Outline for today

Stat155 Game Theory Lecture 26: More Voting. Voting systems. Recall: voting and ranking rules, Arrow’s impossibility theorem Voting rules: Gibbard-Satterthwaite Theorem Properties of voting rules Peter Bartlett Properties of instant runoff voting , rules

December 1, 2016

1 / 31 2 / 31 Recall: Voting and Ranking Recall: Properties of ranking rules

Assumptions There is a setΓ of candidates. Voter i has a preference relation “ ” defined on candidates that is: i 1 Complete: A = B Γ, A B or B A. Unanimity ∀ 6 ∈ i i 2 Transitive: A, B, C Γ, A B and B C implies A C. ∀ ∈ i i i A ranking rule R has the If all voters prefer candidate unanimity property if, for all i, A over B, candidate A Definitions A B, then = R( ,..., ) should be ranked above B. i 1 n A voting rule f maps a preference profile π = ( ,..., ) to a satisfies A B. 1 n winner fromΓ.  A ranking rule R maps a preference profile π = ( ,..., ) to a 1 n social ranking “ ” onΓ, which is another complete, transitive preference relation.

3 / 31 4 / 31 Recall: Properties of ranking rules Recall: Properties of ranking rules

Strategically vulnerable Independence of irrelevant A ranking rule R is strategically This means that Voter i has alternatives (IIA) vulnerable if, for some preference a preference relation i , but Consider two different voter The ranking rule’s relative profile( 1,..., n), some voter by stating an alternative preference profiles( 1,..., n) rankings of candidates A and i and some candidates A, B Γ, preference relation i0, it can and( 0 ,..., 0 ), and define B should depend only on the ∈ swap the ranking rule’s 1 n = R( 1,..., n) and voters’ relative rankings of := R( 1,..., i ,..., n), preference between A and B 0= R( 10 ,..., n0 ). For these two candidates. 0 := R( 1,..., 0,..., n), to make it consistent with i A, B Γ, if, for all i, A i B iff  i . ∈ A i0 B, then A B iff A 0 B. A i B, B A, but A 0 B.    

5 / 31 6 / 31 Recall: Properties of ranking rules Kenneth Arrow

Theorem Any ranking rule R that violates IIA is strategically vulnerable.

Definition

A ranking rule R is a dictatorship if there is a voter i ∗ such that, for any preference profile( ,..., ), = R( ,..., ) has A B iff A B. 1 n 1 n i ∗   Arrow’s Impossibility Theorem For Γ 3, any ranking rule R that satisfies both IIA and unanimity is a | | ≥ (www.nationalmedals.org) dictatorship. Thus, any ranking rule R that satisfies unanimity and is not strategically OR and Economics, Stanford. vulnerable is a dictatorship. Nobel Prize for Economics, 1972. Hence, strategic vulnerability is inevitable. Founder of modern social choice theory.

7 / 31 8 / 31 Outline Strategic vulnerability

Recall: Ranking Voting A ranking rule R is strategically A voting rule f is strategically vulnerable if, for some preference vulnerable if, for some preference Voting systems. profile( ,..., ), some voter profile( 1,..., n), some voter i Recall: voting and ranking rules, Arrow’s impossibility theorem 1 n i and some candidates A, B Γ, and some candidates A, B Γ, Voting rules: Gibbard-Satterthwaite Theorem ∈ ∈ Properties of voting rules := R( ,..., ,..., ), π := ( 1,..., i ,..., n), Properties of instant runoff voting 1 i n Borda count, positional voting rules 0 := R( ,..., 0,..., ), π0 := ( 1,..., 0,..., n), 1 i n i  A B, B A, but A B. A i B, B = f (π), but A = f (π0). i 0   Voter i, by incorrectly reporting preferences, can change the outcome to match his true preferences.

