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Game Theory Social 8 1 Social Choice Theory Game Theory Social 8. Social Choice Theory Choice Theory Introduction Introduction Social Choice Functions Condorcet Social Choice Functions Methods Condorcet Methods Arrow’s Impossibility Theorem Albert-Ludwigs-Universität Freiburg Gibbard- Satterthwaite Theorem Some Positive Results Bernhard Nebel and Robert Mattmüller Summary June 14th, 2016 June 14th, 2016 B. Nebel, R. Mattmüller – Game Theory 3 / 62 Social Choice Theory Social Choice Theory Definition (Social Welfare and Social Choice Function) Social Social Choice Let A be a set of alternatives (candidates) and L be the set of Choice Motivation: Aggregation of individual preferences Theory Theory Introduction all linear orders on A. For n voters, a function Introduction Social Choice Social Choice Examples: Functions Functions Condorcet n Condorcet political elections Methods F : L ! L Methods Arrow’s Arrow’s council decisions Impossibility Impossibility Theorem is called a social welfare function. A function Theorem Eurovision Song Contest Gibbard- Gibbard- Satterthwaite n Satterthwaite Theorem f : L ! A Theorem Question: If voters’ preferences are private, then how to Some Some implement aggregation rules such that voters vote truthfully Positive is called a social choice function. Positive Results Results (no “strategic voting”)? Summary Summary Notation: Linear orders ≺ 2 L express preference relations. a ≺i b : voter i prefers candidate b over candidate a. a ≺ b : candidate b socially preferred over candidate a. June 14th, 2016 B. Nebel, R. Mattmüller – Game Theory 4 / 62 June 14th, 2016 B. Nebel, R. Mattmüller – Game Theory 5 / 62 Social Choice Functions Social Choice Functions Examples Examples Instant runoff voting: Plurality voting (aka first-past-the-post or Social Social each voter submits his preference order winner-takes-all): Choice Choice Theory iteratively candidates with fewest top preferences are Theory only top preferences taken into account Introduction Introduction Social Choice eliminated until one candidate has absolute majority Social Choice Functions Functions candidate with most top preferences wins Condorcet Condorcet Methods Drawback: Tactical voting still possible. Methods Drawback: Wasted votes, compromising, winner only Arrow’s Arrow’s preferred by minority Impossibility Borda count: Impossibility Theorem Theorem each voter submits his preference order over the m Gibbard- Gibbard- Plurality voting with runoff: Satterthwaite candidates Satterthwaite First round: two candidates with most top votes proceed to Theorem if a candidate is in position j of a voter’s list, he gets m − j Theorem Some Some second round (unless absolute majority) Positive points from that voter Positive Second round: runoff Results points from all voters are added Results Drawback: still, tactical voting and strategic nomination Summary candidate with most points wins Summary possible. Drawback: Tactical voting still possible (“Voting opponent down”). June 14th, 2016 B. Nebel, R. Mattmüller – Game Theory 6 / 62 June 14th, 2016 B. Nebel, R. Mattmüller – Game Theory 7 / 62 Social Choice Functions Social Choice Functions Examples Examples 23 voters, candidates a, b, c, d, e. Social Social Choice Choice Theory # voters 8 6 4 3 1 1 Theory Introduction Introduction Social Choice 1st e a b c d d Social Choice Condorcet winner: Functions Functions Condorcet 2nd d b c b c c Condorcet Methods Methods each voter submits his preference order 3rd b c d d a b perform pairwise comparisons between candidates Arrow’s Arrow’s Impossibility 4th c e a a b e Impossibility if one candidate wins all his pairwise comparisons, he is Theorem Theorem the Condorcet winner Gibbard- 5th a d e e e a Gibbard- Satterthwaite Satterthwaite Drawback: Condorcet winner does not always exist. Theorem Theorem Some Plurality voting: candidatee wins (8 votes) Some Positive Positive Results Plurality voting with runoff: Results Summary first round: candidates e (8 votes) and a (6 votes) proceed Summary second round: candidatea (6 + 4 + 3 + 1 = 14 votes) beats candidate e (8 + 1 = 9 votes) June 14th, 2016 B. Nebel, R. Mattmüller – Game Theory 8 / 62 June 14th, 2016 B. Nebel, R. Mattmüller – Game Theory 9 / 62 Social Choice Functions Social Choice Functions Examples Examples 23 voters, candidates a, b, c, d, e. 23 voters, candidates a, b, c, d, e. Social Social Choice # voters 8 6 4 3 1 1 Choice # voters 8 6 4 3 1 1 Theory Theory Introduction 1st e a b c d d 4 points Introduction 1st e a b c d d Social Choice Social Choice Functions 2nd d b c b c c 3 points Functions 2nd d b c b c c Condorcet Condorcet Methods 3rd b c d d a b 2 points Methods 3rd b c d d a b Arrow’s 4th c e a a b e 1 point Arrow’s 4th c e a a b e Impossibility Impossibility Theorem 5th a d e e e a 0 points Theorem 5th a d e e e a Gibbard- Gibbard- Satterthwaite Satterthwaite Theorem Borda