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My notations

My notations. Set of voters {1,..., n } Set of m candidates { a , b , c ...} Preference profile: a vector of rankings. a. b. a. b. a. c. c. c. b. Charles Dodgson. English author and mathematician, better known as Lewis Carroll

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My notations

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  1. My notations • Set of voters {1,...,n} • Set of m candidates {a,b,c...} • Preference profile: a vector of rankings a b a b a c c c b

  2. Charles Dodgson • English author and mathematician, better known as Lewis Carroll • Suggested choosing a candidate “as close as possible” to a Condorcet winner

  3. Dodgson’s rule • Score of x = minimum # of exchanges between adjacent candidates needed to make x a Condorcet winner

  4. Dodgson score example a b e b a b c c c d d a e e d e b c e 3 3 3 2 2 3 3 2 3 3 4 2 3 3 4 3 P(a,b) P(a,c) P(a,d) P(a,e) d d a a c b

  5. Dodgson’s rule • Score of x = minimum # of exchanges between adjacent candidates needed to make x a Condorcet winner • Alternatively: total number of positions that the voters push x • Elect candidate with minimum score

  6. Complexity of Dodgson • Dodgson-Score: given candidate x, a preference profile, and a threshold k, is the Dodgson score of xat most k ? • [Bartholdi et al, SCW 89] Dodgson-Score is NP-complete, Dodgson-Winner is NP-hard • [Hemaspaandra et al., JACM 97] Dodgson-Winner is complete for Parallel access to NP

  7. What can we do? • Heuristics [McCabe-Dansted et al., COMSOC 06; Homan&Hemaspaandra, MFCS 06] • Fixed parameter tractable algorithms [Betzler et al., I&C 10] • Approximation [Cargiannis et al., SODA 09; Caragiannis et al., EC 10] • -approximation = solution that is at most  times the Dodgson score

  8. Greedy algorithm • Given candidate x and pref profile • y is alive if it beats x in pairwise elections, otherwise dead • Cost-effectiveness of push = ratio between # of live candidates overtaken and # of positions pushed • Greedy Algorithm: while  live candidates, perform the most cost-effective push

  9. It’s alive! d d d d d a c b c c d e4 c b a x c c e9 e13 b b e5 x e16 b b e10 e17 a a e6 x e14 a a x e1 e11 e15 e7 e2 e12 e8 e3 x x

  10. Greed pays off • Theorem [Caragiannis et al., SODA 09]: The greedy alg has an approx ratio of Hm-1 • Best ratio possible in polytime • Proof relies on the dual fitting technique [Vaz 01], inspired by constrained set multicover • Primal solution found by algorithm upper-bounded by infeasible dual assignment • Divide dual assignment by Hm-1 and show that shrunk assignment is feasible INF ALG OPT INF/Hm-1 ALG/Hm-1

  11. Approximation algs as voting rules • Does it make sense to approximate a voting rule? • Approximation algorithm is a new voting rule • Should satisfy desirable social choice properties - possibly not satisfied by Dodgson! • E.g., monotonicity • Designing a polytime monotonic O(log m)-approx algorithm is difficult but possible [Caragiannis et al., EC 10]

  12. Greedy is nonmonotonic d d c d d d b a a c b c d e4 c c e9 e13 b c e5 e14 e16 b b e10 e17 e6 e11 e15 x a a b a a e1 e7 e12 x e2 e8 x e3 x x x

  13. Greedy is nonmonotonic d d d a c b e4 c c e9 e13 e5 e14 e16 b b e10 e17 e6 e11 e15 x a a e1 e7 e12 x e2 e8 x e3 x x x

  14. Greedy is nonmonotonic d d d d d a c b c c d e4 c b a x c c e9 e13 b b e5 x e16 b b e10 e17 a a e6 x e14 a a x e1 e11 e15 e7 e2 e12 e8 e3 x x

  15. Computational hardness as a barrier to manipulation

  16. Manipulation b a c c b • Often it is in the voters’ interest to reveal false preferences • Example: Borda • May lead to the election of a socially bad candidate a a d d b c d