9 / 31 10 / 31 Dictatorship Another impossibility theorem

Recall: Arrow’s Impossibility Theorem For Γ 3, any ranking rule R that satisfies unanimity and is not | | ≥ strategically vulnerable is a dictatorship. Ranking Voting A ranking rule R is a dictatorship A voting rule f is a dictatorship if A voting rule f maps from the voters’ preference profile π to the if there is a voter i such that, there is a voter i such that, for ∗ ∗ winnerΓ. for any preference profile any preference profile We say f is onto Γ if, for all candidates A Γ, there is a π satisfying ( 1,..., n), = R( 1,..., n) ( 1,..., n), A = f ( 1,..., n) ∈ f (π) = A. has A B iff A i B. iff for all B = A, A i B. ∗ 6 ∗  If f is not ontoΓ, some candidate is excluded from winning. Voter i ∗ determines the outcome. Gibbard-Satterthwaite Theorem For Γ 3, any voting rule f that is ontoΓ and is not strategically | | ≥ vulnerable is a dictatorship.

11 / 31 12 / 31 Gibbard-Satterthwaite theorem Gibbard-Satterthwaite theorem

A,B A,B A B if f (π{ }) = A, B A if f (π{ }) = B. Proof   By contradiction: Then we can check that: Use f to construct a ranking rule that violates Arrow’s Theorem. 1 If f is ontoΓ and not strategically vulnerable, then for all S Γ, S ⊆ Suppose f is: ontoΓ, not strategically vulnerable, not a dictatorship. f (π ) S so is complete. ∈ 1 S (Otherwise, in the path from a π0 f − (S) to π , some voter switch Define = R(π) via ∈ would demonstrate a strategic vulnerability.)  A,B 2 A B if f (π{ }) = A, Also, is transitive. (The same argument shows that A,B,C A,B f (π{  }) = A implies A B and A C, so cycles are impossible.) B A if f (π{ }) = B, 3 R satisfies unanimity    S (A B implies π A,B = (π A,B ) A , so A B.) where π maintains the order of candidates in S but moves them i { } { } { } above all other candidates in all voters’ preferences. 4 R satisfies IIA. (Similar argument.)  5 Because f is not a dictatorship, R is not a dictatorship. 6 But Arrow’s Theorem shows that this R cannot exist.

13 / 31 14 / 31 Gibbard-Satterthwaite theorem Outline

Voting systems. Theorem Recall: voting and ranking rules, Arrow’s impossibility theorem For Γ 3, any voting rule f that is ontoΓ and is not strategically Voting rules: Gibbard-Satterthwaite Theorem | | ≥ vulnerable is a dictatorship. Properties of voting rules Properties of instant runoff voting Borda count, positional voting rules

15 / 31 16 / 31 Properties of voting systems Properties of voting systems

Symmetry: Permuting voters does not affect the outcome. Reversal symmetry: If A wins for some voter preference profile, A Monotonicity: Changing one voter’s preferences by promoting does not win when the preferences of all voters are reversed. candidate A without changing any other preferences should not Cancellation of ranking cycles: If a set of Γ voters have change the outcome from A winning to A not winning. | | preferences that are cyclic shifts of each other (e.g., A B C, 1 1 Condorcet winner criterion: If a candidate is majority-preferred in B C A, C A B), then removing these voters does not 2 2 3 3 pairwise comparisons with any other candidate, then that candidate affect the outcome. wins. Cancellation of opposing rankings: If two voters have reversed Condorcet loser criterion: If a candidate is preferred by a minority preferences, then removing these voters does not affect the outcome. of voters in pairwise comparisons with all other candidates, then that Consistency: If A wins for voter preference profiles π and π0, A also candidate should not win. wins when these voter preference profiles are combined. The winner always comes from the Smith set (the Participation: If A wins for some voter preference profile, then smallest nonempty set of candidates that are majority-preferred in adding a voter with A B does not change the winner from A to B. pairwise comparisons with any candidate outside the set).

