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

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

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

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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

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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...

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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

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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 ABCD A06040100 B40070100 C60300100 D0000 ABCD A020-20100 B-20040100 C20-400100 D -100 0

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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 G R 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

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Relative advantage and optimality Relative advantage Adv D (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 Adv D (F,G) ≥ 0 F is optimal if it is at least as good as every other voting rule G wrt any distribution D Equivalently, F is optimal if it is at least as good as every other voting rule G on every profile R

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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>Dp(A) = 1/2 30 B>C>A>Dp(B) = 1/4 20 C>A>B>Dp(C) = 1/4 10 C>B>A>Dp(D) = 0 An optimal voting rule (GT) from game theory

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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: GTD Choose candidate with highest probability in optimal mixed strategy

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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)

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

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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)

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How does GT perform against other voting rules?

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How often is winner contained in GT support?

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GTD does slightly better than GT against other common rules

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Open problems Is it possible to modify Schulze so it always chooses a winner in GT support? Can one analytically determine Adv D (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 ?

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Thanks!

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