Projektseminar Computational Social Choice -Eine Einführung- Jörg Rothe & Lena Schend SS 2012, HHU Düsseldorf 4. April 2012

Introduction Social Choice Theory  voting theory  preference aggregation  judgment aggregation Computer Science  artificial intelligence  algorithm design  computational complexity theory - worst-case/average-case complexity - optimization, etc. voting in multiagent systems multi-criteria decision making meta search, etc.... Software agents can systematically analyze elections to find optimal strategies

Introduction Social Choice Theory  voting theory  preference aggregation  judgment aggregation Computational Social Choice Computer Science  artificial intelligence  algorithm design  computational complexity theory - worst-case/average-case complexity - optimization, etc. computational barriers to prevent manipulation control bribery Software agents can systematically analyze elections to find optimal strategies

Computational Social Choice With the power of NP-hardness vulcans have constructed complexity shields to protect elections against many types of manipulation and control.

Computational Social Choice With the power of NP-hardness vulcans have constructed complexity shields to protect elections against many types of manipulation and control. Question: Are worst-case complexity shields enough? Or do they evaporate on "typical elections"?

NP-Hardness Shields Evaporating? NP-hardness shields single-peaked electorates junta distributions approximation experimental analysis

Elections  An election is a pair (C,V) with a finite set C of candidates: a finite list V of voters.  Voters are represented by their preferences over C: either by linear orders: > > > or by approval vectors: (1,1,0,1)  Voting system: determines winners from the preferences

Voting Systems Approval Voting (AV)  votes are approval vectors in v1v v2v v3v v4v v5v v6v6 1001

Voting Systems Approval Voting (AV)  votes are approval vectors in  winners: all candidates with the most approvals v1v v2v v3v v4v v5v v6v ∑ 4324

Voting Systems Approval Voting (AV)  votes are approval vectors in  winners: all candidates with the most approvals winners: v1v v2v v3v v4v v5v v6v ∑ 4324

Voting Systems Positional Scoring Rules (for m candidates)  defined by scoring vector with  each voter gives points to the candidate on position i  winners: all candidates with maximum score Borda:Plurality Voting (PV): k-Approval (m-k-Veto):Veto (Anti-Plurality):

-4:02:23:1 0:4-1:32:2 3:1-2:2 1:32:2 - Voting Systems Pairwise Comparison v 1 : >>> v 3 : > > > v 2 : >>> v 4 : > > > Condorcet: beats all other candidates strictly Copeland : 1 point for victory points for tie Maximin: maximum of the worst pairwise comparison

Voting Systems Round-based: Single Transferable Vote (STV) v 1 : >>> v 2 : > > > v 3 : > >> v 4 : > > > Round 1 over eliminate cand. with lowest plurality score Round 2 over eliminate cand. with lowest plurality score Final Round over eliminate cand. with lowest plurality score

Voting Systems Round-based: Single Transferable Vote (STV) v 1 : >> v 2 : > > v 3 : > > v 4 : > > Round 1 over eliminate cand. with lowest plurality score Round 2 over eliminate cand. with lowest plurality score Final Round over eliminate cand. with lowest plurality score

Voting Systems Round-based: Single Transferable Vote (STV) v 1 : v 2 : v 3 : v 4 : Round 1 over eliminate cand. with lowest plurality score Round 2 over eliminate cand. with lowest plurality score Final Round over … and the winner is…

Voting Systems Level-based: Bucklin Voting (BV) v 1 : > > > v 2 : > > > v 3 : > > > v 4 : > > > v 5 : > > >  5 voters => strict majority threshold is 3 Lvl 11220

Voting Systems Level-based: Bucklin Voting (BV) v 1 : > > > v 2 : > > > v 3 : > > > v 4 : > > > v 5 : > > >  5 voters => strict majority threshold is 3 Lvl Lvl 22233

Voting Systems Level-based: Bucklin Voting (BV) v 1 : > > > v 2 : > > > v 3 : > > > v 4 : > > > Level 2 Bucklin v 5 : > > > winners:  5 voters => strict majority threshold is 3 Lvl Lvl 22233

Voting Systems Level-based: Fallback Voting (FV)  combines AV and BV Candidates: v: {, } | {, } v: > | {, }  Bucklin winners are fallback winners.  If no Bucklin winner exists (due to disapprovals), then approval winners win.

