1. Shahar Dobzinski (Hebrew U) Ariel D. Procaccia (MS Israel R&D Center)

2.  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 Voter 1Voter 2Voter 3 a b c a c b b a c 2

3.  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  i s.t.  <, f(<)=top(< i )  Theorem (Gibbard-Satterthwaite): Let m  3. Any SP and onto rule is dictatorial. 3

4.  [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] 4

5.  “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 5

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

7. Voter 1Voter 2 a a a a c c d d e e c c d d e e b b a a a a b b 7

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