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）的协作设置。合作博弈的解决方案提供了不同的方式在参与者之间分享利润或成本，以确保合作的公平性和稳定性，同时考虑到任何参与者的子群体都有可能组成自己的联盟。这个项目的重点是最普遍的一类合作博弈——整数最大化博弈。在这些博弈中，每个联盟中的参与者需要解决一个整数最大化问题，以实现联盟的最佳利益。这一提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，在合理的大型博弈背景下，如何分配收益。通常情况下，将现实生活中的应用程序描述为合作游戏并不是一项困难的任务。阻碍合作博弈论应用的部分是由于其组合结构而难以进行数值计算。这在整数最大化游戏中尤其如此，因为问题的输入集，即每个联盟可以创造的价值，涉及解决指数数量的整数线性程序。本研究的第一部分提供了在合理大的整数最大化博弈中分配收益的有效算法。此外，还提供了用于计算这些解决方案和展示实际应用程序的开源软件包。这有望将影响扩大到希望将合作博弈论应用于利润/成本共享应用的广泛从业者和学者群体。该项目还旨在研究具有不确定收益的合作博弈。虽然不确定性是大多数决策问题的自然组成部分，但在合作博弈论的文献中，这个问题在很大程度上被忽视了，目前也没有严格的框架来处理这些问题。我们提出了一个新的框架，在面对不确定性时，重新定义稳定和公平等基本概念。

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.