Frequent Manipulability of Elections: The Case of Two Voters

1. Frequent Manipulability of Elections: The Case of Two Voters ShaharDobzinski(Hebrew U) Ariel D. Procaccia (MS Israel R&D Center)

2. Voting: notations • Set of voters {1,...,n} • Set of m alternatives {a,b,c...} • Each i has linear order <i over alternatives • Preference profile: a vector < of rankings a b a b a c c c b

3. The G-S Theorem • Voting rule: a mapping f from preference profiles to alternatives; designates winner • f is strategyproof (SP) if <,i ,<i’ f(<i’,<-i) i f(<) • f is dictatorial if is.t. <, f(<)=top(<i) • Theorem (Gibbard-Satterthwaite): Let m3. Any SP and onto rule is dictatorial.

4. Complexity of Manipulation • [BTT89] Circumvent G-S using Computational Complexity • Many worst-case hardness results • Are there voting rules that are usually hard to manipulate? • Recent typical-case tractability results: • Algorithmic [PR07,CS06,ZPR08] • Descriptive [PR07b,XC08]

5. Noam’s idea • “Randomized algorithm”: choose a random manipulation • Given “reasonable” voting rule, works with polynomially small prob. w.r.t. “many” preference profiles • Good prob. of success by repeating • [FKN08] This is true for neutral voting rules if m=3 • [XC08b] This is true, under arguable conditions on voting rule, for any constant m

6. Our Result • f is -strategyproof if i, Pr[ f(<) <if(<i’,<-i) ]   • f is -dictatorial if i, Pr[ f(<)  top(<i) ]   • f is Pareto-optimal (PO) if [ i, y <ix ] f(<)  y • Main Theorem: Let n=2, m  3. If f is PO and -SP, then f is poly(m)-dictatorial.

7. Sketch of Proof Sketch a a b b a a c c d d e e

8. Concluding Remarks • Comparison with [FKN08] • Future work: Prove general theorem (duh...)