  Evolutionary game theory is a common framework to study the evolution of
cooperation, where it is usually assumed that the same game is played in all
interactions. Here, we investigate a model where the game that is played by two
individuals is uniformly drawn from a sample of two different games. Using the
master equation approach we show that the random mixture of two games is
equivalent to play the average game when (i) the strategies are statistically
independent of the game distribution and (ii) the transition rates are linear
functions of the payoffs. We also use Monte-Carlo simulations in a two
dimensional lattice and mean-field techniques to investigate the scenario when
the two above conditions do not hold. We find that even outside of such
conditions, several quantities characterizing the mixed-games are still the
same as the ones obtained in the average game when the two games are not very
different.
