Abstract:

In many economic settings， who interacts with whom matters. More and more work needs the cooperation of the population. The mutual influence between people forms complex social networks. In recent years， how social networks influence individuals’ behavior has become a hot topic in economics research. The traditional literature on social network analysis mainly considers a single social network of the population. However， people often have different partners when faced with various activities. For example， college teachers must engage both in scientific research and complete teaching tasks. The output of scientific research is closely related to their collaborators’ input， and the efforts of their colleagues in teaching have a significant impact on the quality of education. In general， collaborators in research and teaching are different， which implies teachers in a college have different social ties in these two tasks. A heterogeneous network game model with multiple activities and characterizes the existence condition of the Nash equilibrium is constructed. Two applications of the model are also provided to enhance the economic implications. The first is the pricing problem for customers who gain network externalities from their neighbors who consume the same products. Compared to previous studies， it is assumed that consumers’ network externalities on different products are heterogeneous. The second is the resource allocation problem among agents with varying cooperation relationships in multiple tasks. Each agent’s marginal output depends on how many resources she obtained. The planner could design a resource allocation scheme to induce agents to exert the maximum total effort. Specifically， the content of this article is structured as follows： A model where players have heterogeneous social network relationships on different tasks is firstly built. Referring to the study of Bramoulé et al. （2014）， which connects the network game and the potential function， the maximum condition of the potential function is used to characterize the existence and uniqueness of the Nash equilibrium. This condition is closely related to the smallest eigenvalue of the technology matrix and the largest eigenvalue of the network matrix. Next， I focus on the uniqueness and the existence of the equilibrium. Under some assumptions of the maximum eigenvalue of network matrices， the equilibrium can be expressed in a closed form. Then comparative static analyses of the exogenous parameters of the game is made. Finally， two applications of the theoretical model are provided. In the pricing problem， it is revealed that the optimal monopoly price for each product is half of the customer’s marginal propensity to consume， and the equilibrium oligopoly prices are proportional to a weighted summation of all customers’ marginal consumption propensity. In the resource allocation problem， the planner and agents play a two-stage game. The planner first decides how to allocate resources for different tasks， and each agent then simultaneously selects the effort investment in each task. By backward induction， the planner should allocate resources to agents according to their inter-centrality measure in the network.

Key words: network games, peer effects, multiple activities, heterogeneous networks, potential game