Manipulation and Control for Approval Voting and Other Voting Systems Jörg Rothe Oxford Meeting for COST Action IC1205 on Computational Social Choice April 16, 2013

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

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 NP-hardness complexity shields enough? Or do they evaporate for single-peaked electorates?

NP-Hardness Shields to Protect Elections NP-hardness shields Manipulation & Control in Single-peaked Electorates Elections & Voting Systems Manipulation & Control Proof Sketch: CCAV in Approval

NP-Hardness Shields to Protect Elections NP-hardness shields Manipulation & Control in Single-peaked Electorates Elections & Voting Systems Manipulation & Control Proof Sketch: CCAV in Approval

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

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

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 Hi, I am Ramon Llull. In 1299, I came up with the voting system that these guys now study!

Llull/Copeland Rule For FIFA World Championships or UEFA European Championships: Simply use = 1/3 as the tie value. Difference between the Llull and the Copeland rule? What happens if the head-to-head contest ends with a tie? Llull: Both get 1 point Copeland 0 : Both get 0 points Copeland 0.5 : Both get half a point Copeland : Both get points, for a rational, 0<<1

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.

NP-Hardness Shields to Protect Elections NP-hardness shields Manipulation & Control in Single-peaked Electorates Elections & Voting Systems Manipulation & Control Proof Sketch: CCAV in Approval

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

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

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

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

NP-Hardness Shields for Control

References: Control J. Bartholdi, C. Tovey, and M. Trick: How Hard is it to Control an Election? Mathematical and Computer Modelling, E. Hemaspaandra, L. Hemaspaandra, and J. Rothe: Anyone but Him: The Complexity of Precluding an Alternative. Artificial Intelligence, (AAAI-2005) P. Faliszewski, E. Hemaspaandra, L. Hemaspaandra, and J. Rothe: Llull and Copeland Voting Computationally Resist Bribery and Constructive Control. Journal of Artificial Intelligence Research, (AAAI-2007; AAIM-2008) G. Erdélyi, M. Nowak, and J. Rothe: SP-AV Fully Resists Constructive Control and Broadly Resists Destructive Control. Mathematical Logic Quarterly, (MFCS-2008) G. Erdélyi and J. Rothe: Control Complexity in Fallback Voting. Proceedings of CATS G. Erdélyi, L. Piras, and J. Rothe: The Complexity of Voter Partition in Bucklin and Fallback Voting: Solving Three Open Problems. Proceedings of AAMAS-2011.

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 dont 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 dont want him to be the captain!!

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

NP-Hardness Shields to Protect Elections NP-hardness shields Manipulation & Control in Single-peaked Electorates Elections & Voting Systems Manipulation & Control Proof Sketch: CCAV in Approval

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 voters degree of preference rises to a peak and then falls (or just rises or just falls). A voters 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 voters degree of preference rises to a peak and then falls (or just rises or just falls). A voters > > > 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 voters degree of preference rises to a peak and then falls (or just rises or just falls). A voters > > > 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 voters 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 voters 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 Vs 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 voters 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

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

NP-Hardness Shields to Protect Elections NP-hardness shields Manipulation & Control in Single-peaked Electorates Elections & Voting Systems Manipulation & Control Proof Sketch: CCAV in Approval

A Sample Proof Sketch number of approvals from voters in V for candidates that are votes in W that can be added (with multiplicities)

A Sample Proof Sketch number of approvals from voters in V for candidates that are votes in W that can be added (with multiplicities) Which vote types from W should we add? Especially if they are incomparable?

A Sample Proof Sketch number of approvals from voters in V for candidates that are votes in W that can be added (with multiplicities) Well handle this by a smart greedy algorithm.

A Sample Proof Sketch number of approvals from voters in V for candidates that are votes in W that can be added (with multiplicities) Why are F, C, B, c, f, and j dangerous but the remaining candidates can be ignored?

A Sample Proof Sketch number of approvals from voters in V for candidates that are votes in W that can be added (with multiplicities) First, each added vote will be an interval including p. So drop all others.

A Sample Proof Sketch number of approvals from voters in V for candidates that are votes in W that can be added (with multiplicities) First, each added vote will be an interval including p. So drop all others.

A Sample Proof Sketch number of approvals from voters in V for candidates that are votes in W that can be added (with multiplicities) Now, if adding votes from W causes p to beat c then p must also beat a and b.

A Sample Proof Sketch number of approvals from voters in V for candidates that are votes in W that can be added (with multiplicities) Thus, c is a dangerous rival for p but a and b can safely be ignored.

A Sample Proof Sketch number of approvals from voters in V for candidates that are votes in V that can be added (with multiplicities) Likewise, f is dangerous but d and e can safely be ignored.

A Sample Proof Sketch number of approvals from voters in V for candidates that are votes in V that can be added (with multiplicities) Likewise, j is dangerous but g, h, and i can safely be ignored.

A Sample Proof Sketch number of approvals from voters in V for candidates that are votes in V that can be added (with multiplicities) Hey, why do you do that step by step? Just say j is dangerous and ignore a, …, i.

A Sample Proof Sketch number of approvals from voters in V for candidates that are votes in V that can be added (with multiplicities) No! Look what happens if we add 6 votes of the type with multiplicity 7!

A Sample Proof Sketch votes in W that can be added (with multiplicities) No! Look what happens if we add 6 votes of the type with multiplicity 7!

A Sample Proof Sketch number of approvals from voters in V for candidates that are votes in W that can be added (with multiplicities) OK, thats not illogical. But how does your smart greedy algorithm work?

Smart Greedy Algorithm OK, first I need more space for that!

Smart Greedy Algorithm OK, first I need more space for that! In smart greedy, we eat through all dangerous rivals to the right of p starting with the leftmost: c. To become the unique winner, p must beat c. Only votes in W whose right endpoints fall in can help. Let B be the set of those votes. Choose votes from B starting with the rightmost left endpoint. This is a perfectly safe strategy!

Smart Greedy Algorithm votes in W that can be added (with multiplicities)

Smart Greedy Algorithm votes in B that can be added (with multiplicities)

Smart Greedy Algorithm votes in B that can be added (with multiplicities)

Smart Greedy Algorithm 1 First rival defeated 1 votes in B that can be added (with multiplicities)

Smart Greedy Algorithm OK, first I need more space for that! In smart greedy, we eat through all dangerous rivals to the right of p starting with the leftmost: c. To become the unique winner, p must beat c. Only votes in W whose right endpoints fall in can help. Let B be the set of those votes. Choose votes from B starting with the rightmost left endpoint. This is a perfectly safe strategy! Iterate. If you run out of dangerous candidates on the right of p, mirror image the societal order (i.e., reverse L) and finish off the remaining dangerous candidates until you either succeed or reach the addition limit.

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