Bi-cooperative network games: A link-based allocation rule

We study the notion of a bi-cooperative network game and obtain a link-based allocation rule for the corresponding class. Unlike classical network games where players have single motive of forming (or not forming) a network, a player under a bi-cooperative set up has two more variations: all her links attribute exclusively to her positive or negative contributions, or few links are positive while few others are negative. We extend the position value, an important link-based allocation rule in classical network games, and show that it is the only rule for all the three variations of a bi-cooperative network game.