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We consider an incomplete information network game in which agents' information is restricted only to the identity of their immediate neighbors. Agents form beliefs about the adjacency pattern of others and play a linear-quadratic effort game to maximize interim payoffs. We establish the existence and uniqueness of Bayesian-Nash equilibria in pure strategies. In this equilibrium agents use local information, i.e., knowledge of their direct connections to make inferences about the complementarity strength of their actions with those of other agents which is given by their updated beliefs regarding the number of walks they have in the network. Our model clearly demonstrates how asymmetric information based on network position and the identity of agents affect strategic behavior in such network games. We also characterize agent behavior in equilibria under different forms of ex-ante prior beliefs such as uniform priors over the set of all networks, Erdos-Renyi network generation, and homophilic linkage.

In cooperative games with transferable utilities, the Shapley value is an extreme case of marginalism while the Equal Division rule is an extreme case of egalitarianism. The Shapley value does not assign anything to the non-productive players and the Equal Division rule does not concern itself to the relative efficiency of the players in generating a resource. However, in real life situations neither of them is a good fit for the fair distribution of resources as the society is neither devoid of solidarity nor it can be indifferent to rewarding the relatively more productive players. Thus a trade-off between these two extreme cases has caught attention from many researchers. In this paper, we obtain a new value for cooperative games with transferable utilities that adopts egalitarianism in smaller coalitions on one hand and on the other hand takes care of the players' marginal productivity in sufficiently large coalitions. Our value is identical with the Shapley value on one extreme and the Equal Division rule on the other extreme. We provide four characterizations of the value using variants of standard axioms in the literature. We have also developed a strategic implementation mechanism of our value in sub-game perfect Nash equilibrium.