The methods and results involve linear inequalities but not extrema. The main result is a distinct improvement of D. B. Gillies’ theorem [Ann. Math. Stud. 40, 47–85 (1959; Zbl 0085.13106)] that a positive fraction of all games have unique solutions. There are a number of other results on necessary and/or sufficient conditions for existence and nature of cores or \(k\)-quotas, and for a 1-dimensional core to be a solution. Systematic and effective use is made of the (finite) set of all expressions of \((1,1,\cdots,1)\) as a non-negative combination of independent 0-1 vectors.