Abstract
We present a three-player Bayesian game for which there are no ∈-equilibria in Borel measurable strategies for small enough positive ∈, however there are non-measurable equilibria. The structure of the game employs a nonamenable 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.
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