Voting mechanics clarification

Question

I am reading the following paper: http://www.jstor.org/stable/10.1086/587624 and I am confused about one of the definitions at the start of section III. The auther defines $\bar{U} = \sum_{i=m}^n\lceil U_i \rceil^\varepsilon > 0$ as the minimal sum that party $Y$ must promise to win a majority, but it seems to me as though party $Y$ would only need to increase their votes by $\lceil \frac{n-m}{2} \rceil$ in order to win, so they would only need to offer $\sum_{i=m}^\frac{m+n}{2}\lceil U_i \rceil^\varepsilon = \sum_{i=m}^{\lceil\frac{|N|}{2}\rceil}\lceil U_i \rceil^\varepsilon$ in order to minimize their sum to win a majority, any additional votes after they get the median vote are completely unnecessary, what am i missing?