A network game assigns a level of collectively generated wealth to every network that can form on a given set of players. A variable network game combines a network game with a network formation probability distribution, describing certain restrictions on network formation. Expected levels of collectively generated wealth and expected individual payoffs can be formulated in this setting. We investigate properties of the resulting expected wealth levels as well as the expected variants of well-established network game values as allocation rules that assign to every variable network game a payoff to the players in a variable network game. We establish two axiomatizations of the Expected Myerson Value, originally formulated and proven on the class of communication situations, based on the well-established component balance, equal bargaining power and balanced contributions properties. Furthermore, we extend an established axiomatization of the Position Value based on the balanced link contribution property to the Expected Position Value.

We develop a generalisation of the $$\lambda $$ λ -Core solution for non-cooperative games in normal form. We show that this generalised $$\lambda $$ λ -Core is non-empty for the class of separable games that admit a socially optimal Nash equilibrium. Examples are provided that indicate that non-emptiness of the generalised $$\lambda $$ λ -Core cannot be expected for large classes of normal form games.