Dynamics in network games with local coordination and global congestion effects

Several strategic interactions over social networks display both negative and positive externalities at the same time. E.g., participation to a social media website with limited resources is more appealing the more of your friends participate, while a large total number of participants may slow down the website (because of congestion effects) thus making it less appealing. Similarly, while there are often incentives to choose the same telephone company as the friends and relatives with whom you interact the most frequently, concentration of the market share in the hands of a single firm typically leads to higher costs because of the lack of competition. In this work, we study evolutionary dynamics in network games where the payoff of each player is influenced both by the actions of her neighbors in the network, and by the aggregate of the actions of all the players in the network. In particular, we consider cases where the payoff increases in the number of neighbors who choose the same action (local coordination effect) and decreases in the total number of players choosing the same action (global congestion effect). We study noisy best-response dynamics in networks which are the union of two complete graphs, and prove that the asymptotic behavior of the invariant probability distribution is characterized by two phase transitions with respect to a parameter measuring the relative strength of the local coordination with respect to the global congestion effects. Extensions to random networks with strong community structure are studied through simulations.