  The distributed computation of Nash equilibria is assuming growing relevance
in engineering where such problems emerge in the context of distributed
control. Accordingly, we present schemes for computing equilibria of two
classes of static stochastic convex games complicated by a parametric
misspecification, a natural concern in the control of large-scale networked
engineered system. In both schemes, players learn the equilibrium strategy
while resolving the misspecification: (1) Monotone stochastic Nash games: We
present a set of coupled stochastic approximation schemes distributed across
agents in which the first scheme updates each agent's strategy via a projected
(stochastic) gradient step while the second scheme updates every agent's belief
regarding its misspecified parameter using an independently specified learning
problem. We proceed to show that the produced sequences converge in an
almost-sure sense to the true equilibrium strategy and the true parameter,
respectively. Surprisingly, convergence in the equilibrium strategy achieves
the optimal rate of convergence in a mean-squared sense with a quantifiable
degradation in the rate constant; (2) Stochastic Nash-Cournot games with
unobservable aggregate output: We refine (1) to a Cournot setting where we
assume that the tuple of strategies is unobservable while payoff functions and
strategy sets are public knowledge through a common knowledge assumption. By
utilizing observations of noise-corrupted prices, iterative fixed-point schemes
are developed, allowing for simultaneously learning the equilibrium strategies
and the misspecified parameter in an almost-sure sense.

Optimization and Control