War on Electoral Control AV winners: "chair": knows all preferences v1v v2v v3v v4v v5v v6v ∑ 4324

War on Electoral Control AV winner: "chair": knows all preferences and can change the structure of an election v1v v2v v3v v4v v5v v6v ∑ 2312

War on Electoral Control AV winner: "chair": knows all preferences and can change the structure Other types of control: of an election  adding/partitioning voters  deleting/adding/partitioning candidates v1v v2v v3v v4v v5v v6v ∑ 2312

NP-Hardness Shields for Control Resistance = NP-hardness, Vulnerability = P, Immunity, and Susceptibility

Cope- land Score -4:02:23:12.5 0:4-1:32:20.5 2:23:1-2:22 1:32:2 -1 War on Manipulation Copeland : winner v 1 : >>> v 3 : > > > v 2 : >>> v 4 : > > > I like Spock but I don‘t want him to be the captain!!

Copeland : winner v 1 : >>> v 3 : > > > v 2 : >>> v 4 : > > > assumption:. v 4 knows the other voters‘ votes v 4 lies to make his most preferred candidate win Cope- land Score -4:02:23:12.5 0:4-1:32:20.5 2:23:1-2:22 1:32:2 -1 War on Manipulation I like Spock but I don‘t want him to be the captain!!

Copeland : winners v 1 : >>> v 3 : > > > v 2 : >>> v 4 : > > > Here: unweighted voters, single manipulator. Other types: - coalitional manipulation - weighted voters Cope- land Score -3:12:2 2 1:3- 0 2:23:1-2:22 3:12:2-2 War on Manipulation I like Spock but I don‘t want him to be the captain!!

NP-Hardness Shields for Manipulation Results due to Conitzer, Sandholm, Lang (J.ACM 2007)

NP-Hardness Shields Evaporating? NP-hardness shields single-peaked electorates junta distributions approximation experimental analysis

Junta Distributions of Procaccia and Rosenschein (JAAMAS 2007) are omitted here, as they are a rather technical concept.

NP-Hardness Shields Evaporating? NP-hardness shields single-peaked electorates junta distributions approximation experimental analysis

Experiments Manipulation  testing (heuristic) algorithms for manipulation problem at hand on given elections sample real elections generate random elections: Impartial Culture (IC) Polya-Eggenberger (PE) voters vote independently all preferences are equally likely voters are highly correlated v 1 v 2 v 3... Walsh (IJCAI 2009; ECAI 2010)

Experiments Manipulation  testing (heuristic) algorithms for manipulation problem at hand on given elections sample real elections generate random elections: Impartial Culture (IC) Polya-Eggenberger (PE) voters vote independently all preferences are equally likely voters are highly correlated v 1 v 2 v 3... Walsh (IJCAI 2009; ECAI 2010)

Experiments Manipulation  testing (heuristic) algorithms for manipulation problem at hand on given elections sample real elections generate random elections: Impartial Culture (IC) Polya-Eggenberger (PE) voters vote independently all preferences are equally likely voters are highly correlated v 1 v 2 v 3... Walsh (IJCAI 2009; ECAI 2010)

Experiments Manipulation  testing (heuristic) algorithms for manipulation problem at hand on given elections sample real elections generate random elections: Impartial Culture (IC) Polya-Eggenberger (PE) voters vote independently all preferences are equally likely voters are highly correlated v 1 v 2 v 3... Walsh (IJCAI 2009; ECAI 2010)

Experiments Manipulation  testing (heuristic) algorithms for manipulation problem at hand on given elections sample real elections generate random elections: Impartial Culture (IC) Polya-Eggenberger (PE) voters vote independently all preferences are equally likely voters are highly correlated v 1 v 2 v 3... Walsh (IJCAI 2009; ECAI 2010)

Experiments Manipulation  testing (heuristic) algorithms for manipulation problem at hand on given elections sample real elections generate random elections: Impartial Culture (IC) Polya-Eggenberger (PE) voters vote independently all preferences are equally likely voters are highly correlated v 1 v 2 v 3... Walsh (IJCAI 2009; ECAI 2010)

Experiments Manipulation  Results for STV Single Manipulation:  for up to 128 candidates/voters manipulation has low computational costs (for all voter distributions)  chance of successful manipulation decreases with increasing number of nonmanipulative voters Coalitional Manipulation:  larger coalitions are more likely to be successful  again: computational costs are low for up to 128 candidates/voters  Results for Veto (weighted) if manipulators‘ weights are too big/small => trivial even in critical region: computational costs are low only correlated voters increase computational costs Walsh (IJCAI 2009; ECAI 2010)

NP-Hardness Shields Evaporating? NP-hardness shields single-peaked electorates junta distributions approximation experimental analysis

Approximating Manipulation Before: Is manipulation possible?

