By a theory we mean a subset of propositions that is closed under consequences and conjunctions. When the set of all propositions is a Boolean algebra, theories are filters in this algebra. When there are several theorists, they can also have theories about what theories the other theorists...

We study repeated prize allocation when the discount factors of the agents are unequal. The feasible set of payoffs behaves badly: the Pareto frontier is everywhere discontinuous and there is an open subset of (high) discount factors under which the feasible set is totally disconnected.