When evaluating two students using two types of tests, assigning weights to their scores can be a useful approach. Consider the following scenarios: Suppose student a₁'s scores are (40, 80) and student a₂'s scores are (60, 60). With equal weights of 1: 1, their overall evaluations are the same. With weights of 2: 1, student a₂ outperforms student a₁. With weights of 1: 2, the conclusion is reversed. This observation indicates that one can control the overall evaluation by weights. This paper shows the same is true for not only individuals but also groups (sets). The mathematical foundation for this approach is based on Borsuk's antipodal theorem (1).
Hidefumi Kawasaki (Wed,) studied this question.