  We study stochastic evolution of optional games on simple graphs. There are
two strategies, A and B, whose interaction is described by a general payoff
matrix. In addition there are one or several possibilities to opt out from the
game by adopting loner strategies. Optional games lead to relaxed social
dilemmas. Here we explore the interaction between spatial structure and
optional games. We find that increasing the number of loner strategies (or
equivalently increasing mutational bias toward loner strategies) facilitates
evolution of cooperation both in well-mixed and in structured populations. We
derive various limits for weak selection and large population size. For some
cases we derive analytic results for strong selection. We also analyze strategy
selection numerically for finite selection intensity and discuss combined
effects of optionality and spatial structure.
