Mechanisms for aggregating the preferences of agents in elections need to balance many different considerations, including efficiency, information elicited from agents, and manipulability. We consider the utilitarian social welfare of mechanisms for preference aggregation, measured by the distortion. We show that for a particular input format called threshold approval voting, where each agent is presented with an independently chosen threshold, there is a mechanism with nearly optimal

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... when the number of voters is large. Threshold mechanisms are potentially manipulable, but place a low informational burden on voters. We then consider truthful mechanisms. For the widely-studied class of ordinal mechanisms which elicit the rankings of candidates from each agent, we show that truthfulness essentially imposes no additional loss of welfare. We give truthful mechanisms with distortion O(√m log m) for k-winner elections, and distortion O(√m log m) when candidates have arbitrary costs, in elections with m candidates. These nearly match known lower bounds for ordinal mechanisms that ignore the strategic behavior. We further tighten these lower bounds and show that for truthful mechanisms our first upper bound is tight. Lastly, when agents decide between two candidates, we give tight bounds on the distortion for truthful mechanisms.