Approximating Manipulation Before: Is manipulation possible? Now: How many manipulators are needed? (min!) Approximation Algorithms: -efficient algorithms -do not always find optimal solution -can be analyzed both theoretically and experimentally

Approximating Borda 3x > > > > > > 2x > > > > > > Borda winner manipulators prefer B-Score

Approximating Borda Algorithm for Borda-CCUM : "Reverse" m 1 > > > > > > B-Score Zuckerman, Procaccia & Rosenschein (Artificial Intelligence 2009)

Approximating Borda Algorithm for Borda-CCUM : "Reverse" m 1 > > > > > > B-Score Zuckerman, Procaccia & Rosenschein (Artificial Intelligence 2009)

Approximating Borda Algorithm for Borda-CCUM : "Reverse" m 1 > > > > > > m 2 > > > > > > B-Score Zuckerman, Procaccia & Rosenschein (Artificial Intelligence 2009)

Approximating Borda Algorithm for Borda-CCUM : "Reverse" m 1 > > > > > > m 2 > > > > > > B-Score Zuckerman, Procaccia & Rosenschein (Artificial Intelligence 2009)

Approximating Borda Algorithm for Borda-CCUM : "Reverse" m 1 > > > > > > m 2 > > > > > > m 3 > > > > > > B-Score Zuckerman, Procaccia & Rosenschein (Artificial Intelligence 2009)

Approximating Borda Algorithm for Borda-CCUM : "Reverse" m 1 > > > > > > m 2 > > > > > > m 3 > > > > > > B-Score Zuckerman, Procaccia & Rosenschein (Artificial Intelligence 2009)

Approximating Borda Algorithm for Borda-CCUM : "Reverse" m 1 > > > > > > m 2 > > > > > > m 3 > > > > > > m 4 > > > > > > B-Score Zuckerman, Procaccia & Rosenschein (Artificial Intelligence 2009)

Approximating Borda Algorithm for Borda-CCUM : "Reverse" m 1 > > > > > > m 2 > > > > > > m 3 > > > > > > m 4 > > > > > > B-Score Zuckerman, Procaccia & Rosenschein (Artificial Intelligence 2009)

Approximating Borda Algorithm for Borda-CCUM : "Reverse" m 1 > > > > > > m 2 > > > > > > m 3 > > > > > > m 4 > > > > > > m 5 > > > > > > B-Score Zuckerman, Procaccia & Rosenschein (Artificial Intelligence 2009)

Approximating Borda Algorithm for Borda-CCUM : "Reverse" m 1 > > > > > > m 2 > > > > > > m 3 > > > > > > m 4 > > > > > > m 5 > > > > > > B-Score Zuckerman, Procaccia & Rosenschein (Artificial Intelligence 2009)

Approximating Borda Optimal solution: 4 manipulators m 1 > > > > > > m 2 > > > > > > m 3 > > > > > > m 4 > > > > > > "Reverse" needs one manipulator more than optimal B-Score Zuckerman, Procaccia & Rosenschein (Artificial Intelligence 2009)

Approximation Results  Maximin: factor 2 (twice number of optimal manipulators) factor 5/3 (not better than 3/2 unless P=NP)  Borda: Reverse: additional 1 Largest Fit unbounded additional number Average Fit of manipulators,, and are theoretically incomparable experimental comparison: IC model 76%83%99% PE model 76%43%99% Zuckerman, Lev & Rosenschein (AAMAS 2011) Davies, Katsirelos, Narodytska & Walsh (AAAI 2011)

NP-Hardness Shields Evaporating? NP-hardness shields single-peaked electorates junta distributions approximation experimental analysis

Single-Peaked Preferences  A collection V of votes is said to be single-peaked if there exists a linear order L over C such that each voter‘s „degree of preference“ rises to a peak and then falls (or just rises or just falls). A voter‘s preference curve on galactic taxes low galactic taxes high galactic taxes

 A collection V of votes is said to be single-peaked if there exists a linear order L over C such that each voter‘s „degree of preference“ rises to a peak and then falls (or just rises or just falls). A voter‘s > > > preference curve on galactic taxes low galactic taxes high galactic taxes Single-Peaked Preferences Single-peaked preference consistent with linear order of candidates

 A collection V of votes is said to be single-peaked if there exists a linear order L over C such that each voter‘s „degree of preference“ rises to a peak and then falls (or just rises or just falls). A voter‘s > > > preference curve on galactic taxes low galactic taxes high galactic taxes Single-Peaked Preferences Preference that is inconsistent with this linear order of candidates

Single-Peaked Preferences  A collection V of votes is said to be single-peaked if there exists a linear order L over C such that each voter‘s „degree of preference“ rises to a peak and then falls (or just rises or just falls).  If each vote v i in V is a linear order > i over C, this means that for each triple of candidates c, d, and e: (c L d L e or e L d L c) implies that for each i, if c > i d then d > i e.

