  We investigate two-winner election problem seeking to minimize the social
cost. We are interested in strategy-proof mechanisms where each voter only
reports a single candidate. In our model, candidates and voters are located in
Euclidean space and candidates' locations are known to the mechanism. The
quality of a mechanism is measured by its distortion, defined as the worst-case
ratio between the social cost achieved by the mechanism and the optimal one. We
find that the ratio between the maximum and minimum distances among every two
candidates plays a vital role in the distortion of mechanisms. When there are
three candidates, the problem is solved mainly by previous work. We mainly
focus on the problem with at least four candidates. When voters and candidates
are embedded in 1-dimensional space, we establish several lower bounds of the
distortion. When voters and candidates are embedded in at least 3-dimensional
space, we give a tight bound of the distortion.
