcbPc aPcaPc bPabPa bPcbPc cPacPa aPbaPb cPbcPb cPacPa bPabPa cPbcPb cPacPa aPbaPb bPcbPcbPcbPc aPcaPcaPcaPc aPbaPbaPbaPb cPbcPbcPbcPb aPcaPcaPcaPc bPcbPcbPcbPc bPabPabPabPa aPcaPcaPcaPc bPabPabPabPa bPcbPcbPcbPc cPacPacPacPa cPacPacPacPa cPbcPbcPbcPb aPbaPbaPbaPb bPabPabPabPa cPbcPbcPbcPb cPacPacPacPa cPbcPbcPbcPbcPbcPb cPbcPbcPbcPbcPbcPb cPbcPbcPbcPbcPbcPb cPacPacPacPacPacPa cPacPacPacPacPacPa cPacPacPacPacPacPa aPbaPbaPbaPbaPbaPb aPbaPbaPbaPbaPbaPb aPbaPbaPbaPbaPbaPb aPcaPcaPcaPcaPcaPc aPcaPcaPcaPcaPcaPc aPcaPcaPcaPcaPcaPc Ken Jen Consistency& IIA

28. 28 a b ca c bb a cb c ac a bc b a a b c a c b b a c b c a c a b c b a 111213141516 212223242526 313233343536 414243444546 515253545556 616263646566 aPbaPb aPbaPb bPcbPc aPcaPc aPbaPb cPbcPb aPcaPc bPabPa bPcbPc aPcaPc bPabPa bPcbPc cPacPa aPbaPb cPbcPb cPacPa bPabPa cPbcPb cPacPa aPbaPb bPcbPcbPcbPc aPcaPcaPcaPc aPbaPbaPbaPb cPbcPbcPbcPb aPcaPcaPcaPc bPcbPcbPcbPc bPabPabPabPa aPcaPcaPcaPc bPabPabPabPa bPcbPcbPcbPc cPacPacPacPa cPacPacPacPa cPbcPbcPbcPb aPbaPbaPbaPb bPabPabPabPa cPbcPbcPbcPb cPacPacPacPa cPbcPbcPbcPbcPbcPb cPbcPbcPbcPbcPbcPb cPbcPbcPbcPbcPbcPb cPacPacPacPacPacPa cPacPacPacPacPacPa cPacPacPacPacPacPa bPabPabPabPabPabPa bPabPabPabPabPabPa bPabPabPabPabPabPa aPbaPbaPbaPbaPbaPb aPbaPbaPbaPbaPbaPb aPbaPbaPbaPbaPbaPb aPcaPcaPcaPcaPcaPc aPcaPcaPcaPcaPcaPc aPcaPcaPcaPcaPcaPc Ken Jen Consistency& IIA

29. 29 a b ca c bb a cb c ac a bc b a a b c a c b b a c b c a c a b c b a 111213141516 212223242526 313233343536 414243444546 515253545556 616263646566 aPbaPb aPbaPb bPcbPc aPcaPc aPbaPb cPbcPb aPcaPc bPabPa bPcbPc aPcaPc bPabPa bPcbPc cPacPa aPbaPb cPbcPb cPacPa bPabPa cPbcPb cPacPa aPbaPb bPcbPcbPcbPc aPcaPcaPcaPc aPbaPbaPbaPb cPbcPbcPbcPb aPcaPcaPcaPc bPcbPcbPcbPc bPabPabPabPa aPcaPcaPcaPc bPabPabPabPa bPcbPcbPcbPc cPacPacPacPa cPacPacPacPa cPbcPbcPbcPb aPbaPbaPbaPb bPabPabPabPa cPbcPbcPbcPb cPacPacPacPa cPbcPbcPbcPbcPbcPb cPbcPbcPbcPbcPbcPb cPbcPbcPbcPbcPbcPb cPacPacPacPacPacPa cPacPacPacPacPacPa cPacPacPacPacPacPa bPabPabPabPabPabPa bPabPabPabPabPabPa bPabPabPabPabPabPa bPcbPcbPcbPcbPcbPc bPcbPcbPcbPcbPcbPc bPcbPcbPcbPcbPcbPc aPbaPbaPbaPbaPbaPb aPbaPbaPbaPbaPbaPb aPbaPbaPbaPbaPbaPb The Dictator aPcaPcaPcaPcaPcaPc aPcaPcaPcaPcaPcaPc aPcaPcaPcaPcaPcaPc Ken Jen Consistency& IIA

30. 30 Universal Domain Consistency Pareto Independence of Irrelevant Alternatives (IIA) and that requires only individual rankings Arrow’s Theorem is the Dictator Principle The only principle that satisfies: i.e., where the social ordering simply replicates, across all profiles, some given individual’s ranking.

