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An Optimal Preferential Voting System Based on Game Theory

An Optimal Preferential Voting System Based on Game Theory. Ronald L. Rivest Emily Shen MIT CSAIL COMSOC 2010 September 16, 2010. Voting rules. Vote: ranking of candidates Voting rule: given a profile of votes, output single winner Question: What makes a good voting rule?. Outline.

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An Optimal Preferential Voting System Based on Game Theory

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  1. An Optimal Preferential Voting System Based on Game Theory Ronald L. Rivest Emily Shen MIT CSAIL COMSOC 2010 September 16, 2010

  2. Voting rules • Vote: ranking of candidates • Voting rule: given a profile of votes, output single winner • Question: What makes a good voting rule?

  3. Outline • Quantitative approach • Optimal voting rule (GT) from game theory • Relation to prior work • Simulation results • Open problems

  4. Traditional approach: axiomatic Long list of well-studied properties: Condorcet Majority Consistency Participation Monotonicity … Impossible to satisfy all these criteria Axiomatic approach can give inconclusive advice, depending on perceived relative importance of various criteria

  5. Our approach: quantitative • “A voting rule F is better than G if voters tend to prefer the outcome of F to the outcome of G” • Ultimately, a voting rule aims to satisfy voters’ preferences • We will make this precise...

  6. Quantitative approach • We define a metric to compare any two voting rules • Using this metric, we define a notion of optimality (achievable using game theory) • Similar to decision-theoretic approach of [Lu, Boutilier ’10] • GT is closely related to prior work – more on this later

  7. Notation Profile R = collection of voters’rankings of candidates Preference matrix N:N(x,y) = # voters preferring x to y Margin matrix M:M(x,y) = N(x,y) – N(y,x) 40 A>B>C>D 30 B>C>A>D 20 C>A>B>D 10 C>B>A>D

  8. Game between two voting rules “A voting rule F is better than G if voters tend to prefer the outcome of F to the outcome of G” D = distribution on profiles Choose a profile R from D Play a game GR between F and G: F and G choose their outcomes x = F(R), y = G(R) F wins N(x,y) points, G wins N(y,x) points Net: F wins M(x,y) points, G wins M(y,x) points

  9. Relative advantage and optimality Relative advantage AdvD(F,G) = E[M(x,y) / (# voters)] = expected value of average net fraction of voters won by F over G F is at least as good as G (wrt D) if AdvD(F,G) ≥ 0 F is optimal if it is at least as good as every other voting rule G wrt any distributionD Equivalently, F is optimal if it is at least as good as every other voting rule G on every profile R

  10. An optimal voting rule (GT)from game theory • Use zero-sum two-player game theory to define an optimal voting rule • Payoffs = margins • Compute optimal mixed strategy • Choose winner according to optimal mixed strategy 40 A>B>C>D p(A) = 1/2 30 B>C>A>D p(B) = 1/4 20 C>A>B>D p(C) = 1/4 10 C>B>A>D p(D) = 0

  11. An optimal voting rule must be randomized • When there is a Condorcet winner, we don’t need randomization • If there is a Condorcet cycle, any deterministic rule is sub-optimal • Condorcet cycle can be thought of as “generalized tie” • Cave Creek, AZ ’09: broke tie for city council seat by drawing cards from a shuffled deck • Deterministic variant: GTDChoose candidate with highest probability in optimal mixed strategy

  12. Related work • GT is a special case of “maximal lottery methods” (Fishburn ’84) • GT support ~ bipartisan set for tournament game (LLL ’93) • GT game corresponds to plurality game (LLL ’94)

  13. Properties of GT • Optimality • Condorcet winner/loser • Pareto efficiency • Independence of clones* • Reversal symmetry* • No monotonicity

  14. Simulated election comparisons • Randomly generated 10,000 profiles for 5 candidates, 100 voters, full ballots • “Spherical” distribution • 64% had Condorcet winner • 77% had unique optimal mixed strategy • (Also tried impartial culture distribution)

  15. How does GT perform against other voting rules?

  16. How often is winner contained in GT support?

  17. GTD does slightly better than GT against other common rules

  18. Open problems • Is it possible to modify Schulze so it always chooses a winner in GT support? • Can one analytically determine AdvD(F,G) for some F,G,D? • How sensitive are GT probabilities to input votes?How hard is it to manipulate GT? • Can one lower bound the penalty paid for being deterministic, monotonic, etc. for some D?

  19. Thanks!

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