Cooperative games with nonempty core are called balanced, and the set of balanced games is a polyhedron. Given a game with empty core, we look for the closest balanced game, in the sense of the (weighted) Euclidean distance, i.e., the orthogonal projection of the game on the set of balanced games. Besides an analytical approach which becomes rapidly intractable, we propose a fast algorithm to find the closest balanced game, avoiding exponential complexity for the optimization problem, and being able to run up to 20 players. We show experimentally that the probability that the closest game has a core reduced to a singleton tends to 1 when the number of players grow. We provide a mathematical proof that the proportion of facets whose games have a non-singleton core tends to 0 when the number of players grow, by finding an expression of the aymptotic growth of the number of minimal balanced collections. This permits to prove mathematically the experimental result. Consequently, taking the core of the projected game defines a new solution concept, which we call least square core due to its analogy with the least core, and our result shows that the probability that this is a point solution tends to 1 when the number of players grow.

翻译：具有非空核心的合作博弈称为平衡博弈，平衡博弈的集合构成一个多面体。给定一个核心为空的博弈，我们寻找在（加权）欧几里得距离意义下最接近的平衡博弈，即该博弈在平衡博弈集合上的正交投影。除了计算复杂度迅速变得难以处理的解析方法外，我们提出了一种快速算法来寻找最接近的平衡博弈，该算法避免了优化问题的指数级复杂度，并且能够处理多达20名参与者。实验表明，当参与者数量增加时，最接近博弈的核心缩减为单点集的概率趋近于1。我们通过找到最小平衡集合数量的渐近增长表达式，从数学上证明了当参与者数量增加时，其博弈具有非单点核心的刻面比例趋近于0，从而从数学上验证了实验结果。因此，取投影博弈的核心定义了一个新的解概念，由于其与最小核心的类比，我们称之为最小二乘核心，并且我们的结果表明，当参与者数量增加时，该解为点解的概率趋近于1。