Ship collision avoidance has always been one of the classic topics in the field of marine research. In traditional encounter situations, officers on watch (OOWs) usually use a very high frequency (VHF) radio to coordinate each other. In recent years, with the continuous development of autonomous ships, there will be a mixed situation where ships of different levels of autonomy coexist at the same time. Under such a scenario, different decision makers have different perceptions of the current scene and different decision-making logic, so conventional collision avoidance methods may not be applicable. Therefore, this paper proposes a collaborative collision avoidance strategy for multi-ship collision avoidance under mixed scenarios. It builds a multi-ship cooperative network to determine cooperative objects and timing, at the same time. Based on a cooperative game model, a global collision avoidance responsibility distribution that satisfies group rationality and individual rationality is realised, and finally achieves a collaborative strategy according to the generalised reciprocal velocity obstacle (GRVO) algorithm. Case studies show that the strategy proposed in this paper can make all ships pass each other clearly and safely.

Considering the shortcomings of current methods for real-time resolution of two-aircraft flight conflicts, a geometric optimal conflict resolution and recovery method based on the velocity obstacle method for two aircraft and a cooperative conflict resolution method for multiple aircraft are proposed. The conflict type was determined according to the relative position and velocity of the aircraft, and a corresponding conflict mitigation strategy was selected. A resolution manoeuvre and a recovery manoeuvre were performed. On the basis of a two-aircraft conflict resolution model, a multi-aircraft cooperative conflict resolution game was constructed to identify an optimal solution for maximising group welfare. The solution and recovery method is simple and effective, and no new flight conflicts are introduced during track recovery. For multi-aircraft conflict resolution, an equilibrium point that maximises the welfare function of the group was identified, and thus, an optimal strategy for multi-aircraft conflict resolution was obtained.

The main goal is to illustrate that the so-called indirect function of a cooperative game in characteristic function form is applicable to determine the nucleolus for a subclass of coalitional games called compromise stable transferable utility (TU) games. In accordance with the Fenchel–Moreau theory on conjugate functions, the indirect function is known as the dual representation of the characteristic function of the coalitional game. The key feature of a compromise stable TU game is the coincidence of its core with a box prescribed by certain upper and lower core bounds. For the purpose of the determination of the nucleolus, we benefit from the interrelationship between the indirect function and the prekernel of coalitional TU games. The class of compromise stable TU games contains the subclasses of clan games, big boss games and $1$- and $2$-convex $n$-person TU games. As an adjunct, this paper reports the indirect function of clan games for the purpose of determining their nucleolus.