  Distributed Nash equilibrium seeking for games in uncertain networked systems
without a prior knowledge about control directions is explored in this paper.
More specifically, the dynamics of the players are supposed to be first-order
or second-order systems in which the control directions are unknown and there
are parametric uncertainties. To achieve Nash equilibrium seeking in a
distributed way, Nussbaum function based strategies are proposed through
separately designing an optimization module and a state regulation module. The
optimization module generates a reference trajectory, that can search for the
Nash equilibrium, for the state regulation module. The state regulator is
designed to steer the players' actions to the reference trajectory. An adaptive
law is included in the state regulation module to compensate for the uncertain
parameter in the players' dynamics and the Nussbaum function is included to
address the unavailability of the control directions. Fully distributed
implementations of the proposed algorithms are discussed and investigated.
Through our analytical explorations, we show that the proposed seeking
strategies can drive the players' actions to the Nash equilibrium
asymptotically without requiring the homogeneity of the players' unknown
control directions based on Barbalat's lemma. A numerical example is given to
support the theoretical analysis of the proposed algorithms.