17 / 31 18 / 31 Outline Properties of voting systems

Instant runoff voting Voters provide a ranking of the candidates. Voting systems. If only one candidate remains, return that candidate. Recall: voting and ranking rules, Arrow’s impossibility theorem Otherwise: Voting rules: Gibbard-Satterthwaite Theorem 1 Eliminate the candidate that is top-ranked by the fewest voters. Properties of voting rules 2 Drop that candidate’s preferences from voters’ rankings. Properties of instant runoff voting 3 Use instant runoff voting on the remaining candidates with the Borda count, positional voting rules reassigned preferences.

Properties of instant runoff voting Symmetry? (Permuting voters does not affect the outcome.)

19 / 31 20 / 31 Properties of voting systems Properties of voting systems

Properties of instant runoff voting Properties of instant runoff voting Monotonicity? (Changing one voter’s preferences by promoting candidate A without changing any other preferences does not change Condorcet winner criterion? (If a candidate is majority-preferred in the outcome from A winning to A not winning.) pairwise comparisons with any other candidate, then that candidate wins.)

1st 2nd 3rd 1st 2nd 3rd 1st 2nd 3rd 30% ABC 30% ABC 30% ABC 35% BCA 45% BCA 45% CBA 10% CBA 25% CAB 25% BAC 25% CAB C eliminated in round 1. B eliminated in round 1. A eliminated in round 1. A wins. A wins. B wins. When 10% of voters move B above C, it changes the outcome from B B is preferred over any other candidate. but A wins. winning to A winning.

21 / 31 22 / 31 Properties of voting systems Properties of voting systems

Properties of instant runoff voting Smith criterion? (The winner always comes from the Smith set—the smallest nonempty set of candidates that are majority-preferred in pairwise comparisons with any candidate outside the set.) Properties of instant runoff voting Condorcet loser criterion? (If a candidate is preferred by a minority 1st 2nd 3rd of voters in pairwise comparisons with all other candidates, then that 30% ABC candidate should not win.) 45% CBA 25% BAC B eliminated in round 1. A wins. The Smith set is B . { } (Notice that a rule that violates the Condorcet winner criterion violates the Smith criterion for some preference profile with a singleton Smith set.)

23 / 31 24 / 31 Properties of voting systems Properties of voting systems

Properties of instant runoff voting Reversal symmetry? (If A wins for some voter preference profile, A does not win when the preferences of all voters are reversed. Properties of instant runoff voting Cancellation of ranking cycles? 1st 2nd 3rd 1st 2nd 3rd Cancellation of opposing rankings? 30% ABC 30% CBA Consistency? 45% BCA 45% ACB Participation? 25% CAB 25% BAC C eliminated in round 1. B eliminated in round 1. A wins. A wins. When the preferences of all voters are reversed, A still wins.

25 / 31 26 / 31 Outline Properties of voting systems

Borda count Voters rank candidates from1 to N (where N = Γ ). | | A candidate that is ranked in ith position is assigned N i + 1 points. − The candidate with the largest total wins. Voting systems. Recall: voting and ranking rules, Arrow’s impossibility theorem Voting rules: Gibbard-Satterthwaite Theorem Jean-Charles de Borda Properties of voting rules Properties of instant runoff voting Borda count, positional voting rules 1733-1799. French naval commander, scientist, inventor: ballistics, mapping and surveying instruments, pumps, metric trigonometric tables.

(wikipedia.org)

27 / 31 28 / 31 Properties of voting systems Properties of voting systems

Positional voting rules Define a a a . Properties of positional voting rules 1 ≥ 2 ≥ · · · ≥ N For each candidate, assign ai points for each voter that assigns that Symmetry? candidate rank i. Monotonicity? The candidate with the largest total wins. Condorcet winner criterion? e.g., Borda count: N, N 1,..., 1. Cancellation of ranking cycles? − e.g., Plurality:1 , 0,..., 0. e.g., :1 , 1,..., 1, 0,..., 0.

29 / 31 30 / 31 Outline

Voting systems. Recall: voting and ranking rules, Arrow’s impossibility theorem Voting rules: Gibbard-Satterthwaite Theorem Properties of voting rules Properties of instant runoff voting Borda count, positional voting rules

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