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How “impossible” is it to design a Voting rule? Angelina Vidali University of Athens

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individuals (voters) alternatives N = f 1 ;:::; n g A = f a 1 ;:::; a m g The setting u i ( a ) > u i ( b ) Voter i prefers a to b voter i u i u = ( u 1 ;:::; u n ) a preference profile abcabc

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Voting rule vs Social welfare function A Voting rule is a mapping it chooses the “winning alternative” f : U ! A set of all (strict) preference profiles u set of alternatives A Social welfare functionis a mapping it chooses the “social ranking ” set of all (weak) preference profiles u Gibbard-Satterthwaite ‘70s Arrow ‘50s f : U ! U ¤

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a beats b in a pairwise election if the majority of voters prefers a to b a is a Condorcet winner if a beats any other candidate in a pairwise election 4 Condorcet rule

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c b a a c b b a c 5 a a a b b b c c c a a c c b b the Condorcet paradox voter 1 voter 2 voter 3

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Some voting rules Plurality vote: a single winner is chosen by having more votes than any other individual representative. Borda vote: The Kth-ranked alternative gets a score of N-K. All scores are summed and the candidate with the highest total score wins. Instant Runoff Voting: If no candidate receives a majority of first preference rankings, the candidate with the fewest number of votes is eliminated and that candidate's votes redistributed to the voters' next preferences among the remaining candidates. This process is repeated until one candidate has a majority of votes.

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A manipulable voting rule If individual i reports a false preference profile instead of his true preference the outcome of the elections is better for him. voter i KKE ΠΑΣΟΚ ΝΔ If I vote ΚΚΕ the outcome f(u) will be either ΝΔ or ΠΑΣΟΚ if I vote ΠΑΣΟΚ the outcome will be ΠΑΣΟΚ. It is better for me to report ΠΑΣΟΚ. non-manipulable=strategyproof u i ( f ( u 0 i ; u ¡ i )) > u i ( f ( u ))

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The voter who lies determines the winner in a tie! Ties is the most tricky part… voter 1 KKE ΠΑΣΟΚ ΝΔ voter 2 KKE ΠΑΣΟΚ ΝΔ KKE ΠΑΣΟΚ ΝΔ KKE ΠΑΣΟΚ ΝΔ voter 3voter 4 KKE ΠΑΣΟΚ ΝΔ voter 5

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A dictator The dictators top alternative is the outcome of the elections. u i ( f ( u )) ¸ u i ( a ) f ora ll a 2 A an d u 2 U

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Neutral The names of the candidates don’t matter. i.e. f commutes with permutations of [m] Example: b is the winner if all voters a b in their preference profiles then a is the winner. f ( ¼ ( u )) = ¼ ( f ( u ))

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Monotonicity “If our preference for the government increases it is reelected” Let f be a strategyproof voting rule, f(u)=a. As long as, for all voters, the alternatives that were worse than a in u, remain worse in v the allocation remains the same. …even if one of the black elements moves below a as long as the red elements stay below a the outcome remains a u = 0 B B B B B @ a 1 ¢¢¢ a 2... a a 3 a 4 1 C C C C C A 0 B B B B B @ a 1 ¢¢¢ a... a 3 a 2 a 4 1 C C C C C A

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Pareto Optimality “If everybody prefers a to b then b is not elected.” Pareto Optimality follows from Monotonicity u = 0 B B B B B B @ a......... a... a... a b b... a b...... bb...... 1 C C C C C C A

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Gibbard (‘73)-Satterthwaite (‘75) theorem If the number of alternatives then a voting rule that is strategyproof and onto is dictatorial. n ¸ 3 follows from Arrow’s impossibility theorem (1951) using the correspondence between: Independence of Irrelevant Alternatives and strategyproofness

