Characteristic functions in cooperative differential games on networks

Abstract

In the paper, a class of cooperative differential games on networks is considered. In such games, the new characteristic function is introduced based on the possibility of stopping interaction by players outside the coalition in each time instant or imposing sanction on players from the coalition. This gives the real possibility for the computation of characteristic function. Thus, the characteristic function is evaluated along the cooperative trajectory. It measures the worth of coalitions under the process of cooperation instead of under minimax confrontation or the Nash non-cooperative stance. The approach essentially simplifies the construction of the characteristic function and cooperative solutions such as the Shapley value, Core, τ-value and others. Also, it is proved that the proposed characteristic function is convex, time consistent, and as a result, the Shapley value belongs to the Core and is time consistent. Also, a modification of the dynamic game on networks, namely, dynamic network game with partner sets is considered. In this case, payoffs of a given player depend on his actions and the actions of the players from his partner set. Using previous ideas, the special type of characteristic function is introduced, and cooperative solutions are proposed.

1. Introduction Recently, many interesting problems have been modelled with the help of dynamic games on networks. Among them the transportation problem or problem of influence in social networks. We have to mention the first researches in the field of dynamic network games ([2]), ([22]), ([8]), ([27]), ([7]), ([10]). An obvious continuation of research in the field of dynamic games is to expand them to the class of cooperative dynamic games on networks (the following papers should be noted ([11]), ([3]), and the papers ([24]), ([16]), ([14]) where the new characteristic function in differential cooperative network game was introduced in a special case when the payoffs of players depend only upon their actions and actions of neighbours in the network. Different properties of the cooperative solutions of dynamic network games are investigated in ([23,25,26]). When we speak about cooperation, we have to define the characteristic function. There are many different ways to define the characteristic function in the game. In the paper, we review the results we have obtained in recent years, which relate to differential games on networks. Namely, we introduce different types of characteristic functions based on the assumption that players can stop interactions with other players at any time instant or pose sanctions on players in the coalition. For each case we present an example and construct cooperative solutions: the Shapley value and -value.