Network games are an important class of games that model agent interactions in networked systems, where players are situated at the nodes of a graph and their payoffs depend on the actions taken by their neighbors. We extend the classical framework by considering a game model where the strategies are positive semidefinite matrices having trace one. These (continuous) games can serve as a simple model of quantum strategic interactions. We focus on the zero-sum case, where the sum of all players' payoffs is equal to zero. We establish that in this class of games, Nash equilibria can be characterized as the projection of a spectrahedron, that is, the feasible region of a semidefinite program. Furthermore, we demonstrate that determining whether a game is a semidefinite network game is equivalent to deciding if the value of a semidefinite program is zero. Beyond the zero-sum case, we characterize Nash equilibria as the solutions of a semidefinite linear complementarity problem.

翻译：网络博弈是一类重要的博弈模型，用于刻画网络化系统中智能体之间的交互：玩家位于图的节点上，其收益取决于邻居节点的策略选择。我们通过考虑策略为迹为1的半定正定矩阵的博弈模型，对经典框架进行了扩展。这些（连续）博弈可视为量子战略互动的简化模型。我们重点研究零和情形——即所有玩家收益之和为零的情形。研究表明，在此类博弈中，纳什均衡可表征为谱面体（即半定规划的可行域）的投影。进一步，我们证明判定一个博弈是否为半定网络博弈等价于判定某个半定规划的值是否为零。在零和情形之外，我们将纳什均衡刻画为半定线性互补问题的解。