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Independence of Irrelevant Alternatives The social relative ranking of two alternatives a, b depends only on their relative ranking. If one candidate dies the choice to be made among the set S of surviving candidates should be independent of the the preferences of individuals for candidates not in S. [See: Social Choice and Individual Values, K.J. Arrow p.26]

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a voting method that violates IIA The alternative with the highest weighted sum of votes wins. voter 1voter 2 cdcd voter 3 cdacda weight 4 3 2 1 At first a is chosen a:4+4+2=10 b:7 c:8 d:6 but if b leaves: a:10 c:10 d:7 we get a tie between a and c b b a cdcd b a see: www.scorevoting.netwww.scorevoting.net

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The proof of G-S theorem for n=2 voters So a wins in both u and v. voter 1 becomes a dictator for a. c cannot win (Pareto Optimality) Assume a wins (w.l.o.g.) Then b wins (monotonicity) cdc u = 0 @ a b b a cc 1 A v = 0 @ a b b c c a 1 A c cannot win (Pareto Optimal.) Suppose b wins

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The proof of G-S theorem for n=2 voters (2) Repeat for every pair of alternatives {a,b} A 1 ={x| player 1 is a dictator for x} A 2 ={y| player 2 is a dictator for y} because: if it had two distinct elements then one of them should belong to A 1 or A 2. Finally some A i = all the element belong to A j and j is the dictator. j A n( A 1 \ A 2 )j · 1 ;

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Towards a Quantitative version Gibbard-Satherwaite theorem: “Every non-trivial (=non dictatorial) voting rule is strategically vulnerable.” How often? For what fraction of profiles does such a manipulation exist?

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Impartial culture assumption Voters vote independently and randomly We draw independently and uniformly a random ranking for each voter possible rankings for voter i: m! P(each ranking)= 1/m!

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Manipulation power of a voter: M i (f) The manipulation power M i ( f ), of voter i on the social choice function f, is the probability that : if voter i reports a chosen uniformly at random this is a profitable manipulation of f for voter i. u 0 i $i$$i$ What is the probability I can gain something by just drawing one of the m alternatives randomly and reporting this instead of my true preference? individual i

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ε-strategyproof P [ f ( u 0 i ; u ¡ i ) > f ( u )] · ² The manipulation power M i ( f ), of voter i on the social choice function f, satisfies M i ( f ) < ² =

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δ-far from dictarorship The distance between two functions f,g is f is δ-far from dictatorship, if for any dictatorship g ¢ ( f ; g ) = P u 2 U [ f ( u ) 6 = g ( u )] ¢ ( f ; g ) > ±

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Quantitative version of G-S theorem [Friedgut-Kalai-Nisan FOCS’08] For every >0 if f is a voting rule for n voters is neutral among 3 alternatives -far from dictatorship, then one of the voters has a non-negligible manipulation power of. ( 1 n ) ²²

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Quantitative version of G-S theorem [Xia-Conitzer EC’08] - a list of assumptions… homogeneity anonymity non-imposition a canceling out condition there exists a stable profile that is still stable after one given alternative is uniformly moved to different positions + (they argue that many known voting rules satisfy them) + for arbitrarily many alternatives and players

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Quantitative version of G-S theorem 2 voters [Dobzinski-Proccacia WINE’08] For if a voting rule f for 2 voters is Pareto optimal (annoying condition!) among at least 3 alternatives with manipulation power < then f is -far from dictatorship. ² ² < 1 32 m 9 16 m 8 ² no neutrality assumption here! m can also be greater than 3

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Some open problems Quantitative version of G-S theorem for more than 2 voters (with less conditions!) What about the impartial culture assumption? is it plausible? Find quantitative versions of known mechanism design results: straptegyproof ε-strategyproof

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Endnote "Most systems are not going to work badly all of the time, all I proved is that all can work badly at times." K. J. Arrow …or do they work badly most of the time???

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MAT 105 Spring 2008. There are many more methods for determining the winner of an election with more than two candidates We will only discuss a few.

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