### Jorge Cortés

#### Professor

Distributed convergence to Nash equilibria in two-network
zero-sum games

B. Gharesifard, J. Cortés

Automatica 49 (6) (2013), 1683-1692

### Abstract

This paper considers a class of strategic scenarios in
which two networks 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. For the case when the
interaction topology of each network is undirected, we
synthesize a distributed saddle-point strategy and
establish its convergence to the Nash equilibrium for
the class of strictly concave-convex and locally
Lipschitz objective functions. We also show that this
dynamics does not converge in general if the topologies
are directed. This justifies the introduction, in the
directed case, of a generalization of this distributed
dynamics which we show converges to the Nash equilibrium
for the class of strictly concave-convex differentiable
functions with locally Lipschitz gradients. The
technical approach combines tools from algebraic graph
theory, nonsmooth analysis, set-valued dynamical
systems, and game theory.

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Mechanical and Aerospace Engineering,
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