1. How Should Presidents Be Elected? E. Maskin Institute for Advanced Study

2. 2 Voting rule method for choosing candidate on basis of –ballot (set of available candidates) –voters’ preferences (rankings) over candidates prominent examples –plurality rule (MPs in Britain, members of Congress in U.S.) choose candidate ranked first by more voters than any other –majority rule (Condorcet Method) choose candidate preferred by majority to each other candidate

3. 3 −run-off voting (presidential elections in France) choose candidate ranked first by more voters than any other, unless gets less than majority among top 2 candidates, choose one preferred by majority −rank-order voting (Borda Count) candidate assigned 1 point every time voter ranks her first, 2 points every time ranked second, etc. choose candidate with lowest vote total −utilitarianism choose candidate that maximizes sum of voters’ utilities

4. 4 Which voting rule to adopt? Voting theorist’s answer: –specify what one wants (axioms) –see which rules satisfy the axioms But basic negative results: Arrow Impossibility Theorem Gibbard-Satterthwaite Theorem –if 3 or more candidates, no voting rule satisfies set of compelling axioms

5. 5 Prompts question: Which voting rule(s) satisfy axioms most often? try to answer question today –based on series of papers with P. Dasgupta

6. 6 X = finite set of conceivable candidates society = continuum of voters [0,1] – – reason for continuum clear soon utility function for voter i –restrict attention to strict utility functions profile

7. 7 voting rule F − winning candidates (could be ties) Example – plurality rule

8. 8 Axioms –that is, there is clear-cut winner but that is too much to ask for e.g., with plurality rule –exact ties are possible might be 2 candidates who are both ranked first the most still, with large number of voters, tie very unlikely similarly with majority rule, rank-order voting, etc. continuum helps formalize this

9. 9 –fraction of electorate who prefer x to y –profile is regular if proportions of voters preferring one candidate to another fall outside exceptional set: Generic Decisiveness (GD): There exists finite S such that

10. 10 if everybody prefers x to y, then y should not win when x is on the ballot winner should not depend on which voter has which preferences –names don’t matter –all voters treated symmetrically all candidates treated symmetrically

11. 11 only preference rankings matter –strength of preference not captured rules out utilitarianism reflects standard idea that preference strength has no operational meaning if only one “good” (no choice experiment) implied by later axiom could be replaced by cardinality – (as in von Neumann-Morgenstern preferences) what’s important: no interpersonal comparisons –such comparisons not operational

12. 12 next axiom most controversial but –has compelling rationale –invoked by Arrow (1951) and Nash (1950) then –if x elected and then some non-elected candidate removed from ballot, x still elected –no “spoilers” (e.g. Nader in 2000 U.S. presidential election, Le Pen in 2002 French presidential election) –Nash formulation

13. 13 plurality rule violates I so do rank-order voting and run-off voting majority rule satisfies I but violates GD

14. 14 Theorem 1: No voting rule satisfies GD, P, A, N, O, and I –analog of Arrow Impossibility Theorem But natural follow-up question: Which voting rule (s) satisfies axioms as often as possible? –have to clarify “as often as possible”

15. 15 F – – –

16. 16 i.e.,

17. 17 Let S be exceptional for F. Because S finite, there exists n such that −if divide population into n equal groups −assign everybody in a group same utility function −resulting profile regular w.r.t. S.

18. 18 Consider – – then

19. 19 – Continuing iteratively

20. 20 Conversely, suppose F does not

21. 21 Consider – –

22. 22 Final Axiom: Nonmanipulability (NM): then –the members of coalition C can’t all gain from misrepresenting

23. 23 Lemma: If F satisfies NM and I, F satisfies O – – – – – NM rules out utilitarianism

24. 24 But majority rule also violates NM – –one possibility – –

25. 25 Theorem 3: There exists no voting rule satisfying GD, P,A,N,I and NM Proof: similar to that of Gibbard-Satterthwaite Theorem

26. 26 –

27. 27 Conversely, suppose Proof: From NM and I, if F works very well on U, F must be ordinal Hence result follows from –Theorem 2 –

28. 28 Let’s drop I –most controversial –GS again F works nicely on U if satisfies GD,P,A,N,NM on U

29. 29 Theorem 5: Suppose F works nicely on U, Proof: – – –

30. 30 Striking that the 2 longest-studied voting rules (Condorcet and Borda) are also only two that work nicely on maximal domains