Some finite and symmetric two-player games have no (pure or mixed) symmetric Nash equilibrium when played by partly morally motivated players. The reason is that the "right thing to do" may be not to randomize. We analyze this issue both under complete information between equally moral players and under incomplete information between arbitrarily moral players. We provide necessary and sufficient conditions for the existence of equilibrium and illustrate the results with examples and counter-examples.
Economic Theory, 2020, forthcoming