Motivated by the markets operating on fast time scales, we present a framework for online coalitional games with time-varying coalitional values and propose real-time payoff distribution mechanisms. Specifically, we design two online distributed algorithms to track the Shapley value and the core, the two most widely studied payoff distribution criteria in coalitional game theory. We show that the payoff distribution trajectory resulting from our proposed algorithms converges to a neighborhood of the time-varying solutions. We adopt an operator-theoretic perspective to show the convergence of our algorithms. Numerical simulations of a real-time local electricity market and cooperative energy forecasting market illustrate the performance of our algorithms: {the difference between online payoffs and static payoffs (Shapley and the core) to the participants is little; online algorithms considerably improve the scalability of the mechanism with respect to the number of market participants.

翻译：受快速时间尺度市场运作的启发，本文提出了一种具有时变联盟值的在线合作博弈框架，并设计了实时收益分配机制。具体而言，我们构建了两种在线分布式算法，以追踪合作博弈论中两种最广泛研究的收益分配准则——沙普利值和核心解。研究表明，所提算法产生的收益分配轨迹收敛到时变解的邻域内。我们采用算子理论视角证明了算法的收敛性。通过实时本地电力市场与协作能源预测市场的数值仿真，验证了算法的性能：在线收益与静态收益（沙普利值和核心解）之间的参与者差异极小；在线算法显著提升了机制相对于市场参与者数量的可扩展性。