  17. Gibbard-Satterthwaite Theorem • If m=2, Plurality is nonmanipulable • Let m3. The following properties are incompatible: • Onto: any candidate can be elected • Nondictatorship: there is no single voter who dictates the outcome of the election • Nonmanipulability

  18. Circumventing Gibbard-Satterthwaite • Mechanism Design: aligning incentives using money • Restricting preferences

  19. Single Peaked Preferences • A grocery store is being built. Each voter (resident of the street) wants it as close as possible to his own house. Need to choose a spot • Suggestion: choose the median peak • Onto and nondictatorial • This is also the Condorcet winner • The median is nonmanipulable!

  20. Circumventing Gibbard-Satterthwaite • Mechanism Design: assuming money is available and preferences are quasi-linear • Restricting preferences • Computational Complexity

  21. Complexity of manipulation • Manipulation is always possible in theory • But can we design voting rules where it is difficult in practice? • Are there “reasonable” voting rules where manipulation is a hard computational problem?

  22. The computational problem c c b b • R-Manipulation problem: • Given votes of nonmanipulators and a preferred candidate p • Can manipulator cast vote that makes p (uniquely) win under R? • Example: Borda d d c a a d b a

  23. A greedy algorithm c c b b • Algorithm: • Rank p in first place • While there are unranked candidates: • If there is a candidate that can be placed in next spot without preventing p from winning, place this candidate. • Otherwise return `false’. d d c a a d b b b b a

  24. Example: Copeland a b e b a c c d b d e a e c d e a c c b b d b a e b b b d

  25. When does the alg work? • Theorem [Bartholdi et al., SCW 89]: Let R be a rule s.t.  function s(<,x) such that: • For every < chooses a candidate that maximizes s(<,x) • {y: y < x}  {y: y <‘ x}  s(x,<)  s(x,<‘) Then the alg always decides R-Manipulation correctly • Captures: • All scoring rules, e.g., Borda • Copeland: s is number of pairwise elections x wins • Maximin: s is the worst pairwise election of x

  26. Voting rules that are hard to manipulate • Natural rules • Copeland with second order tie breaking [Bartholdi et al., SCW 89] • STV [Bartholdi&Orlin, SCW 91] • Ranked Pairs [Xia et al., IJCAI 09] • Order pairwise elections by decreasing strength of victory • Successively lock in results of pairwise elections unless it leads to a cycle • Winner is the top ranked candidate in final order

  27. Example: Ranked Pairs a b 6 12 4 8 10 d c 2

  28. Voting rules that are hard to manipulate • Natural rules • Copeland with second order tie breaking [Bartholdi et al., SCW 89] • STV [Bartholdi&Orlin, SCW 91] • Ranked Pairs [Xia et al., IJCAI 09] • Order pairwise elections by decreasing strength of victory • Successively lock in results of pairwise elections unless it leads to a cycle • Winner is the top ranked candidate in final order • Can also “tweak” easy to manipulate voting rules [Conitzer&Sandholm, IJCAI 03]

  29. Coalitional manipulation • R-Unweighted-Coalitional-Manipulation (UCM) problem: • Given votes of nonmanipulators and a preferred candidate p • Can k manipulators cast votes that make p (uniquely) win under R? • R-WCM: the same with weights • Voters can be weighted by shares in a company or seats in an assembly • WCM is NP-complete in a variety of voting rules, even for a constant m [Conitzer et al., JACM 07]

  30. Example: WCM in veto • We want: given the nonmanipulators’ votes • … it is NP-hard to find votes for the manipulators to achieve their objective • Simple example: veto rule, 3 candidates • Suppose, from the given votes, p has received t-1 more vetoes than a, and t-1 more than b • The manipulators’ combined weight is 2t • The only way for p to win is if the manipulators veto a with t weight, and b with t weight • But this is doing PARTITION  NP-hard!