count: Theorem Instant runoff voting: Some Cand. a: 8 · 0 + 6 · 4 + 4 · 1 + 3 · 1 + 1 · 2 + 1 · 0 = 33 pts Some Positive Positive First elimination: d Results Cand. b: 8 · 2 + 6 · 3 + 4 · 4 + 3 · 3 + 1 · 1 + 1 · 2 = 62 pts Results Second elimination: b Summary Cand. c: 8 · 1 + 6 · 2 + 4 · 3 + 3 · 4 + 1 · 3 + 1 · 3 = 50 pts Summary Third elimination: a Cand. d: 8 · 3 + 6 · 0 + 4 · 2 + 3 · 2 + 1 · 4 + 1 · 4 = 46 pts · · · · · · Nowc has absolute majority and wins. Cand. e: 8 4 + 6 1 + 4 0 + 3 0 + 1 0 + 1 1 = 39 pts Candidateb wins. June 14th, 2016 B. Nebel, R. Mattmüller – Game Theory 10 / 62 June 14th, 2016 B. Nebel, R. Mattmüller – Game Theory 11 / 62 Social Choice Functions Social Choice Functions Examples Examples 23 voters, candidates a, b, c, d, e. 23 voters, candidates a, b, c, d, e. Social Social # voters 8 6 4 3 1 1 Choice # voters 8 6 4 3 1 1 Choice 1st e a b c d d Theory 1st e a b c d d Theory Introduction Introduction Social Choice Social Choice 2nd d b c b c c Functions 2nd d b c b c c Functions Condorcet Condorcet 3rd b c d d a b Methods 3rd b c d d a b Methods 4th c e a a b e Arrow’s 4th c e a a b e Arrow’s Impossibility Impossibility 5th a d e e e a Theorem 5th a d e e e a Theorem Gibbard- Gibbard- Condorcet winner: Ex.: a ≺i b 16 times, b ≺i a 7 times Satterthwaite Plurality voting: candidatee wins. Satterthwaite Theorem Theorem a b c d e Plurality voting with runoff: candidatea wins. Some Some a – 0 0 0 1 Positive Instant runoff voting: candidatec wins. Positive Results Results b 1 – 1 1 1 − candidateb wins. Summary Borda count / Condorcet winner: candidateb wins. Summary c 1 0 – 1 1 Different winners for different voting systems. d 1 0 0 – 0 Which voting system to prefer? Can even strategically e 0 0 0 1 – choose voting system! June 14th, 2016 B. Nebel, R. Mattmüller – Game Theory 12 / 62 June 14th, 2016 B. Nebel, R. Mattmüller – Game Theory 13 / 62 Condorcet Paradox Condorcet Methods Why Condorcet Winner not Always Exists Example: Preferences of voters 1, 2 and 3 on candidates a, b and c. Social Social Choice Choice Theory Theory a ≺1 b ≺1 c Introduction Introduction Social Choice Social Choice ≺ ≺ Functions Definition Functions b 2 c 2 a Condorcet Condorcet Methods Methods ≺ ≺ A Condorcet method return a Condorcet winner, if one exists. c 3 a 3 b Arrow’s Arrow’s Impossibility Impossibility Theorem Theorem Then we have cyclical preferences. Gibbard- One particular Condorcet method: the Schulze method. Gibbard- Satterthwaite Satterthwaite a b c Theorem Relatively new: Proposed in 1997 Theorem Some Some a – 0 1 Positive Already many users: Debian, Ubuntu, Pirate Party, . Positive b 1 – 0 Results Results c 0 1 – Summary Summary a ≺ b, b ≺ c, c ≺ a: violates transitivity of linear order consistent with these preferences. June 14th, 2016 B. Nebel, R. Mattmüller – Game Theory 14 / 62 June 14th, 2016 B. Nebel, R. Mattmüller – Game Theory 15 / 62 Schulze Method Schulze Method Notation: d(X;Y) = number of pairwise comparisons won by X against Y Social Social Choice Choice Theory Theory Definition Introduction Definition Introduction Social Choice Social Choice For candidates X and Y, there exists a path C1;:::;Cn Functions Functions Condorcet Let p(X;Y) be the maximal value z such that there exists a Condorcet between X and Y of strength z if Methods path of strength z from X to Y, and p(X;Y) = 0 if no such path Methods Arrow’s Arrow’s C1 = X, Impossibility exists. Impossibility Theorem Theorem Cn = Y, Gibbard- Then, the Schulze winner is the Condorcet winner, if it exists. Gibbard- Satterthwaite Otherwise, a potential winner is a candidate a such that Satterthwaite d(Ci ;Ci+1) > d(Ci+1;Ci ) for all i = 1;:::;n − 1, and Theorem Theorem Some p(a;X) ≥ p(X;a) for all X =6 a. Some d(Ci ;Ci+1) ≥ z for all i = 1;:::;n − 1 and there exists Positive Positive Results Tie-Breaking is used between potential winners. Results j = 1;:::;n − 1 s.t. d(Cj ;Cj+1) = z Summary Summary Example: path of strength 3. 8 5 3 a b c d June 14th, 2016 B. Nebel, R. Mattmüller – Game Theory 16 / 62 June 14th, 2016 B. Nebel, R. Mattmüller – Game Theory 17 / 62 Schulze Method Schulze Method Example Example # voters 3 2 2 2 # voters 3 2 2 2 Social Social 1st a d d c Choice 1st a d d c Choice Theory Theory 2nd b a b b Introduction 2nd b a b b Introduction Social Choice Social Choice 3rd c b c d Functions 3rd c b c d Functions Condorcet Condorcet 4th d c a a Methods 4th d c a a Methods Arrow’s Arrow’s Impossibility Impossibility Theorem Theorem As a graph: Is there a Condorcet winner? Gibbard- Weights d(X;Y): Path strengths p(X;Y): Gibbard- Satterthwaite Satterthwaite Theorem a b c d 5 a b c d Theorem a b c d a b Some a – 5 5 3 a – 5 5 5 Some a – 1 1 0 Positive Positive Results b 4 – 7 5 5 b 5 – 7 5 Results b 0 – 1 1 6 7 Summary c 4 2 – 5 5 c 5 5 – 5 Summary c 0 0 – 1 5 d 1 0 0 – d 6 4 4 – d c d 6 5 5 – No! Potential winners: b and d.
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