Campaign Management via Bribery Piotr Faliszewski AGH University of Science and Technology, Poland Joint work with Edith Elkind and Arkadii Slinko

◦ Manipulation ◦ Control ◦ Bribery COMSOC and Voting Computational social choice - group decision making

Bribery Bribery ◦ Invest money to change votes ◦ Some votes are cheaper than others ◦ Want to spend as little as possible Campaign management ◦ Invest money to change voters’ minds ◦ Some voters are easier to convince ◦ The campaign should be as cheap as possible vs Campaign Management

Agenda Introduction ◦ Standard model of elections ◦ Election systems Swap bribery ◦ Cost model ◦ Basic problems ◦ Complexity of swap bribery Shift bribery ◦ Why useful? ◦ Algorithms for shift bribery Conlusions and open problems

Election Model Election E = (C,V) ◦ C – the set of candidates ◦ V – the set of voters A candidate set

Election Model Election E = (C,V) ◦ C – the set of candidates ◦ V – the set of voters A vote (preference order) >>>

Election Model Election E = (C,V) ◦ C – the set of candidates ◦ V – the set of voters >>> >>> >>> Borda count = 6 = 5 = 4 = 3 Many other elections systems studied! E.g, Plurality, k-approval, maximin, Copeland

Bribery Models Standard bribery ◦ Payment ==> full control over a vote Nonuniform bribery ◦ Payment depends on the amount of change Problem: How to represent the prices?

Swap Bribery Price function π for each voter. >>> π(, ) = 5

Swap Bribery Price function π for each voter. >>> π(, ) = 2 π(, ) = 5

Swap Bribery Price function π for each voter. Swap bribery problem ◦ Given: E = (C,V), price function for each voter ◦ Question: What is the cheapest sequence of swaps that makes our guy a winner? >>> π(, ) = 2

Questions About Swap Bribery Price of reaching a given vote? Swap bribery and other voting problems? Complexity of swap bribery? >>>>>> Voting problem Swap bribery <m<m

Relations Between Voting Problems

The Complexity of Swap Bribery Voting rule Swap bribery PluralityP VetoP k-approvalNP-com BordaNP-com MaximinNP-com CopelandNP-com Limit the number of candidates? Limit the number of voters? Limit the types of swaps?

Shift Bribery Allowed swaps: ◦ Have to involve our candidate Realistic? ◦ As bribery: Yes ◦ Also: as a campaigning model! Gain in complexity?

Voting rule Swap briberyShift bribery The Complexity of Swap Bribery PluralityP P VetoP P k-approvalNP-com P BordaNP-com NP-com MaximinNP-com NP-com CopelandNP-com NP-com

Voting rule Swap briberyShift briberyApprox.ratio The Complexity of Swap Bribery PluralityP P ― VetoP P ― k-approvalNP-com P ― BordaNP-com NP-com 2 MaximinNP-com NP-comO(logm) CopelandNP-com NP-comO(m)

Voting rule Swap briberyShift briberyApprox.ratio The Complexity of Swap Bribery PluralityP P ― VetoP P ― k-approvalNP-com P ― BordaNP-com NP-com 2 MaximinNP-com NP-comO(logm) CopelandNP-com NP-comO(m) Single algorithm for all scoring protocols, even if weighted!

The Algorithm Why 2-approximation? >>> αiαi α i+1

The Algorithm Why 2-approximation? >>> αiαi α i+1 gains α i+1 – α i points loses α i+1 – α i points Getting 2x the points for than the best bribery gives is sufficient to win

The Algorithm Why 2-approximation? >>> αiαi α i+1 gains α i+1 – α i points loses α i+1 – α i points Getting 2x the points for than the best bribery gives is sufficient to win Operation of the algorithm 1.Guess a cost k 2.Get most points for at cost k 3.Guess a cost k’ <= k 4.Get most points for at cost k’ This is a 2-approximation… but works in polynomial time only if prices are encoded in unary

Why Does the Algorithm Work? Operation of the algorithm 1.Guess a cost k 2.Get most points for p at cost k 3.Guess a cost k’ <= k 4.Get most points for p at cost k’ How much does optimal solution shift candidate p in each vote? O – the optimal solution  gives p some T points v1v1 v5v5 v3v3 v4v4 v2v2

Why Does the Algorithm Work? How much does optimal solution shift candidate p in each vote? O – the optimal solution  gives p some T points v1v1 v5v5 v3v3 v4v4 v2v2

Why Does the Algorithm Work? How much does optimal solution shift candidate p in each vote? O – the optimal solution  gives p some T points v1v1 v5v5 v3v3 v4v4 v2v2 S – solution that gives most points at cost k

Why Does the Algorithm Work? How much does optimal solution shift candidate p in each vote? O – the optimal solution  gives p some T points v1v1 v5v5 v3v3 v4v4 v2v2 S – solution that gives most points at cost k min(O,S) – min shift of the two in each vote gives some D points to p Now it is possible to complete min(O,S) in two independent ways: 1.By continuing as S does (getting at least T-D points extra) 2.By continuing as O does (getting T-D points extra)

Why Does the Algorithm Work? How much does optimal solution shift candidate p in each vote? Now it is possible to complete min(O,S) in two independent ways: 1.By continuing as S does (getting at least T-D points extra) 2.By continuing as O does (getting T-D points extra) After we perform shifts from min(O,S), there is a way to make p win by shifts that give him T-D points Thus, any shift that gives him 2(T-D) points, makes him a winner. It is easy to find these 2(T-D) points. We’re done! v1v1 v5v5 v3v3 v4v4 v2v2

The Algorithm (General Case) 2-approximation algorithm for unary prices 2+ ε -approximation scheme for any prices 2-approximation algorithm for any prices Scaling argument + twists Bootstrapping-flavored argument

The Algorithm Why 2-approximation? >>> αiαi α i+1 gains α i+1 – α i points loses α i+1 – α i points Operation of the algorithm 1.Guess a cost k 2.Get most points for at cost k 3.Guess a cost k’ <= k 4.Get most points for at cost k’ Is this algorithm still a 2- approximation? Unclear!

Conclusions Swap bribery ◦ Interesting model ◦ Many hardness results ◦ Connection to possible winner Special cases ◦ Fixed #candidates, fixed #voters  boring ◦ Shift bribery  Realistic  Lowers the complexity  Interesting approximation issues