  31. Hardness of UCM • R-UCM is known to be NP-complete under: • Copeland [Faliszewski et al., AAMAS 08,10] • Maximin [Xia et al., IJCAI 09] • Weird scoring rule [Xia et al., EC 10] • Even with only two manipulators! • Open problem: complexity of UCM under Borda

  32. Beyond worst-case hardness • Results such as NP-hardness suggest that the runtime of any successful manipulation algorithm grows dramatically on some instances • But there may be algorithms that usually solve the problem • Can we design rules where manipulable instances are usually hard to solve? • Uh, no?

  33. Three approaches • The “fraction of manipulators” approach [Procaccia&Rosenschein, AAMAS 07; Xia&Conitzer, EC 08; Walsh, IJCAI 09] • The “axiomatic” approach [Conitzer&Sandholm, AAAI 06; Friedgut et al., FOCS 08; Xia&Conitzer, EC 08; Dobzinski&Procaccia, WINE 08] • The “Window of error” approach [Procaccia&Rosenschein, JAIR 07; Zuckerman et al., AIJ 09; Xia et al., EC 10]

  34. What is a “window of error”? • We will focus on WCM and UCM • We will consider algorithms that can incorrectly decide instances • Return “false” when there is a successful manipulation • “Window of error” = instances that are incorrectly decided by the algorithm • Can we say something that would hold for many distributions over instances?

  35. Window of error illustrated

  36. The greedy algorithm revisited • Reminder: R-WCM problem • Given weighted votes of nonmanipulators and a preferred candidate p • Can weighted manipulators cast votes that make p (uniquely) win under R? • Greedy algorithm for WCM under scoring rules [Procaccia&Rosenschein, JAIR 07]: each manipulator ranks p first, and the other candidates by inverse score • One manipulator  coincides with previous alg

  37. Example: Algorithm is correct 40 0 10 20 30 10 5 5 40 0 10 20 30 40 0 10 20 30

  38. Example: Algorithm is wrong 40 0 10 20 30 10 5 40 0 10 20 30 40 0 10 20 30

  39. Theorem: WCM in Borda • Theorem [Zuckerman et al., AIJ 09]: • If the alg returns “true” then there is a successful manipulation • If the algincorrectly returns “false” then it would find a successful manipulation given an extra manipulator with max weight • In UCM this is just one extra manipulator

  40. Example for the theorem 40 0 10 20 30 10 10 5 40 0 10 20 30 40 0 10 20 30

  41. Generalization • Zuckerman et al. design algorithms for specific voting rules • Xia et al. [EC 10] give a more general but weaker theorem for scoring rules, using a connection to scheduling

  42. Junta Distributions • Cool animation! • Procaccia and Rosenschein[JAIR 07]:find hard distributions over instances

  43. Other work • Control [Bartholdi et al., 92; Faliszewski et al., JAIR 09] • Bribery [Faliszewski et al, JAIR 09b, Elkind et al., SAGT 09] • Manipulation in multi-winner elections [Meir et al., JAIR 08] • ...

  44. Getting involves in this community • Community mailing list: https://lists.duke.edu/sympa/subscribe/comsoc • Computational Social Choice (COMSOC) workshop http://ccc.cs.uni-duesseldorf.de/COMSOC-2010/

  45. A few useful overviews • Y. Chevaleyre, U. Endriss, J. Lang, and N. Maudet. A Short Introduction to Computational Social Choice. In Proc. 33rd Conference on Current Trends in Theory and Practice of Computer Science (SOFSEM-2007), LNCS 4362, Springer-Verlag, 2007. • V. Conitzer. Making decisions based on the preferences of multiple agents. Communications of the ACM, 53(3):84–94, 2010. • V. Conitzer. Comparing Multiagent Systems Research in Combinatorial Auctions and Voting. To appear in the Annals of Mathematics and Artificial Intelligence. • P. Faliszewski, E. Hemaspaandra, L. Hemaspaandra, and J. Rothe. A richer understanding of the complexity of election systems. In S. Ravi and S. Shukla, editors, Fundamental Problems in Computing: Essays in Honor of Professor Daniel J. Rosenkrantz, chapter 14, pages 375–406. Springer, 2009. • P. Faliszewski and A. D. Procaccia. AI's War on Manipulation: Are We Winning? To appear in AI Magazine.

  46. Thank You!

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