### Jorge Cortés

#### Professor

Distributed convergence to Nash equilibria by adversarial
networks with undirected topologies

B. Gharesifard, J. Cortés

Proceedings of the American Control Conference,
Montréal, Canada, 2012, pp. 5881-5886

### Abstract

This paper considers a class of strategic scenarios in
which two undirected 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 one. We synthesize
a distributed saddle-point algorithm that is
implementable via local interactions and establish its
convergence to the set of Nash equilibria for a class of
strictly concave-convex and locally Lipschitz objective
functions. Our algorithm synthesis builds on a
continuous-time optimization strategy for finding the
set of minimizers of a sum of convex functions in a
distributed way. As a byproduct, we show that this
strategy can be itself cast as a saddle-point dynamics
and use this fact to establish its asymptotic
convergence properties. 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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