▁ ▁is ▁manip ul able ▁if ▁and ▁only ▁if ▁there ▁exists ▁a ▁profile ▁ ▁where ▁some ▁vot er ▁, ▁by ▁replacing ▁her ▁ball ot ▁ ▁with ▁another ▁ball ot ▁, ▁can ▁get ▁an ▁outcome ▁that ▁she ▁pre fers ▁( in ▁the ▁sense ▁of ▁). ▁ ▁We ▁denote ▁by ▁ ▁the ▁image ▁of ▁, ▁i . e . ▁the ▁set ▁of ▁possible ▁out comes ▁for ▁the ▁election . ▁For ▁example , ▁we ▁say ▁that ▁ ▁has ▁at ▁least ▁three ▁possible ▁out comes ▁if ▁and ▁only ▁if ▁the ▁cardinal ity ▁of ▁ ▁is ▁ 3 ▁or ▁more . ▁ ▁We ▁say ▁that ▁ ▁is ▁dict atorial ▁if ▁and ▁only ▁if ▁there ▁exists ▁a ▁vot er ▁ ▁who ▁is ▁a ▁dict ator , ▁in ▁the ▁sense ▁that ▁the ▁winning ▁alternative ▁is ▁always ▁her ▁most - lik ed ▁one ▁among ▁the ▁possible ▁out comes ▁regardless ▁of ▁the ▁prefer ences ▁of ▁other ▁vot ers . ▁If ▁the ▁dict ator ▁has ▁several ▁ex - a equ o ▁most - lik ed ▁alternatives ▁among ▁the ▁possible ▁out comes , ▁then ▁the ▁winning ▁alternative ▁is ▁simply ▁one ▁of ▁them . ▁ ▁Ex amples ▁ ▁Serial ▁dict ator ship ▁ ▁The ▁serial ▁dict ator ship ▁is ▁defined ▁as ▁follows . ▁If ▁vot er ▁ 1 ▁has ▁a ▁unique ▁most - lik ed ▁candidate , ▁then ▁this ▁candidate ▁is ▁elected . ▁Otherwise , ▁possible ▁out comes ▁are ▁restricted ▁to ▁the ▁most - lik ed ▁candidates , ▁whereas ▁the ▁other ▁candidates ▁are ▁elimin ated . ▁Then ▁vot er ▁ 2 ' s ▁ball ot