TITLE Completion of Gibbard-Satterthwaite impossibility theorem; range voting and voter honesty AUTHOR Warren D. Smith DATE August 2006 ABSTRACT Let $S$ be a reasonable'' single-winner voting system. (The precise definition of reasonable'' will vary from theorem to theorem and is not stated in this abstract.) Then (a) For each $C \ge 3$: if $S$ is based on rank-order ballots (with equalities either permitted or forbidden) then there exist $C$-candidate election situations with complete information'' (i.e, the voter knows everybody else's votes) in which voting honestly is not best voting strategy. (b) If $S$ is range voting, then in every $C$-candidate election situation ($C \ge 1$) with complete information, and also in every $C$-candidate election situation with incomplete information ($1 \le C \le 3$), there is a semi-honest'' vote (i.e, in which the $<$, $>$, and $=$ relations among the candidate-scores are valid for a \emph{limit} of scores obeying the honest relations) which is strategically best. (c) If $S$ is based on either rank-order ballots (with equalities either permitted or forbidden) or candidate-scoring range vote'' type ballots where each candidate is rated with a real number, then for each $C \ge 4$ there exist $C$-candidate election situations with incomplete information in which no \emph{semi}-honest vote is best voting strategy. Part (a) is Gibbard \& Satterthwaite's impossibility theorem. These results show a sense in which range voting is a best possible deterministic single-winner voting system. These theorems also hold for certain classes of probabilistic voting systems (in which chance plays a role in determining the winner) but not all. We conclude by introducing and beginning the study of the Nash model'' of voter honesty. KEYWORDS Voter honesty, semi-honesty, strategy, strong Nash equilibria, Nash model.