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

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

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...

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

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 A B C D0000 ABCD A B C D

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

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

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

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

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)

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

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)

How does GT perform against other voting rules?

How often is winner contained in GT support?

GTD does slightly better than GT against other common rules

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 ?

Thanks!