A Bayesian game without ϵ-equilibria

We present a three player Bayesian game for which there are no ϵ-equilibria in Borel measurable strategies for small enough positive ϵ, however there are non-measurable equilibria. The structure of the game employs a non- amenable semi-group action corresponding to the knowledge of the players. The equilibrium property is related to the proper colouring of graphs and the Borel chromatic number; but rather than keeping adjacent vertices coloured differently there are algebraic conditions relating to the topology of the space and some ergodic operators.