Weighted position value for Network games

In Network games under cooperative framework, the position value is a link based allocation rule. It is obtained from the Shapley value of an associated cooperative game where the links of the network are considered players. The Shapley value of each of the links is then divided equally among the players who form those links. The inherent assumption is that the value is indifferent to the weights of the players in the network. Depending on how much central a player is in the network, or the ability of making links with other players etc., for example, players can be considered to have weights. Thus, in such situations, dividing the Shapley value equally among the players can be an over-simplistic notion. We propose a generalised version of the position value: the weighted position value that allocates the Shapley shares proportional to the players' weights. These weights of the players are exogenously given. We provide two axiomatic characterizations of our value. Finally, a bidding mechanism is formulated to show that any sub-game perfect equilibrium (SPE) of this mechanism coincides with the weighted position value.