Scoring systems, where each candidate gets a number of points from each voter based on the candidate’s position in the voter’s vote, are the most important broad class of voting rules. For example, plurality elections are scoring systems in which one gives one’s favorite candidate one point and gives each of the other candidates zero points. Each vector that satisfies certain sanity constraints is considered to define a scoring system, and many concrete scoring systems have been studied over the years.

In most papers, the scoring system is not itself fluid but rather provides the election framework. However, this paper looks at the complexity of determining--given as input the votes, the candidates, and a designated candidate p--whether a scoring system exists in which p is the sole winner of the election. The paper uses linear programming to establish that this problem has a polynomial-time solution in a variety of domains. The paper also proposes the more difficult question of the complexity of determining, for the case in which there already is a scoring system S tentatively in place, whether there exists a scoring system that is “not too far” from S and under which p is the sole winner. This paper’s direction and results will be of great interest to all who are interested in computational social choice theory.