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How to Rig Elections and Competitions

How to Rig Elections and Competitions. Outline. A common way to aggregate agents preferences - voting But it can be manipulated! Current models assume perfect information We analyzed a model of imperfect information Theoretically, manipulation is hard: NP-Complete proofs

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How to Rig Elections and Competitions

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  1. How to Rig Elections and Competitions

  2. Outline • A common way to aggregate agents preferences - voting • But it can be manipulated! • Current models assume perfect information • We analyzed a model of imperfect information • Theoretically, manipulation is hard: NP-Complete proofs • Practically it is not: Heuristics and experimental evaluation • Conclusion, Future work

  3. Why Voting? • Preference aggregation - to a socially desirable decision • Not only agents… • Human decisions among a collection of alternatives • Sport tournaments • But it can be manipulated if we have perfect information • By a single voter or a coalition of voters • By the election officer

  4. Our Model - Imperfect Information • Agenda rigging by the election officer • Linear order • Tree order • Complexity results • Verifying an agenda is in P • Finding a perfect agenda seems to be hard • Empirical results • Easy-to-implement heuristics performed well • Tested against random and real data • Motivation - better design of voting protocols

  5. Basic Definitions • Set of outcomes/candidates, • Imperfect information ballot matrix M, • Voting trees A C B A C B A C A B C D C D

  6. Linear Order Tree • Verifying an agenda is O(n2) - with dynamic programming • A candidate can only benefit by going late in an order • And this is not true for general trees! • Finding an order for a particular candidate to win: • With a non-zero probability - O(n2) • With at least a certain probability

  7. Weak Version of Rigged Agenda • A run - • An order agenda together with the intended outcomes • P[r | M] - a probability that the run will result in our candidate • WIIRA Problem - • Set of candidates, Ω • Imperfect information matrix, M • A favored candidate, ω* • A probability p • Is there a run in which the winner is ω*, s.t. P[r | M] ≥ p ? • NP-C, from the k-HCA problem on tournaments

  8. Balanced Tree Order • Verifying an agenda is O(n2) • Finding an order for a particular candidate to win: • With a non-zero probability - open problem [Lang07] • With at least a certain probability - NP-C [Vu08]

  9. From Theory to Practice “Concern: It might be that there are effective heuristics to manipulate an election, even though manipulation is NP-complete. Discussion: True. The existence of effective heuristics would weaken any practical import of our idea. It would be very interesting to find such heuristics.” [Bartholdi89]

  10. Linear Order – Conversion • Probabilistic data  deterministic data • Simple convert • Threshold convert • Not better than random order

  11. General Approach • Heuristics: • Far adversary . . . . • Best win • Local search • Random order • Two data sets: • Random data- uniform and normal distribution • Real data - 29 teams from the NBA, top 13 tennis players 1 2 m-2 m-1 ω*

  12. Linear Order- Normal Distribution

  13. Linear Order- 13 Tennis Players

  14. Balanced Tree- Normal Distribution

  15. Balanced Tree- 13 Tennis Players

  16. Balanced Tree- 29 NBA Teams

  17. Conclusion • Two voting protocols with incomplete information • Agenda verification is in P • Manipulation by the officer seems to be hard • Far adversary, best win and local search can help a lot! • Future: other voting protocols {hazonn,sarit}@cs.biu.ac.il, {mjw,ped}@csc.liv.ac.uk

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