The shield that never was: Societies with single-peaked preferences are more open to manipulation and control

Much work has been devoted, during the past 20 years, to using complexity to protect elections from manipulation and control. Many “complexity shield” results have been obtained—results showing that the attacker’s task can be made NP-hard. Recently there has been much focus on whether such worst-case hardness protections can be bypassed by frequently correct heuristics or by approximations. This paper takes a very different approach: We argue that when electorates follow the canonical political science model of societal preferences the complexity shield never existed in the first place. In particular, we show that for electorates having single-peaked preferences, many existing NP-hardness results on manipulation and control evaporate.