1. The Complexity of Elections: A New Domain for Heuristic Computation Piotr Faliszewski AGH University of Science and Technology, Kraków, Poland faliszew@agh.edu.pl

2. Why Are Elections Important? –Politics Electing leaders Deciding policies, laws … –Business settings Stock holders make decisions in companies University Senates, hiring decisions Competitions (e.g., racing, Eurovision) –Multiagent systems Meta-search engines Planning … Elections aggregate the opinions of the voters

3. Computational issues in elections Winner determination Running the election Possible winner Campaign management Cheating in elections Manipulation bribery control Gibbard- Satterthwaite Theorem

4. Complexity Barrier Approach Complexity barrier approach: If manipulating elections is hard, then we can ignore the fact that it is in principle possible. Approach initiated by Bartholdi, Tovey, and Trick in the late 80s and the early 90s Model: Represent each cheating strategy as a computational decision problem.

5. Complexity Barrier: Results Effects of complexity barrier research –Dozens of computational problems identified –Multiple standard election systems analyze –Quite thorough understanding of worst case complexity of elections Complications… –We would like some of the problems to be efficiently computable Determining winners Organizing a campaign –Worst-case analysis seems problematic… Room for heuristic computation!

6. Agenda Why study elections? Formal model –What are elections? –How to model votes? –What voting rules to use? Case study: Plurality bribery –Simplest case –Deterministic solution Case study: Campaign management –Campaign management as bribery –Why is it good to study? What has been done? Where to take data from? Conclusions and comments

7. How to Study Elections? We need to answer several questions: –What are elections? –How can we model voters’ preferences? –What election rules are we interested in? A mathematical object, a pair E = (C,V). C – set of candidates, V – set of voters A single top guy? A ranking of candidates? Numerical values, “utility” a voter derives from having a particular candidate elected? An election rule is a function that given an election returns the set of winners.

8. C = {,,,, } V = {>>>>, >>>>, >>>> } Election Model: Example Election E = (C,V) C – candidate set V – voter set Who is the winner? –Plurality –Veto –Borda –k-approval –Approval –Copeland –Llull –Dodgson –Kemeny –Young –…–… families of scoring protocols

9. C = {,,,, } V = {>>>>, >>>>, >>>> } Election Model: Example Election E = (C,V) C – candidate set V – voter set Who is the winner? –Borda count α = ( 4, 3, 2, 1, 0 ) score( ) = 13 score( ) = 9 score( ) = 8 score( ) = 12

10. C = {,,,, } V = {>>>>, >>>>, >>>> } Election Model: Example Election E = (C,V) C – candidate set V – voter set Who is the winner? –Copeland

11. Agenda Why study elections? Formal model –What are elections? –How to model votes? –What voting rules to use? Case study: Plurality bribery –Simplest case –Deterministic solution Case study: Campaign management –Campaign management as bribery –Why is it good to study? Conclusions and comments

12. Computational issues in elections Winner determination Running the election Possible winner Campaign management Cheating in elections Manipulation bribery control

13. Bribery Name: Ɛ -bribery. Given: An election E = (C,V), a candidate p in C, and an integer k. Question: Can we make p a winner via modifying votes of at most k voters? Many flavors of the bribery problem. - Ɛ -bribery - Ɛ -$bribery - Ɛ -weighted-bribery - Ɛ -weighted-$bribery Notation: Ɛ – an election system

14. Bribery in Plurality Setting –We are given: An election E = (C,V) A preferred candidate p A budget k –Question: Can we make p a winner by bribing voters of cost at most k? Observation: –It is sufficient to only look at each voter’s most preferred candidate unpricedpriced unweighted weighted

15. Bribery in Plurality Example: –What is the cheapest way to make Alice a winner? Setting –We are given: An election E = (C,V) A preferred candidate p A budget k –Question: Can we make p a winner by bribing voters of cost at most k?

16. Bribery in Plurality Example: –What is the cheapest way to make Alice a winner? Setting –We are given: An election E = (C,V) A preferred candidate p A budget k –Question: Can we make p a winner by bribing voters of cost at most k?

17. Bribery in Plurality Example: –What is the cheapest way to make Alice a winner? Setting –We are given: An election E = (C,V) A preferred candidate p A budget k –Question: Can we make p a winner by bribing voters of cost at most k?

18. Bribery in Plurality Example: –What is the cheapest way to make Alice a winner? Setting –We are given: An election E = (C,V) A preferred candidate p A budget k –Question: Can we make p a winner by bribing voters of cost at most k?

