Outline for today Stat155 Game Theory Lecture 26: More Voting. Peter Bartlett December 1, 2016 1 / 31 2 / 31 Recall: Voting and Ranking Recall: Properties of ranking rules Assumptions There is a set Γ of candidates. Voter i has a preference relation i defined on candidates that is: 1 Complete: A B Γ, A i B or B i A. 2 Transitive: A, B, C Γ, A i B and B i C implies A i C. Definitions A voting rule f maps a preference profile π = ( 1,..., n ) to a winner from Γ. A ranking rule R maps a preference profile π = ( 1,..., n ) to a social ranking on Γ, which is another complete, transitive preference relation. Unanimity A ranking rule R has the unanimity property if, for all i, A i B, then = R( 1,..., n ) satisfies A B. If all voters prefer candidate A over B, candidate A should be ranked above B. 3 / 31 4 / 31
Recall: Properties of ranking rules Recall: Properties of ranking rules Strategically vulnerable A ranking rule R is strategically vulnerable if, for some preference profile ( 1,..., n ), some voter i and some candidates A, B Γ, := R( 1,..., i,..., n ), := R( 1,..., i,..., n ), A i B, B A, but A B. This means that Voter i has a preference relation i, but by stating an alternative preference relation i, it can swap the ranking rule s preference between A and B to make it consistent with i. Independence of irrelevant alternatives (IIA) Consider two different voter preference profiles ( 1,..., n ) and ( 1,..., n), and define = R( 1,..., n ) and = R( 1,..., n). For A, B Γ, if, for all i, A i B iff A i B, then A B iff A B. The ranking rule s relative rankings of candidates A and B should depend only on the voters relative rankings of these two candidates. 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 ( 1,..., n ), = R( 1,..., n ) has A B iff A i B. Arrow s Impossibility Theorem For Γ 3, any ranking rule R that satisfies both IIA and unanimity is a dictatorship. Thus, any ranking rule R that satisfies unanimity and is not strategically vulnerable is a dictatorship. Hence, strategic vulnerability is inevitable. OR and Economics, Stanford. Nobel Prize for Economics, 1972. Founder of modern social choice theory. (www.nationalmedals.org) 7 / 31 8 / 31
Outline Strategic vulnerability Recall: Ranking A ranking rule R is strategically vulnerable if, for some preference profile ( 1,..., n ), some voter i and some candidates A, B Γ, := R( 1,..., i,..., n ), := R( 1,..., i,..., n ), A i B, B A, but A B. Voting A voting rule f is strategically vulnerable if, for some preference profile ( 1,..., n ), some voter i and some candidates A, B Γ, π := ( 1,..., i,..., n ), π := ( 1,..., i,..., n ), A i B, B = f (π), but A = f (π ). Voter i, by incorrectly reporting preferences, can change the outcome to match his true preferences. 9 / 31 10 / 31 Dictatorship Another impossibility theorem Ranking A ranking rule R is a dictatorship if there is a voter i such that, for any preference profile ( 1,..., n ), = R( 1,..., n ) has A B iff A i B. Voter i determines the outcome. Voting A voting rule f is a dictatorship if there is a voter i such that, for any preference profile ( 1,..., n ), A = f ( 1,..., n ) iff for all B A, A i B. Recall: Arrow s Impossibility Theorem For Γ 3, any ranking rule R that satisfies unanimity and is not strategically vulnerable is a dictatorship. A voting rule f maps from the voters preference profile π to the winner Γ. We say f is onto Γ if, for all candidates A Γ, there is a π satisfying f (π) = A. If f is not onto Γ, some candidate is excluded from winning. Gibbard-Satterthwaite Theorem For Γ 3, any voting rule f that is onto Γ and is not strategically vulnerable is a dictatorship. 11 / 31 12 / 31
Gibbard-Satterthwaite theorem Gibbard-Satterthwaite theorem A B if f (π {A,B} ) = A, B A if f (π {A,B} ) = B. Proof By contradiction: Use f to construct a ranking rule that violates Arrow s Theorem. Suppose f is: onto Γ, not strategically vulnerable, not a dictatorship. Define = R(π) via A B if f (π {A,B} ) = A, B A if f (π {A,B} ) = B, where π S maintains the order of candidates in S but moves them above all other candidates in all voters preferences. Then we can check that: 1 If f is onto Γ and not strategically vulnerable, then for all S Γ, f (π S ) S so is complete. (Otherwise, in the path from a π f 1 (S) to π S, some voter switch would demonstrate a strategic vulnerability.) 2 Also, is transitive. (The same argument shows that f (π {A,B,C} ) = A implies A B and A C, so cycles are impossible.) 3 R satisfies unanimity (A i B implies π {A,B} = (π {A,B} ) {A}, so A B.) 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 Theorem For Γ 3, any voting rule f that is onto Γ and is not strategically vulnerable is a dictatorship. 15 / 31 16 / 31
