GibbardSatterthwaite【荐】.pptVIP

更多资源/ Corollary Let P be a sensitive scoring protocol. Then SCWM in P is NP-Hard (with m?3) Sketch: If CWM is NP-Hard, then SCWM is as well as partitioning does not depend on generating tally from votes Jeremy Clark * Proposition 2 Let P be a sensitive scoring protocol. Then ?* is a junta distribution for SCWM in P with C={p,c1,c2,…,cm-1} and m=O(1). Where ?* is the following distribution: Independently randomly choose w(v) from [0,1] (with discrete precision). Independently randomly choose S[ci] from [W,(m-1)W]. Jeremy Clark * Is this Junta? Hard? Yes Balance? Authors calculate bounds using Chernoff’s bounds Dichotomy? First discrete step is non-negligible Symmetry? Invariant to candidates Refinement? 2nd ranked candidate will at least tie p Jeremy Clark * Greedy Algorithm Sort candidates from lowest score to highest Choose p as first choice, and rest in sorted order Recalculate scores and repeat for each vote When finished, return true iff p has highest score Jeremy Clark * Example Borda: 3,2,1,0, n=5 S[Con] = 20 S[Lib] = 19 S[NDP] = 17 S[Gre] = 10 ? p Jeremy Clark * Example S[Con] = 20 S[Lib] = 19 S[NDP] = 17 S[Gre] = 10 t1 : GreNDPLibCon Jeremy Clark * Example S[Con] = 20 + 0 = 20 S[Lib] = 19 + 1 = 20 S[NDP] = 17 + 2 = 18 S[Gre] = 10 + 3 = 13 t1 : GreNDPLibCon Jeremy Clark * Example S[Con] = 20 S[Lib] = 20 S[NDP] = 18 S[Gre] = 13 Jeremy Clark * Example S[Con] = 20, 20 , 20 , 21 , 23 , 23 S[Lib] = 19, 20 , 21 , 21 , 22 , 24 S[NDP] = 17, 18 , 20 , 22 , 22 , 23 S[Gre] = 10, 13 , 16 , 19 , 22 , 25 t1 : GreNDPLibCon t2 : GreNDPLibCon t3 : GreNDPConLib t4 : GreConLibNDP t5 : GreLibNDPCon Jeremy Clark * Greedy Properties Greedy is P-time Greedy never issues false positives Greedy does issue false negatives, however these are bounded to Pr[err]?1/p(n) Therefore Greedy is deterministic heuristic polynomial time Jeremy Clark * Theorem Let P be a sensitive scoring protocol. If m=O(1) then P, with candidates C={p,c1,c2,…,cm-1}, is susceptible to SCWM. Th