31. 31 If, for some x and y, xPy at some profile where everyone in V ranks x above y and everyone else ranks y above x..... then, for any w and z, wPz at any profile where everyone in V ranks w above z, irrespective of how anyone else ranks w and z. Given the four conditions (U, C, P, IIA) We have already shown this in the 2-person, 3-alternative, strict ranking case. Generalising this to n-persons, m-alternatives is not very difficult.

32. Arrow’s Impossibility Theorem We have proved it in the context of a two-person society, but as John Bone says, it is not difficult to extend it to a larger society. This is what he does in the next three slides. It is not necessary for you to know this in detail, just understand the principle. If you would like another proof of the theorem, you might like this by Kevin Feasel which I have shamelessy downloaded from the Internet. This follows after the generalisation to n people.

33. 33 There must be some x and y, and some individual J, such that xPy at some profile where J ranks x above y and everyone else ranks y above x. Given the four conditions (U, C, P, IIA) Consider any set V of individuals such that aPb at some profile where everyone in V ranks a above b and everyone else ranks b above a. (We know there must be such a V, since from Pareto one such V is the set of all individuals.) From the previous slide, we know that therefore aPb at any profile where everyone in V ranks a above b. Such as.. Suppose that V contains more than one person.

34. 34 where V 1 and V 2 are subsets of V (i.e. everyone in V is in exactly one of V 1 and V 2, and no-one else is). everyone in V 1 everyone in V 2 abc cab everyone elsebca aPb implies that for any c either aPc or cPb. It follows that there is a set smaller than V, and a pair {x,y}, such that xPy at some profile where everyone in that set ranks x above y and everyone else ranks y above x. Since we can make the same argument for any V containing more than one person, it follows that the smallest such V is an individual (let’s say J).

35. 35 If there is any profile in which Jen ranks x above y and Ken ranks y above x, then from IIA: and at which,xPy at every profile in which Jen ranks x above y and Ken ranks y above x xPy such as at: Jen Ken xyz yzx at which, from Pareto and Consistency:xPz So, again from IIA, also at:xPz Jen Ken wxz zwx wPzat which, from Pareto and Consistency:

36. Arrow's Impossibility Theorem Kevin Feasel December 10, 2006 http://36chambers.wordpress.com/arrow/

37. The Rules 2 individuals with 3 choices (x, y, z) --> 6 profiles for each. – x > y > z, x > z > y, y > x > z, y > z > x, z > x > y, z > y > x – 36 profiles in all for our two-person example (6 * 6)

38. The 36 Profiles

39. Explanation Of Symbols Colored Text – Blue – First choice for both – Green – Second choice for both – Red – Third choice for both – Black – Inconsistent choices (e.g., in profile 10, Z > Y for person A but Y > Z for person B) Numbers along the left-hand side depict the #1, #2, and #3 choices for person A. Numbers along the bottom depict the #1, #2, and #3 choices for person B. Each profile has a number (e.g., 1 and 10 here). How to read this: for profile 10, person A (left-hand side) likes Z > X > Y. Person B (bottom) prefers X > Y > Z.

40. The Assumptions Completeness – All profiles must be solvable. In this case, we have 3 choices among 2 people, so 36 total sets of preferences could arise. We must have a solution for each one of the 36.

41. The Assumptions Unanimity – If all individuals agree on a single position, that position will be guaranteed. – In our case, if an option is red, both individuals rank this option as the #3 choice, so it will end up as #3 in the preference ranking setup.

42. The Assumptions Independence of Irrelevant Alternatives – The relationship between X and Y should be independent of Z. In general terms, the relationship between two elements will not change with the addition of another element. – Example: in profile 25, X > Y for person A (X > Y > Z) and for person B (X > Z > Y). Adding in option Z, we assume, does not alter this relationship. – Example: in profile 26, Y > X for person A (Y > Z > X) and for person B (Z > Y > X). If we removed option Z, we would still expect Y > X for both.

43. Proving The Final Rule Non-dictatorship. We want to understand whether we can come up with a social rule which follows the rules of completeness, unanimity, and independence of irrelevant alternatives, and which is simultaneously non-dictatorial. “Dictatorial” here means that all 36 profiles will match one person's profiles exactly. In other words, all social choices will precisely match the individual's preferences.

