  In this paper we introduce the novel framework of distributionally robust
games. These are multi-player games where each player models the state of
nature using a worst-case distribution, also called adversarial distribution.
Thus each player's payoff depends on the other players' decisions and on the
decision of a virtual player (nature) who selects an adversarial distribution
of scenarios. This paper provides three main contributions. Firstly, the
distributionally robust game is formulated using the statistical notions of
$f$-divergence between two distributions, here represented by the adversarial
distribution, and the exact distribution. Secondly, the complexity of the
problem is significantly reduced by means of triality theory. Thirdly,
stochastic Bregman learning algorithms are proposed to speedup the computation
of robust equilibria. Finally, the theoretical findings are illustrated in a
convex setting and its limitations are tested with a non-convex non-concave
function.
