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We introduce a new notion of connectivity, what we call weak connectivity, in a directed network where communication is one-way, and show that weak connectivity is equivalent to the usual concept of connectivity if the outdegree of each node is at most one, referred as the [DC] condition. Based on weak connectivity, we define an allocation rule in a directed network by applying the Shapley value type of consideration. We show that the allocation rule is the unique allocation rule satisfying component efficiency and equal bargaining power under the [DC] condition. If the [DC] condition does not hold, it fails to satisfy component efficiency, but can be shown to be the only allocation rule that satisfies equal bargaining power and quasi-component efficiency which is a weaker property.

We analyse a private firm's decision of whether to refuse to sell to a particular group of consumers whose interaction with other consumers generates negative externalities. The literature has rarely incorporated this motive directly into the firm's profit‐maximisation problem. Discriminatory refusal‐to‐sell policies can increase profits and consumer utility among those affected by the negative externality. Of course it also reduces utility among consumers who are refused, raising the possibility of an indeterminate effect on social welfare. We obtain a stark and rather surprising result: The refusal‐to‐sell policy is socially optimal whenever it is individually optimal for a profit‐maximising firm to adopt such a policy. No legislation or regulation is required from a social‐welfare perspective (under the assumptions used in the specification of the social welfare function). We prove this result analytically for the case of linear demand functions. Numerical simulations show that the result also holds for constant‐price‐elasticity demand functions.

In this paper, we examine the profitability of a merger when agents interact on a network. We assume that the value function is sizedependent, i.e. the value of a coalition does not depend on the identities of the coalition but only on each size of its components. Under the assumption that the third difference of the value function with respect to the size of each component is nonnegative, which is a weaker condition of increasing complementarity by Segal in 2003, we show that a merger of any subset of agents cannot be profitable if the underlying network is cycle-complete.