Mean field games model the strategic interaction among a large number of players by reducing the problem to two entities: the statistical distribution of all players on the one hand and a representative player on the other. The master equation, introduce by Lions, models this interaction in a single equation, whose independent variables are time, state, and distribution. It can be viewed as a nonlinear transport equation on an infinite dimensional space. Solving this transport equation by the method of characteristics is essentially equivalent to finding the unique Nash equilibrium. When the equilibrium is not unique, we seek selection principles, i.e. how to determine which equilibrium players should follow in practice. A natural question, from the mathematical point of view, is whether entropy solutions can be used as a selection principle. We will examine certain classes of mean field games to show that the question is rather subtle and yields both positive and negative results.