Computational Aspects of Approval Voting and Declared-Strategy Voting

# Computational Aspects of Approval Voting and Declared-Strategy Voting

## Computational Aspects of Approval Voting and Declared-Strategy Voting

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1. Computational Aspects of Approval Voting and Declared-Strategy Voting Dissertation defense 17 April 2008 Rob LeGrand Washington University in St. Louis Computer Science and Engineering legrand@cse.wustl.edu Ron Cytron Steven Brams Jeremy Buhler Robert Pless Itai Sened Aaron Stump

2. Themes of research • Approval voting systems • Susceptibility to insincere strategy • encouraging sincere ballots • Evaluating effectiveness of various strategies • Internalizing insincerity • separating strategy from indication of preferences • Complex voting protocols • complexity of finding most effective ballot • complexity of calculating the outcome

3. What is “manipulation”? • Broadly, effective influence on election outcome • Election officials can . . . • exclude/include alternatives [Nurmi ’99] • exclude/include voters [Bartholdi, Tovey & Trick ’92] • choose election protocol [Saari ’01] • Alternatives may be able to . . . • drop out to avoid a vote-splitting effect • Voters can . . . • find the ballot that is likeliest to optimize the outcome • This last sense is what we mean

4. Let’s vote! 45 voters A C B 35 voters B C A 20 voters C B A (1st) (2nd) (3rd) sincere preferences

5. Plurality voting 45 voters A C B 35 voters B C A 20 voters C B A sincere ballots A: 45 votes B: 35 votes C: 20 votes “zero-information” result

6. Plurality voting 45 voters A C B 35 voters B C A 20 voters C B A ballots so far ? A: 45 votes B: 35 votes C: 0 votes election state

7. Plurality voting 45 voters A C B 35 voters B C A 20 voters C B A strategic ballots insincerity! B: 55 votes A: 45 votes C: 0 votes final election state [Gibbard ’73] [Satterthwaite ’75]

8. Manipulation decision problem 45 voters A C B 35 voters B C A 20 voters C B A ballot sets BV BU B: 55 votes A: 45 votes C: 0 votes election state

9. Manipulation decision problem Existence of Probably Winning Coalition Ballots (EPWCB) INSTANCE: Set of alternatives A and a distinguished member a of A; set of weighted cardinal-ratings ballots BV; the weights of a set of ballots BU which have not been cast; probability QUESTION: Does there exist a way to cast the ballots BU so that a has at least probability of winning the election with the ballots ? • My generalization of problems from the literature: [Bartholdi, Tovey & Trick ’89] [Conitzer & Sandholm ’02] [Conitzer & Sandholm ’03]

10. Manipulation decision problem Existence of Probably Winning Coalition Ballots (EPWCB) INSTANCE: Set of alternatives A and a distinguished member a of A; set of weighted cardinal-ratings ballots BV; the weights of a set of ballots BU which have not been cast; probability QUESTION: Does there exist a way to cast the ballots BU so that a has at least probability of winning the election with the ballots ? • These voters have maximum possible information • They have all the power (if they have smarts too) • If this kind of manipulation is hard, any kind is

11. Manipulation decision problem Existence of Probably Winning Coalition Ballots (EPWCB) INSTANCE: Set of alternatives A and a distinguished member a of A; set of weighted cardinal-ratings ballots BV; the weights of a set of ballots BU which have not been cast; probability QUESTION: Does there exist a way to cast the ballots BU so that a has at least probability of winning the election with the ballots ? • This problem is computationally easy (in P) for: • plurality voting [Bartholdi, Tovey & Trick ’89] • approval voting

12. Manipulation decision problem Existence of Probably Winning Coalition Ballots (EPWCB) INSTANCE: Set of alternatives A and a distinguished member a of A; set of weighted cardinal-ratings ballots BV; the weights of a set of ballots BU which have not been cast; probability QUESTION: Does there exist a way to cast the ballots BU so that a has at least probability of winning the election with the ballots ? • This problem is computationally infeasible (NP-hard) for: • Hare (single-winner STV) [Bartholdi & Orlin ’91] • Borda [Conitzer & Sandholm ’02]

13. What can we do to make manipulation hard? • One approach: “tweaks” [Conitzer & Sandholm ’03] • Add an elimination round to an existing protocol • Drawback: alternative symmetry (“fairness”) is lost • What if we deal with manipulation by embracing it? • Incorporate strategy into the system • Encourage sincerity as “advice” for the strategy

