A game with no Bayesian approximate equilibria

Simon (2003) presented an example of a 3-player Bayesian game with no Bayesian equilibria, leaving open the question of whether or not there are games with no Bayesian approximate equilibria. We present an example of a Bayesian game with two players, two actions and a continuum of states that possesses no Bayesian approximate equilibria, thus resolving the question. As a side benefit we also have for the first time an an example of a 2-player Bayesian game with no Bayesian equilibria and an example of a strategic-form game with no approximate Nash equilibria. To achieve this we show a close relationship between strategic considerations in overlapping generations games and certain Bayesian games and then make use of an example by Y. Levy of an overlapping generations game with no stationary equilibria.