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How Hard Is It To Manipulate Voting?

How Hard Is It To Manipulate Voting?. Edith Elkind, Princeton Helger Lipmaa, HUT. Manipulation: Example. 99 voters, 3 candidates ( L iberal, U ltra-liberal, R oyalist). 49 voters prefer L to U to R. 48 voters prefer U to L to R.

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How Hard Is It To Manipulate Voting?

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  1. How Hard Is It To Manipulate Voting? Edith Elkind, Princeton Helger Lipmaa, HUT

  2. Manipulation: Example • 99 voters, 3 candidates (Liberal, Ultra-liberal, Royalist). • 49 voters prefer L to U to R. • 48 voters prefer U to L to R. • 2 voters (Edith and Helger) prefer R to U to L. • Aggregation rule: Plurality • each voter casts a vote for one candidate. • the candidate with the largest number of votes wins. • draws are resolved by a coin toss.

  3. If Edith and Helger cooperate, they can guarantee that U is elected What Will Edith and Helger Do? L: 49 votes U: 48 votes R > U > L If I vote for R, L will get elected, so I’d rather vote for U

  4. Why Manipulation Is Bad • Aggregation rules are designed with certain social welfare criteria in mind. • Misrepresentation of preferences results in a suboptimal choice w.r.t. these criteria. • Also, election results do not reflect true distribution of preferences in the society: • maybe, in fact, 20% of the U.S. population prefer Nader to Gore to Bush?

  5. 2nd round L > U 49 votes U wins U > L 50 votes What If We Change Aggregation Rule? • Single Transferable Vote: • This time, Edith and Helger are better off voting honestly, but this will not always be the case… • Other popular voting schemes (Borda, Copeland) suffer from the same problem 1st round L > U > R 49 votes U > L > R 48 votes R > U > L 2 votes

  6. Formal Setup • n voters, m candidates. • Preference of a voter i is a permutation pi of {1, …, m} (best to worst). • Aggregation rule S: (p1, …,pn) → cj. • A voter i can manipulate S if there is • a preference vector p = (p1,…,pi, …,pn) • a permutation pi’ s.t. S(p1,…,pi’, …,pn) >iS(p). Theorem (Gibbard-Satterthwaite, 1971): every non-dictatorial aggregation rule with ≥3 candidates is manipulable.

  7. How Do We Get Around The Impossibility Result? • We cannot make manipulation impossible… • But we can try to make it hard! • Some aggregation rules (e.g., STV) are NP-hard to manipulate (Bartholdi, Orlin 1991) • these rules may not reflect the welfare goals (why is there so many voting rules out there?) • does NP-hardness suffice? • want security against coalitions, not just a single voter.

  8. P r e r o u n d Alice Bob Do most voters prefer Alice to Bob? Alice Diana Ernest Original protocol Adding a Preround (Conitzer-Sandholm) Carl Diana Ernest Frank • Retains some of the intuitive properties of the original protocol. • Is hard to manipulate for many classical protocols: • First pair candidates, then elicit votes: NP-hard • First elicit votes, then randomly pair candidates: #P-hard • Interleave pairing and vote elicitation: PSPACE-hard

  9. Issues With Preround Scheduling • Can we trust the election officials to randomly pair the candidates? • Can we achieve average-case hardness? E.g., can we make manipulation as hard as inverting one-way functions? • What if manipulating voters can collude?

  10. Alice Bob …. Alice ↕ Frank XOR 01101… 11101… Bob Carl …. Bob ↕ Ernest 11100… OWF CarlAlice …. Carl ↕ Diana 10111… 00100… Preround schedule Our Protocol: Preround With a Twist Idea: use votes themselves to select preround schedule.

  11. Key Properties • No trusted source of randomness is needed. • Manipulating this protocol is as hard as inverting the underlying one-way function • even for a large coalition of conspiring would-be manipulators • Proof proceeds by constructing a vector of honest voters’ preferences s.t. there is a unique preround schedule that makes manipulators’ protégé a winner • still a worst-case construction…

  12. Average-Case Hardness? • Can we make manipulation hard for an overwhelming fraction of honest voters preferences? • (Partial) negative result: there is a vector of preferences such that: • If everyone votes honestly, under any preround schedule candidate P wins • There is a manipulation by a single voter that under any preround schedule results in a draw between P and another candidate

  13. Open Problems • Average-case hardness seems hard to achieve using a preround. Other approaches? • What is the maximum fraction of manipulators we can tolerate? • we can prove security against 1/6 of all voters. Is it optimal? • Our results are for specific protocols. More general proofs?

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