Volume 7 - Number 2 | July

How to reach a joint decision with the smallest need for compromise

Sofia Holguin, and Olga Kosheleva


Usually, people's interests do not match perfectly. So when several people need to make a joint decision, they need to compromise. The more people one has to coordinate the decision with, the fewer chances that each person's preferences will be properly taken into account. Therefore, when a large group of people need to make a decision, it is desirable to make sure that this decision can be reached by dividing all the people into small-size groups so that this decision can reach a compromise between the members of each group. The study's objective is to analyze when such a compromise is possible. Design/methodology/approach
In this paper, the authors use a recent mathematical result about convex sets to analyze this problem and to come up with an optimal size of such groups. Findings
The authors find the smallest group size for which a joint decision is possible. Specifically, the authors show that in situations where each alternative is characterized by n quantities, it is possible to have a joint decision if the participants are divided into groups of size n -- and, in general, no such decision is possible if the participants are divided into groups of size n -- 1. Originality/value
The main novelty of this paper is that, first, it formulates the problem, which, to the best of the authors’ knowledge, was never formulated in this way before, and, second, that it provides a solution to this problem.


  1. Bárány, I. (2022), “Helly-type problems”, Bulletin (New Series) of the American Mathematical Society, Vol. 59 No. 4, pp. 471-502.
  2. Birch, B.J. (1959), “On 3N points in a plane”, Proceedings of the Cambridge Philosophical Society, Vol. 55, pp. 289-293.
  3. Frick, F. and Soberón, P. (2020), “The topological Tverberg problem beyond prime powers”, posted on arXiv arXiv:2005.05251, available at: https://arxiv.org/abs/2005.05251
  4. Rockafeller, R.T. (1997), Convex Analysis, Princeton University Press, Princeton, NJ.