Strategic Coalitions in Stochastic Games

The article introduces a notion of a stochastic game with failure states and proposes two logical systems with modality "coalition has a strategy to transition to a non-failure state with a given probability while achieving a given goal." The logical properties of this modality depend on whether the modal language allows the empty coalition. The main technical results are a completeness theorem for a logical system with the empty coalition, a strong completeness theorem for the logical system without the empty coalition, and an incompleteness theorem which shows that there is no strongly complete logical system in the language with the empty coalition.1. Introduction In this article we study coalition power in stochastic games. An example of such a game is the road situation depicted in Figure 1. In this situation, self-driving car a is trying to pass self-driving car b . Unexpectedly, a truck moving in the opposite direction appears on the road. For the sake of simplicity, we assume that cars a and b have only three strategies: slowdown (), maintain the current speed (0), and accelerate (). We also assume that the truck is too heavy to significantly change the speed before a possible collision. The diagram in Figure 2 describes probabilities of different outcomes of all possible combinations of actions of cars a and b . This diagram has five states: state p is the current ("passing") state of the system.