14. Declared-Strategy Voting [Cranor & Cytron ’96] rational strategizer cardinal preferences ballot election state outcome

15. Declared-Strategy Voting [Cranor & Cytron ’96] sincerity strategy rational strategizer cardinal preferences ballot election state outcome • Separates how voters feel from how they vote • Levels playing field for voters of all sophistications • Aim: a voter needs only to give sincere preferences

16. What is a declared strategy? A: 0.0 B: 0.6 C: 1.0 cardinal preferences A: 0 B: 1 C: 0 declared strategy voted ballot A: 45 B: 35 C: 0 current election state • Captures thinking of a rational voter

17. Can DSV be hard to manipulate? DSV can be made to be NP-hard to manipulate in the EPWCB sense. [LeGrand ’08] Proof by reduction: • Simulate Hare by using particular declared strategy in DSV • Hare is NP-hard to manipulate[Bartholdi & Orlin ’91] • If this DSV system were easy to manipulate, then Hare would be • DSV can be made NP-hard to manipulate So why use “tweaks”? (DSV is better!)

18. Favorite vs. compromise, revisited 45 voters A C B 35 voters B C A 20 voters C B A ballots so far ? A: 45 votes B: 35 votes C: 0 votes election state

19. Approve both! 45 voters A C B 35 voters B C A 20 voters C B A strategic ballots insincerity avoided B: 55 votes A: 45 votes C: 20 votes final election state

20. Approval voting [Ottewell ’77] [Weber ’77] [Brams & Fishburn ’78] • Allows approval of any subset of alternatives • Single alternative with most votes wins • Used historically [Poundstone ’08] • Republic of Venice 1268-1789 • Election of popes 1294-1621 • Used today [Brams ’08] • Election of UN secretary-general • Several academic societies, including: • Mathematical Society of America • American Statistical Association

21. Strands of research

22. Strands of research

23. Strands of research

24. Approval ratings

25. Approval ratings • Aggregating film reviewers’ ratings • Rotten Tomatoes: approve (100%) or disapprove (0%) • Metacritic.com: ratings between 0 and 100 • Both report average for each film • Reviewers rate independently

26. Approval ratings • Online communities • Amazon: users rate products and product reviews • eBay: buyers and sellers rate each other • Hotornot.com: users rate other users’ photos • Users can see other ratings when rating • Can these “voters” benefit from rating insincerely?

27. Approval ratings

28. Average of ratings outcome: data from Metacritic.com: Videodrome (1983)

29. Average of ratings outcome: Videodrome (1983)

30. Another approach: Median outcome: Videodrome (1983)

31. Another approach: Median outcome: Videodrome (1983)

32. Another approach: Median • Immune to insincerity [LeGrand ’08] • voter i cannot obtain a better result by voting • if , increasing will not change • if , decreasing will not change • Allows tyranny by a majority • no concession to the 0-voters

33. Average with Declared-Strategy Voting? • So Median is far from ideal—what now? • try using Average protocol in DSV context • But what’s the rational Average strategy? • And will an equilibrium always be found? rational strategizer cardinal preferences ballot election state outcome

34. Equilibrium-finding algorithm Videodrome (1983)

35. Equilibrium-finding algorithm

36. Equilibrium-finding algorithm

37. Equilibrium-finding algorithm

38. Equilibrium-finding algorithm

39. Equilibrium-finding algorithm • Is this algorithm is guaranteed to find an equilibrium? equilibrium!

40. Equilibrium-finding algorithm • Is this algorithm is guaranteed to find an equilibrium? • Yes! [LeGrand ’08] equilibrium!

41. Expanding range of allowed votes • These results generalize to any range [LeGrand ’08]

42. Multiple equilibria can exist • Will multiple equilibria will always have the same average? outcome in each case:

43. Multiple equilibria can exist • Will multiple equilibria will always have the same average? • Yes! [LeGrand ’08] outcome in each case:

44. Average-Approval-Rating DSV outcome: Videodrome (1983)

45. Average-Approval-Rating DSV • AAR DSV is immune to insincerity in general [LeGrand ’08] outcome:

46. Evaluating AAR DSV systems • Expanded vote range gives wide range of AAR DSV systems: • If we could assume sincerity, we’d use Average • Find AAR DSV system that comes closest • Real film-rating data from Metacritic.com • mined Thursday 3 April 2008 • 4581 films with 3 to 44 reviewers per film • measure root mean squared error

47. Evaluating AAR DSV systems minimum at

48. Evaluating AAR DSV systems