Accurate risk quantification and reachability analysis are crucial for safe control and learning, but sampling from rare events, risky states, or long-term trajectories can be prohibitively costly. Motivated by this, we study how to estimate the long-term safety probability of maximally safe actions without sufficient coverage of samples from risky states and long-term trajectories. The use of maximal safety probability in control and learning is expected to avoid conservative behaviors due to over-approximation of risk. Here, we first show that long-term safety probability, which is multiplicative in time, can be converted into additive costs and be solved using standard reinforcement learning methods. We then derive this probability as solutions of partial differential equations (PDEs) and propose Physics-Informed Reinforcement Learning (PIRL) algorithm. The proposed method can learn using sparse rewards because the physics constraints help propagate risk information through neighbors. This suggests that, for the purpose of extracting more information for efficient learning, physics constraints can serve as an alternative to reward shaping. The proposed method can also estimate long-term risk using short-term samples and deduce the risk of unsampled states. This feature is in stark contrast with the unconstrained deep RL that demands sufficient data coverage. These merits of the proposed method are demonstrated in numerical simulation.

A key issue in cooperative game theory is coalitional stability, usually captured by the notion of the core---the set of outcomes that are resistant to group deviations. However, some coalitional games have empty cores, and any outcome in such a game is unstable. We investigate the possibility of stabilizing a coalitional game by using subsidies. We consider scenarios where an external party that is interested in having the players work together offers a supplemental payment to the grand coalition, or, more generally, a particular coalition structure. This payment is conditional on players not deviating from this coalition structure, and may be divided among the players in any way they wish. We define the cost of stability as the minimum external payment that stabilizes the game. We provide tight bounds on the cost of stability, both for games where the coalitional values are nonnegative (profit-sharing games) and for games where the coalitional values are nonpositive (cost-sharing games), under natural assumptions on the characteristic function, such as superadditivity, anonymity, or both. We also investigate the relationship between the cost of stability and several variants of the least core. Finally, we study the computational complexity of problems related to the cost of stability, with a focus on weighted voting games.