Network games are widely used as a model for selfish resource-allocation problems. In the classical model, each player selects a path connecting her source and target vertices. The cost of traversing an edge depends on the {\em load}; namely, number of players that traverse it. Thus, it abstracts the fact that different users may use a resource at different times and for different durations, which plays an important role in determining the costs of the users in reality. For example, when transmitting packets in a communication network, routing traffic in a road network, or processing a task in a production system, actual sharing and congestion of resources crucially depends on time.
In \cite{AGK17}, we introduced {\em timed network games}, which add a time component to network games. Each vertex $v$ in the network is associated with a cost function, mapping the load on $v$ to the price that a player pays for staying in $v$ for one time unit with this load. Each edge in the network is guarded by the time intervals in which it can be traversed, which forces the players to spend time in the vertices. In this work we significantly extend the way time can be referred to in timed network games. In the model we study, the network is equipped with {\em clocks}, and, as in timed automata, edges are guarded by constraints on the values of the clocks, and their traversal may involve a reset of some clocks. We argue that the stronger model captures many realistic networks. The addition of clocks breaks the techniques we developed in \cite{AGK17} and we develop new techniques in order to show that positive results on classic network games carry over to the stronger timed setting.
定价时间博弈（Priced Timed Games，PTG）是在定价时间自动机的无限图上进行的两人零和博弈，两人轮流选择过渡以优化到达目标状态的成本。Bouyer et al.和Alfred，Bernadsky和Madhusudan独立地提出了在价格上的一定发散限制下求解具有非负价格的PTG的算法。Brihaye、Bruyere和Raskin后来证明了在没有这种发散限制的情况下，最优策略综合问题的不可判定性，从而为这种限制提供了正当理由。具有一个时钟的PTG的这个问题长期以来一直被证明是在多项式时间中，然而由Hansen、Ibsen-Jensen和Miltersen提出的当前最著名的算法是指数算法。我们通过研究具有负价格和正价格的PTG来扩展这一图景。我们完善了最优策略综合问题的不可判定性结果，并显示了几个变种的最优可达性成本目标，包括可达性成本，时间有界的可达性成本，重复可达性成本目标的不可判定性。我们还确定了一个子类的双值价格率，并给出了一个伪多项式算法，部分回答了一个时钟PTG的复杂性的猜想。
Non-Blind Strategies in Timed Network Congestion Games
网络拥塞博弈是一种用于推理网络中路由问题的方便模型：代理必须从源节点移动到目标节点，同时避免拥塞，其成本取决于使用同一链路的玩家数量。在过去的40年里，网络拥塞博弈得到了广泛的研究，而其扩展与时间限制被认为是最近。网络拥塞博弈的大多数结果都考虑了盲策略：它们是静态的，不适应其他参与者选择的策略。我们扩展了[Bertrand et al.，动态网络拥塞游戏。FSTTCS'20]到定时网络拥塞游戏，其中边缘的可用性取决于（离散）时间。我们证明，计算纳什均衡满足一定的约束的总成本（特别是计算最好和最坏的纳什均衡），并计算社会最优，可以在指数空间中实现。如果所有参与者都有相同的源和目标，则可以在多项式空间中计算社会最优值。
