An ongoing scope of research in multi-objective Bayesian optimization is to extend its applicability to a large number of objectives. Recovering the set of optimal compromise solution generally requires lots of observations while being less inter-pretable, since this set tends to grow larger with the number of objectives. We thus propose to focus on a choice of a specific solution originating from game theory, the Kalai-Smorodinski solution, that possesses attractive properties. We further make it insensitive to a monotone transformation of the objectives by considering the objectives in the copula space. A tailored algorithm is proposed to search for the solution, which is tested on a synthetic problem.