On a relation associated with decision making by majority voting. (Q1432628)

Let \(X\) be a non-empty set and let \(D_1, \ldots, D_m \subset X\). The author considers the system of inclusions \(x \in D_j\) for \(j \in \{1, \ldots, m\}\). (1) Such a system is said to be consistent and \(x\) is said to be its solution if \(x \in \cap_{j=1}^{m} D_j\). One of the approaches to the analysis of such systems when they are inconsistent is based on the idea of majority vote and is related to the consideration of the so-called committee solutions. A finite sequence \(Q = (x^1, \ldots, x^q)\) is called a majority committee of \(q\) elements of system (1) if \(| \{i ; x^i \in D_j\}| > \frac{q}{2}\) for each \(j \in \{1, \ldots, m\}\). For an inconsistent system the number of elements in a minimal committee can be viewed as a measure of inconsistency in the system. In Theorem 1 the author answers the following question. Let system (1) have a committee of \(q\) elements. How small with respect to the cardinality of the entire system can the cardinality of its largest sub-system solvable by a committee of \(k\) elements be? The author associates a game \(\Gamma\) with the given problem and proves in Theorem 2 that \(\Gamma\) is solvable in mixed strategies.