1‐concave basis for TU games and the library game

The study of 1‐convex/1‐concave TU games possessing a nonempty core and for which the nucleolus is linear was initiated by Driessen and Tijs (1983) and Driessen (1985). However, until recently appealing abstract and practical examples of these classes of games were missing. The paper solves these drawbacks. We introduce a 1‐concave basis for the entire space of all TU games wherefrom it follows that every TU game is either 1‐convex/1‐concave or is a sum of 1‐convex and 1‐concave games. Thus we may conclude that the classes of 1‐convex/1‐concave games constitute rather considerable subsets in the entire game space. On the other hand, an appealing practical example of 1‐concave game has cropped up in Sales’s study (2002) of Catalan university library consortium for subscription to journals issued by Kluwer publishing house. The so‐called library game turns out to be decomposable into suitably chosen 1‐concave games of the basis mentioned above.