Campaign Management via Bribery

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

2. COMSOC and Voting • Manipulation • Control • Bribery Computational social choice- group decision making

3. vsCampaign Management 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

4. Agenda • Introduction • Standard model of elections • Election systems • Swapbribery • Cost model • Basic problems • Complexity of swapbribery • Shift bribery • Whyuseful? • Algorithms for shiftbribery • Conlusions and openproblems

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

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

7. Election Model Borda count • Election E = (C,V) • C – the set of candidates • V – the set of voters = 6 3 2 1 0 Many otherelections systems studied! E.g, Plurality, k-approval, maximin, Copeland = 5 > > > = 4 > > > = 3 > > >

8. BriberyModels • Standard bribery • Payment ==> full control over a vote • Nonuniform bribery • Payment depends on the amount of change Problem:How to represent the prices?

9. SwapBribery • Price function πfor each voter. > > > π( , ) = 5

10. SwapBribery • Price function πfor each voter. > > > π( , ) = 5 π( , ) = 2

11. SwapBribery • Price function πfor 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

12. QuestionsAboutSwapBribery • Price of reaching a given vote? • Swap bribery and other voting problems? • Complexity of swap bribery? > > > > > > Swapbribery Voting problem <m

13. RelationsBetweenVotingProblems

14. TheComplexity of SwapBribery Voting rule Swap bribery Plurality P Veto P k-approval NP-com Borda NP-com Maximin NP-com Copeland NP-com Limit thetypes of swaps? Limit thenumber of voters? Limit thenumber of candidates?

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

16. TheComplexity of SwapBribery Voting rule Swap bribery Shift bribery Plurality P P Veto P P k-approval NP-com P Borda NP-com NP-com Maximin NP-com NP-com Copeland NP-com NP-com

17. TheComplexity of SwapBribery Voting rule Swap bribery Shift bribery Approx.ratio Plurality P P ― Veto P P ― k-approval NP-com P ― Borda NP-com NP-com 2 Maximin NP-com NP-com O(logm) Copeland NP-com NP-com O(m)

18. TheComplexity of SwapBribery Voting rule Swap bribery Shift bribery Approx.ratio Plurality P P ― Veto P P ― k-approval NP-com P ― Borda NP-com NP-com 2 Maximin NP-com NP-com O(logm) Copeland NP-com NP-com O(m) Single algorithm for all scoring protocols, even if weighted!

19. TheAlgorithm Why 2-approximation? > > > αi+1 αi

20. TheAlgorithm Why 2-approximation? > > > αi+1 αi gains αi+1 – αi points loses αi+1 – αi points Getting 2x the points for than the best bribery gives is sufficient to win

21. TheAlgorithm Why 2-approximation? • Operation of thealgorithm • Guess a cost k • Get most points for atcost k • Guess a cost k’ <= k • Get most points for atcost k’ > > > αi+1 αi gains αi+1 – αi points loses αi+1 – αi points Thisis a 2-approximation… but worksinpolynomial time onlyifpricesareencodedinunary Getting 2x the points for than the best bribery gives is sufficient to win

22. WhyDoestheAlgorithmWork? How much does optimal solution shift candidate p in each vote? v1 v2 v3 v4 v5 O – the optimal solution  gives p some T points • Operation of thealgorithm • Guess a cost k • Get most points for patcost k • Guess a cost k’ <= k • Get most points for patcost k’

23. WhyDoestheAlgorithmWork? How much does optimal solution shift candidate p in each vote? v1 v2 v3 v4 v5 O – the optimal solution  gives p some T points

24. WhyDoestheAlgorithmWork? How much does optimal solution shift candidate p in each vote? v1 v2 v3 v4 v5 O – the optimal solution  gives p some T points S – solution that gives most points at cost k

25. WhyDoestheAlgorithmWork? How much does optimal solution shift candidate p in each vote? v1 v2 v3 v4 v5 O – the optimal solution  gives p some T points 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 Nowitispossible to complete min(O,S) intwo independent ways: By continuing as S does (gettingatleastT-Dpoints extra) By continuing as O does (gettingT-Dpoints extra)

26. WhyDoestheAlgorithmWork? How much does optimal solution shift candidate p in each vote? v1 v2 v3 v4 v5 Nowitispossible to complete min(O,S) intwo independent ways: By continuing as S does (gettingatleastT-Dpoints extra) By continuing as O does (gettingT-Dpoints extra) After we performshiftsfrom min(O,S), thereis a way to make p win by shiftsthatgivehimT-Dpoints Thus, anyshiftthatgiveshim 2(T-D) points, makeshim a winner. Itiseasy to findthese 2(T-D) points. We’redone!

27. TheAlgorithm (General Case) 2-approximation algorithm for unaryprices Scaling argument + twists 2+ε-approximationscheme for anyprices Bootstrapping-flavored argument 2-approximation algorithm for anyprices

28. TheAlgorithm Why 2-approximation? • Operation of thealgorithm • Guess a cost k • Get most points for atcost k • Guess a cost k’ <= k • Get most points for atcost k’ > > > αi+1 αi gains αi+1 – αi points loses αi+1 – αi points Is this algorithm still a 2-approximation? Unclear!

29. 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