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The probability of safe manipulation Mark C. Wilson www.cs.auckland.ac.nz/mcw/blog/ (joint with Reyhaneh Reyhani) Department of Computer Science University of Auckland COMSOC, D usseldorf, 2010-09-16 Mark C. Wilson Outline


  1. The probability of safe manipulation Mark C. Wilson www.cs.auckland.ac.nz/˜mcw/blog/ (joint with Reyhaneh Reyhani) Department of Computer Science University of Auckland COMSOC, D¨ usseldorf, 2010-09-16 Mark C. Wilson

  2. Outline Preliminaries Safe manipulation Algorithms for positional scoring rules Further discussion Mark C. Wilson

  3. Outline What Google thinks this talk is about Mark C. Wilson

  4. Preliminaries Basic setup ◮ A set C of alternatives (candidates) of size m , and a set V of voters, of size n . Mark C. Wilson

  5. Preliminaries Basic setup ◮ A set C of alternatives (candidates) of size m , and a set V of voters, of size n . ◮ Each voter v has a type (sincere preference) and submits an expressed preference. These are permutations L v of the candidates. Mark C. Wilson

  6. Preliminaries Basic setup ◮ A set C of alternatives (candidates) of size m , and a set V of voters, of size n . ◮ Each voter v has a type (sincere preference) and submits an expressed preference. These are permutations L v of the candidates. ◮ A profile is a function V → T . A voting situation is a multiset from T with total weight n . Mark C. Wilson

  7. Preliminaries Basic setup ◮ A set C of alternatives (candidates) of size m , and a set V of voters, of size n . ◮ Each voter v has a type (sincere preference) and submits an expressed preference. These are permutations L v of the candidates. ◮ A profile is a function V → T . A voting situation is a multiset from T with total weight n . ◮ The positional scoring rule determined by a vector w with w 1 ≥ w 2 ≥ · · · ≥ w m − 1 ≥ w m assigns the usual score � | c | := |{ v ∈ V | L v = t }| w L − 1 v ( c ) . t ∈T Mark C. Wilson

  8. Preliminaries Basic setup ◮ A set C of alternatives (candidates) of size m , and a set V of voters, of size n . ◮ Each voter v has a type (sincere preference) and submits an expressed preference. These are permutations L v of the candidates. ◮ A profile is a function V → T . A voting situation is a multiset from T with total weight n . ◮ The positional scoring rule determined by a vector w with w 1 ≥ w 2 ≥ · · · ≥ w m − 1 ≥ w m assigns the usual score � | c | := |{ v ∈ V | L v = t }| w L − 1 v ( c ) . t ∈T ◮ In this talk tiebreaking is mostly not relevant, so we ignore it completely. Mark C. Wilson

  9. Preliminaries Manipulation ◮ Standard social choice definition: a voter expresses an insincere preference to achieve a better outcome than otherwise, assuming other voters vote sincerely. This is individual manipulation. Mark C. Wilson

  10. Preliminaries Manipulation ◮ Standard social choice definition: a voter expresses an insincere preference to achieve a better outcome than otherwise, assuming other voters vote sincerely. This is individual manipulation. ◮ Coalitional manipulation occurs when a subset X of V all simultaneously adopt the above strategy. Their expressed preferences need not be the same, nor their sincere preferences. However all must (weakly) prefer the new outcome to the sincere one. Mark C. Wilson

  11. Preliminaries Manipulation ◮ Standard social choice definition: a voter expresses an insincere preference to achieve a better outcome than otherwise, assuming other voters vote sincerely. This is individual manipulation. ◮ Coalitional manipulation occurs when a subset X of V all simultaneously adopt the above strategy. Their expressed preferences need not be the same, nor their sincere preferences. However all must (weakly) prefer the new outcome to the sincere one. ◮ There is no claim that such strategic voting will take place, just that there is incentive to consider it. Mark C. Wilson

  12. Preliminaries Difficulties with coalitional manipulation ◮ How do coalition members identify each other? Mark C. Wilson

  13. Preliminaries Difficulties with coalitional manipulation ◮ How do coalition members identify each other? ◮ How do coalition members communicate? Mark C. Wilson

  14. Preliminaries Difficulties with coalitional manipulation ◮ How do coalition members identify each other? ◮ How do coalition members communicate? ◮ How do coalition members compute their joint strategy? Mark C. Wilson

  15. Preliminaries Difficulties with coalitional manipulation ◮ How do coalition members identify each other? ◮ How do coalition members communicate? ◮ How do coalition members compute their joint strategy? ◮ How do coalition members enforce the strategy? Mark C. Wilson