44. Solving The Problem 6 preference sets are already solved, thanks to unanimity. In profile 1, for example, all parties agree that X > Y > Z, so X > Y > Z is the social rule. These six completed profiles have the “>” highlighted in yellow.

45. Solving The Problem

46. In addition to this, we know that any profile with colored text is also solved—both people agree on where to place this option. So these can all be filled in as well.

47. Solving The Problem

48. There are two remaining sets of relationships, as shown in profile 23. – a > b: in this example, Y > Z for both individuals. Y is the #2 choice for person A and Z #3, whereas Y is the #1 choice for person B and Z #2. In a case such as this, we know that Y > Z in the social preferences because both people prefer Y to Z. – a ? b: in this example, Z > X for person B, but X > Z for person A. A's preferences are X > Y > Z and B's are Y > Z > X. In this type of situation, we do not yet know if Z or X will be socially preferred because there is no agreement between individuals.

49. Solving The Problem In this case, because we know that Y > Z socially, we can put that down. But we do not know if Y > Z > X, Y > X > Z, or X > Y > Z, so we will not write these down just yet, but once things start clearing up, these rules will be used. The key here is that Z is above and to the right of Y, which means that there is agreement that Y > Z. However, Z is above and to the left of X, meaning there is conflict. Y is also above and to the left of X, so there is another conflict, due to the fact that X > Y > Z for person A, but Y > Z > X for person B.

50. The Single Choice If we look at profile 7, there is a single choice to be made. – Person A (left-hand column): X > Z > Y – Person B (bottom row): X > Y > Z We know that X will be the #1 preference, but we have to decide whether Y > Z or Z > Y here. We can choose either, but let us pick that Y > Z. In other words, we support person B's preference here. – This means that the social preference will be X > Y > Z. In addition, because of the Independence of Irrelevant Alternatives axiom, any time we see a conflict between Y and Z similar to the one in profile 7, we know to choose Y > Z. Let us fill in the chart with this new information...

51. Results

52. Explanation The profiles which have been newly solved (old solutions are colored green; new are yellow) are profiles 7, 8, 9, 10, 17, and 33. 7, 8, 17, and 33 were solved due to knowing one position already because of unanimity. 9 was solved because both individuals support Z > X, so we know that Y > Z > X there. 10 was solved because both individuals support X > Y, so we know that the result must be X > Y > Z. In doing this, we also have a new rule: in 33, X is above and to the left of Y, and X > Y. Because of Independence of Irrelevant Alternatives, we can state that X > Y whenever X is above and to the left of Y, like in 33. This will allow us to fill in more profiles, which we will do now.

53. Step 2 Results

54. Step 2 Results Explanation We have 7 new completed profiles: 28, 30, 31, 32, 34, 35, 36. In addition, we have two new rules, as determined by profile 28. When X is above and to the left of Z, X > Z. Also, when Z is above and to the left of Y, Z > Y Let us see how many new solutions we can find knowing that: – Y above and to the left of Z --> Y > Z – X above and to the left of Y --> X > Y – X above and to the left of Z --> X > Z – Z above and to the left of Y --> Z > Y

55. Step 3 Results

56. Step 3 Results Explanation Now we have profiles 13, 14, 15, 16, 24, 25, 26, 27, and 29 filled in. And, once more, we have another rule, determined from 27: when Y is above and to the left of X, Y > X. This will lead to yet more results.

57. Step 4 Results

58. Step 4 Results Explanation Now profiles 11, 12, 18, 19, 22, and 23 are filled in. And, from profile 22, we can determine that when Z is above and to the left of X, Z > X. This results in:

59. Step 5 Results

60. Step 5 Results Explanation What happened? – In all 36 cases, person B (bottom row) matches up exactly with the social preferences. In other words, person B is our dictator. How did this happen? – We made one decision and used unanimity (in cases where X, Y, or Z was above and to the right) and independence of irrelevant alternatives (where X, Y, or Z was abo ve and to the left) to solve the rest.

61. What Does This Mean? Any complete social preference order which obeys both unanimity and independence of irrelevant alternatives will be dictatorial (as described on slide 8). A non-dictatorial social preference ranking would require dropping one of the three other desirable conditions. Normally, completeness and the independence of irrelevant alternatives are dropped.

62. Practical Results Cannot develop a rational social preference function like what we have for individual preferences No voting mechanism will simultaneously satisfy completeness, unanimity, independence of irrelevant alternatives, and non-dictatorship.

63. Arrow’s Impossibility Theorem So what? Is this profound or trivial? What implications does it have for Social Choice? Are his conditions too strong? What happens if we change/relax them? We shall see in the next lecture. And what do we do next? Give up?