Weighted timed games are two-player zero-sum games played in a timed automaton equipped with integer weights. We consider optimal reachability objectives, in which one of the players, that we call Min, wants to reach a target location while minimising the cumulated weight. While knowing if Min has a strategy to guarantee a value lower than a given threshold is known to be undecidable (with two or more clocks), several conditions, one of them being divergence, have been given to recover decidability. In such weighted timed games (like in untimed weighted games in the presence of negative weights), Min may need finite memory to play (close to) optimally. This is thus tempting to try to emulate this finite memory with other strategic capabilities. In this work, we allow the players to use stochastic decisions, both in the choice of transitions and of timing delays. We give a definition of the expected value in weighted timed games. We then show that, in divergent weighted timed games as well as in (untimed) weighted games (that we call shortest-path games in the following), the stochastic value is indeed equal to the classical (deterministic) value, thus proving that Min can guarantee the same value while only using stochastic choices, and no memory.

翻译：加权时间博弈是在配备整数权重的定时自动机中进行的两人零和博弈。我们考虑最优可达性目标，其中被称为Min的一方希望到达目标位置，同时最小化累积权重。尽管已知判断Min是否拥有保证值低于给定阈值的策略是不可判定的（在具有两个或更多时钟的情况下），已有若干条件（其中之一是发散性）被提出以恢复可判定性。在此类加权时间博弈中（类似于存在负权重的非定时加权博弈），Min可能需要有限记忆来实现（接近）最优策略。因此，尝试用其他策略能力来模拟这种有限记忆颇具吸引力。在本研究中，我们允许参与者使用随机决策，包括在转移选择和时序延迟选择中。我们给出了加权时间博弈中期望值的定义。随后证明，在发散加权时间博弈以及（非定时的）加权博弈（下文称为最短路径博弈）中，随机值确实等于经典（确定性）值，从而证明Min仅通过随机选择（无需记忆）即可保证相同的值。