Cooperative Game Theory: New Mathematical and Algorithmic Approaches.

合作博弈论是博弈论的一个分支，它提供了一个概念上简单而直观的数学框架来模拟涉及多个决策者(玩家)的协作环境。合作博弈的解决方案提供了不同的方式来在玩家之间分享利润或成本，同时确保协作的公平性和稳定性，同时考虑到任何玩家小组都可以选择组成自己的联盟。本项目的重点是最一般的合作对策--整数最大化对策。这些博弈出现在每个联盟中的玩家需要解决整数最大化问题以实现其联盟的最大利益的情况下。这项拟议的研究解决了一个基本问题，即在存在不确定性的新范式下，在合理大型博弈的背景下，如何分配收益。通常，在相关的情况下，将现实生活中的应用程序制定为合作游戏并不是一项困难的任务。阻碍合作博弈理论使用的部分是由于其组合结构而难以进行解的数值计算。这在整数最大化游戏中尤其如此，其中问题的输入集，即每个联盟可以创建的值，涉及求解指数级大的整数线性规划。提出的研究的第一部分提供了在相当大的整数最大化博弈中的收益分配的有效算法。此外，还提供了一个用于计算这些解决方案和展示现实世界应用程序的开放源码软件包。这有望将影响扩大到广泛的从业者和学者群体，他们希望将合作博弈论应用于利润/成本分担应用。该项目还旨在研究收益不确定的合作博弈。虽然不确定性是大多数决策问题的自然组成部分，但在合作博弈理论的文献中，这个问题在很大程度上被忽视了，目前还没有处理这些问题的严格框架。我们提出了一个新的框架，在这个框架中，面对不确定性，稳定和公平等基本概念得到了重新定义。

Cooperative game theory is a branch of game theory that offers a conceptually simple and intuitive mathematical framework to model collaborative settings involving multiple decision makers (players). Solutions of cooperative games offer different ways to share the profit or cost among the players in a way that ensures the fairness and stability of the collaboration, while considering the possibility that any subgroup of players has the option to form their own coalition. The focus of this project is on the most generic class of cooperative games - the integer maximisation games. These games arise in settings where the players in each coalition need to solve an integer maximisation problem to achieve the best interests of their coalition. This proposed research addresses a fundamental question of how to distribute payoff under a new paradigm with the presence of uncertainty and in the context of reasonably large games. Often, formulating a real-life application as a cooperative game, where relevant, is not a difficult task. The part that discourages the use of cooperative game theory is the difficulty in undertaking numerical computation of the solutions due to their combinatorial structures. This is particularly true in integer maximisation games where the set of inputs of the problem, i.e., the value that each coalition can create, involves solving an exponentially large number of integer linear programs. The first part of the proposed research provides efficient algorithms for payoff allocation in reasonably large integer maximisation games. In addition, an open-source software package for computing these solutions and showcase real-world applications is made available. This promises to extend the impact to wide groups of practitioners and academics who want to apply cooperative game theory to profit-/cost-sharing applications. The proposed project also aims to study cooperative games with uncertain payoffs. While uncertainty is a natural part of most decision-making problems, the issue has been largely ignored in the literature of cooperative game theory and there is currently no rigorous framework for handling these. We propose a new framework where fundamental concepts such as stability and fairness are redefined in the face of uncertainty.