19. Bribery in Plurality Setting –We are given: An election E = (C,V) A preferred candidate p A budget k –Question: Can we make p a winner by bribing voters of cost at most k? unpricedpriced unweighted weighted

20. Computational Complexity NP –Class of problems whose solutions are of polynomial size, and are verifiable in polynomial time Some problems in NP –SAT – is a given Boolean formula satisfiable? –Given a sequence of integers (s 1, …, s n ), is there a subsequence that sums up to a given value K? –Does a graph have a clique of size n? NP-com: An NP problem is NP-complete if all NP problems reduce to it Reduction between problems –A, B – two problems –A reduces to B if there is a polynomial-time computable function f such that x in A  f(x) in B AB f f

21. plurality-weighted-$bribery  NP-com Proof: Reduction from the subset-sum problem Input: s 1, s 2, …, s n Subset-Sumplurality-weighted-$bribery 1 5 6 10 12 Question: Is there a subsequence that sums up to exactly half. 5 1 6 10 12 k = 17 =34

22. plurality-weighted-$bribery  NP-com Proof: Reduction from the subset-sum problem Input: s 1, s 2, …, s n Subset-Sumplurality-weighted-$bribery 1 5 6 10 12 Question: Is there a subsequence that sums up to exactly half. 5 1 6 10 12 k = 17 =34

23. Bribery in Plurality Setting –We are given: An election E = (C,V) A preferred candidate p A budget k –Question: Can we make p a winner by bribing voters of cost at most k? unpricedpriced unweighted weighted

24. Weighted Bribery in Plurality plurality-weighted-bribery –Now voters have weights What algorithms can we use? –Heuristics: Heaviest voter first Winner’s heaviest voter first

25. Bribery in Plurality Setting –We are given: An election E = (C,V) A preferred candidate p A budget k –Question: Can we make p a winner by bribing voters of cost at most k? unpricedpriced unweighted weighted

26. Agenda Why study elections? Formal model –What are elections? –How to model votes? –What voting rules to use? Case study: Plurality bribery –Simplest case –Deterministic solution Case study: Campaign management –Campaign management as bribery –Why is it good to study? What has been done? Where to take data from? Conclusions and comments

27. Computational issues in elections Winner determination Running the election Possible winner Campaign management Cheating in elections Manipulation bribery control

28. Bribery vs Campaign Management 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

29. 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?

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

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

32. 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

33. 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

34. Relations Between Voting Problems

35. Results on Swap Bribery Swap Bribery is easy for: –Plurality –Veto Swap bribery is NP-hard for: –Borda –Copeland –Bucklin –Maximin –Ranked pairs –Plurality with runoff –STV –… Swap bribery can model winner problems: –Kemeny –Dodgson

36. Why Study Swap Bribery? Swap bribery is hard for almost all voting systems Swap bribery is a generalization of most important problems

37. Agenda Why study elections? Formal model –What are elections? –How to model votes? –What voting rules to use? Case study: Plurality bribery –Simplest case –Deterministic solution Case study: Campaign management –Campaign management as bribery –Why is it good to study? What has been done? Where to take data from? Conclusions and comments

38. Dealing with Complexity Approximation algorithms –Some success for manipulation –Limited types of swap bribery Parameterized complexity attacks –Mostly smart brute-force algorithms Heuristic attacks –So far, only on manipulation instances –Limited type of algorithms (ad-hoc approaches) –Manipulation  solved (Toby Walsh) –others  untouched

39. Where to Take Data From? Generate data Impartial culture (votes chosen Independently at random) Polya-Eggenberger Urn Model An urn with all m! votes. Pick a vote at random, return it + a additional copies of the vote Real Elections - difficult to obtain - additional sampling And many others: E.g., uniform single-peaked votes…

40. Agenda Why study elections? Formal model –What are elections? –How to model votes? –What voting rules to use? Case study: Plurality bribery –Simplest case –Deterministic solution Case study: Campaign management –Campaign management as bribery –Why is it good to study? Conclusions and comments

41. Conclusions and Comments Conclusions –Elections are a rich source of computational issues –Abundance of worst-case complexity results for many settings –Very few experimental results –Very few heuristic solutions –Swap bribery can be a natural problem to attack with heuristic methods  most general problem studied.

42. Conclusions and Comments Where to get more information about the field? –Book chapter: A Richer Understanding of the Complexity of Election Systems (Faliszewski, Hemaspaandra, Hemaspaandra, Rothe) –Upcoming AI Magazine paper: AI’s War on Manipulation: Are We Winning? (Faliszewski, Procaccia) –Upcoming CACM paper: Using Complexity to Protect Elections (Faliszewski, Hemaspaandra, Hemaspaandra) Google search –By problem: complexity + voting + {manipulation, bribery, swap bribery, possible winner, control} –By people: Conitzer, Procaccia, Walsh, Hemaspaandra, Elkind, Slinko, Faliszewski

43. Conclusions and Comments Where is this kind of work published? –Conferences AAAI (AAAI Conference on Artificial Intelligence) AAMAS (Autonomous Agents and Multiagent Systems) EC (ACM Conference on Electronic Commerce) WINE (Workshop on Internet and Network Economics) SAGT (Symposium on Algorithmic Game Theory) ADT (Algorithmic Decision Theory) –Journals Artificial Intelligence Journal of AI Research Social Choice and Welfare Autonomous Agents and Multiagent Systems Mathematical Social Sciences

44. Thank you! I am sorry, but we have voted, and we are all in favor.