M. Studeny, V. Kratochvil, J. Vomlel: On irreducible min-balanced set systems. In Symbolic and Quantitative Approaches to Reasoning with Uncertainty (G. Kern-Isberner, Z. Ognjanovic eds.), Lecture Notes in Artificial Intelligence 1126, Springer-Verlag, Cham 2019, pp. 444-454.

Non-trivial minimal balanced systems (= collections) of sets are known to characterize through their induced linear inequalities the class of the so-called balanced (coalitional) games. In a recent paper a concept of an irreducible min-balanced (= minimal balanced) system of sets has been introduced and the irreducible systems have been shown to characterize through their induced inequalities the class of totally balanced games. In this paper we recall the relevant concepts and results, relate them to various contexts and offer a catalogue of permutational types of non-trivial min-balanced systems in which the irreducible systems are indicated. The present catalogue involves all types of such systems on sets with at most 5 elements; it has been obtained as a result of an alternative characterization of min-balanced systems.

AMS classification 91A12 68R05 68T30

balanced set system
irreducible min-balanced system
totally balanced games
exact games

A pdf version of a preprint (299kB) is available.
Moreover, the result of the work is also an interactive electronic catalogue of min-balanced system on at most 5 variables.

The manuscript builds on the following papers: