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

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. 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 L > U > R U > L > R R > U > L 1 st round 49 votes 48 votes 2 votes L > U U > L 2 nd round 49 votes 50 votes U wins

6. Formal Setup n voters, m candidates. Preference of a voter i is a permutation  i of {1, …, m} (best to worst). Aggregation rule S: (  1, …,  n ) → c j. A voter i can manipulate S if there is –a preference vector  = (  1,…,  i, …,  n ) –a permutation  i ’ s.t. S(  1,…,  i ’, …,  n ) > i S(  ). 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. Adding a Preround (Conitzer- Sandholm) AliceBob CarlDianaErnestFrank 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 Original protocol AliceDiana Ernest PreroundPreround Do most voters prefer Alice to Bob?

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