Symmetry: Permuting voters does not affect the outcome. Monotonicity: Changing one voter s preferences by promoting candidate A without changing any other preferences should not change the outcome from A winning to A not winning. Condorcet winner criterion: If a candidate is majority-preferred in pairwise comparisons with any other candidate, then that candidate wins. Condorcet loser criterion: If a candidate is preferred by a minority of voters in pairwise comparisons with all other candidates, then that candidate should not win. 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). Reversal symmetry: If A wins for some voter preference profile, A does not win when the preferences of all voters are reversed. Cancellation of ranking cycles: If a set of Γ voters have preferences that are cyclic shifts of each other (e.g., A 1 B 1 C, B 2 C 2 A, C 3 A 3 B), then removing these voters does not affect the outcome. Cancellation of opposing rankings: If two voters have reversed preferences, then removing these voters does not affect the outcome. Consistency: If A wins for voter preference profiles π and π, A also wins when these voter preference profiles are combined. Participation: If A wins for some voter preference profile, then adding a voter with A B does not change the winner from A to B. Outline 17 / 31 18 / 31 Instant runoff voting Voters provide a ranking of the candidates. If only one candidate remains, return that candidate. Otherwise: 1 Eliminate the candidate that is top-ranked by the fewest voters. 2 Drop that candidate s preferences from voters rankings. 3 Use instant runoff voting on the remaining candidates with the reassigned preferences. Symmetry? (Permuting voters does not affect the outcome.) 19 / 31 20 / 31
Monotonicity? (Changing one voter s preferences by promoting candidate A without changing any other preferences does not change the outcome from A winning to A not winning.) 45% B C A 25% C A B C eliminated in round 1. 35% B C A 10% C B A 25% C A B A eliminated in round 1. B wins. When 10% of voters move B above C, it changes the outcome from B winning to A winning. Condorcet winner criterion? (If a candidate is majority-preferred in pairwise comparisons with any other candidate, then that candidate wins.) 45% C B A 25% B A C B eliminated in round 1. B is preferred over any other candidate. but 21 / 31 22 / 31 Condorcet loser criterion? (If a candidate is preferred by a minority of voters in pairwise comparisons with all other candidates, then that candidate should not win.) 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.) 45% C B A 25% B A C B eliminated in round 1. 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
Reversal symmetry? (If A wins for some voter preference profile, A does not win when the preferences of all voters are reversed. 45% B C A 25% C A B 30% C B A 45% A C B 25% B A C C eliminated in round 1. B eliminated in round 1. When the preferences of all voters are reversed, A still wins. Cancellation of ranking cycles? Cancellation of opposing rankings? Consistency? Participation? Outline 25 / 31 26 / 31 Borda count Voters rank candidates from 1 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. Jean-Charles de Borda 1733-1799. French naval commander, scientist, inventor: ballistics, mapping and surveying instruments, pumps, metric trigonometric tables. (wikipedia.org) 27 / 31 28 / 31
Positional voting rules Define a 1 a 2 a N. For each candidate, assign a i points for each voter that assigns that candidate rank i. The candidate with the largest total wins. e.g., Borda count: N, N 1,..., 1. e.g., Plurality: 1, 0,..., 0. e.g., Approval voting: 1, 1,..., 1, 0,..., 0. Properties of positional voting rules Symmetry? Monotonicity? Condorcet winner criterion? Cancellation of ranking cycles? 29 / 31 30 / 31 Outline 31 / 31