Distributed convergence to Nash equilibria by adversarial networks with directed topologies

This paper considers a class of strategic scenarios in which two cooperative groups of agents have opposing objectives with regards to the optimization of a common objective function. In the resulting zero-sum game, individual agents collaborate with neighbors in their respective network and have only partial knowledge of the state of the agents in the other network. We consider scenarios where the interaction topology within each cooperative network is given by a strongly connected and weight-balanced directed graph. We introduce a provably-correct distributed dynamics which converges to the set of Nash equilibria when the objective function is strictly concave-convex, differentiable, with globally Lipschitz gradient. The technical approach combines tools from algebraic graph theory, dynamical systems, convex analysis, and game theory.