  16. Safe manipulation The concept of safe manipulation ◮ A voter of type t (the leader) announces that (s)he will in fact express the preference t ′ . Mark C. Wilson

  17. Safe manipulation The concept of safe manipulation ◮ A voter of type t (the leader) announces that (s)he will in fact express the preference t ′ . ◮ We assume that only voters of type t hear this message, and other voters vote sincerely. The type t voters can either vote as t or t ′ . Let x denote the number who switch to t ′ . Mark C. Wilson

  18. Safe manipulation The concept of safe manipulation ◮ A voter of type t (the leader) announces that (s)he will in fact express the preference t ′ . ◮ We assume that only voters of type t hear this message, and other voters vote sincerely. The type t voters can either vote as t or t ′ . Let x denote the number who switch to t ′ . ◮ The announced vote is safe if for all x , the outcome is never worse for these voters. In particular this applies to the maximal manipulation, where all voters of type t switch. Note that a voter who ranks the sincere winner lowest can never vote unsafely. Mark C. Wilson

  19. Safe manipulation The concept of safe manipulation ◮ A voter of type t (the leader) announces that (s)he will in fact express the preference t ′ . ◮ We assume that only voters of type t hear this message, and other voters vote sincerely. The type t voters can either vote as t or t ′ . Let x denote the number who switch to t ′ . ◮ The announced vote is safe if for all x , the outcome is never worse for these voters. In particular this applies to the maximal manipulation, where all voters of type t switch. Note that a voter who ranks the sincere winner lowest can never vote unsafely. ◮ If in addition there is some x for which the outcome is better for these voters, the profile is safely manipulable by type t in direction t ′ . Mark C. Wilson

  20. Safe manipulation Safe manipulation nonexample ◮ Let m = 5 and use w = (55 , 39 , 33 , 21 , 0) . Suppose that there are 3 voters of each possible type, and 1 extra voter of type 12345. The sincere winner is alternative 1. Mark C. Wilson

  21. Safe manipulation Safe manipulation nonexample ◮ Let m = 5 and use w = (55 , 39 , 33 , 21 , 0) . Suppose that there are 3 voters of each possible type, and 1 extra voter of type 12345. The sincere winner is alternative 1. ◮ If 1 type 53124 voter votes instead as 35241, alternative 2 wins; if 2 switch, alternative 3 wins; if 3 switch, alternative 4 wins. Mark C. Wilson

  22. Safe manipulation Safe manipulation nonexample ◮ Let m = 5 and use w = (55 , 39 , 33 , 21 , 0) . Suppose that there are 3 voters of each possible type, and 1 extra voter of type 12345. The sincere winner is alternative 1. ◮ If 1 type 53124 voter votes instead as 35241, alternative 2 wins; if 2 switch, alternative 3 wins; if 3 switch, alternative 4 wins. ◮ Thus such voters can both undershoot and overshoot in the same profile. Mark C. Wilson

  23. Safe manipulation Previous work ◮ Slinko and White showed that the analogue of the Gibbard-Satterthwaite theorem holds for safe manipulation. They asked about the probability that safe manipulation would succeed. Mark C. Wilson

  24. Safe manipulation Previous work ◮ Slinko and White showed that the analogue of the Gibbard-Satterthwaite theorem holds for safe manipulation. They asked about the probability that safe manipulation would succeed. ◮ Hazon and Elkind studied the complexity of safe manipulation (COMSOC 2010, Tuesday). Their main relevant results: Mark C. Wilson

  25. Safe manipulation Previous work ◮ Slinko and White showed that the analogue of the Gibbard-Satterthwaite theorem holds for safe manipulation. They asked about the probability that safe manipulation would succeed. ◮ Hazon and Elkind studied the complexity of safe manipulation (COMSOC 2010, Tuesday). Their main relevant results: ◮ The results are strongly determined by the complexity of the tiebreaking algorithm. Mark C. Wilson

  26. Safe manipulation Previous work ◮ Slinko and White showed that the analogue of the Gibbard-Satterthwaite theorem holds for safe manipulation. They asked about the probability that safe manipulation would succeed. ◮ Hazon and Elkind studied the complexity of safe manipulation (COMSOC 2010, Tuesday). Their main relevant results: ◮ The results are strongly determined by the complexity of the tiebreaking algorithm. ◮ (IsSafe) Given t, t ′ , and an anonymous rule, it is decidable in polynomial time whether safe manipulation is possible. Mark C. Wilson

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