Single-Peaked Preferences  A collection V of votes is said to be single-peaked if there exists a linear order L over C such that each voter‘s „degree of preference“ rises to a peak and then falls (or just rises or just falls).  If each vote v i in V is a linear order > i over C, this means that for each triple of candidates c, d, and e: (c L d L e or e L d L c) implies that for each i, if c > i d then d > i e.  Bartholdi & Trick (1986); Escoffier, Lang & Öztürk (2008): Given a collection V of linear orders over C, in polynomial time we can produce a linear order L witnessing V‘s single- peakedness or can determine that V is not single-peaked.

 A collection V of votes is said to be single-peaked if there exists a linear order L over C such that each voter‘s „degree of preference“ rises to a peak and then falls (or just rises or just falls). Single-peaked w.r.t. this order? v1v no v2v yes v3v no v4v yes v5v no v6v Single-Peaked Approval Vectors

 Removing NP-hardness shields:  3-candidate Borda  veto  every scoring protocol for  -candidate 3-veto,  Leaving them in place:  STV (Walsh AAAI 2007)  4-candidate Borda  5-candidate 3-veto  Erecting NP-hardness shields:  Artificial election system with approval votes, for size-3-coalition unweighted manipulation Results due to Faliszewski, Hemaspaandra, Hemaspaandra & Rothe (Information & Computation 2011) GeneralSingle-peaked Constructive Coalitional Weighted Manipulation

 Removing NP-hardness shields:  Approval  Constructive control by adding voters  Constructive control by deleting voters  Plurality  constructive control by adding candidates  destructive control by adding candidates  constructive control by deleting candidates  destructive control by deleting candidates Results due to Faliszewski, Hemaspaandra, Hemaspaandra & Rothe (2011)  Brandt, Brill, Hemaspaandra & Hemaspaandra (AAAI 2010) achieved similar results for other voting systems as well (e.g., for systems satisfying the weak Condorcet criterion) and also for constructive control by partition of voters. GeneralSingle-peaked Control for Single-Peaked Electorates

More Results on Single-Peaked Preferences  Faliszewski, Hemaspaandra, Hemaspaandra & Rothe (2011) also prove a dichotomy result for the scoring protocol CCWM is NP-complete if and in P otherwise.  Brandt, Brill, Hemaspaandra & Hemaspaandra (AAAI 2010) generalize this dichotomoy to scoring protocols with any fixed number of candidates.  Mattei (ADT 2011) empirically investigates huge data sets from real-world elections (drawn from the Netflix Prize) and observes that single-peaked preferences very rarely occur in practice.  Faliszewski, Hemaspaandra & Hemaspaandra (TARK 2011) study manipulative attacks in nearly single-peaked electorates.

NP-Hardness Shields Evaporating? NP-hardness shields single-peaked electorates junta distributions approximation experimental analysis

Experiments Control  same approach as for manipulation  testing (heuristic) algorithms for control problem at hand on given elections sample real elections generate random elections: Impartial Culture (IC) Two Mainstreams (TM) voters vote independently all preferences are equally likely voters are correlated

Experiments Control  same approach as for manipulation  testing (heuristic) algorithms for control problem at hand on given elections sample real elections generate random elections: Impartial Culture (IC) Two Mainstreams (TM) voters vote independently all preferences are equally likely voters are correlated

Experiments Control  same approach as for manipulation  testing (heuristic) algorithms for control problem at hand on given elections sample real elections generate random elections: Impartial Culture (IC) Two Mainstreams (TM) voters vote independently all preferences are equally likely voters are correlated v 1 v 2 v 3 v 4...

Experiments Control Observations:  destructive control shows more yes-instances (up to 100%) and lower computational costs  DCPV-TP in FV

Experiments Control Observations:  destructive control shows more yes-instances (up to 100%) and lower computational costs  CCPV-TP in FV

Experiments Control Observations:  destructive control shows more yes-instances (up to 100%) and lower computational costs  FV and BV show similar tendencies  voter control in PV has lower computational costs  deleting/adding voters show similar tendencies  for constructive control: voter control shows more yes-instances than candidate control  as expected: more yes-instances in the IC model than in the TM model

Thank you very much! That‘s typical for you humans! Please wait until the talk ist finished before you start asking questions!