Self-organized criticality is an important framework for understanding the emergence of scale-free natural phenomena. Cellular automata provide simple interesting models in which to study self-organized criticality. We consider the dynamics of a new class of cellular automata which are constructed as natural spatial extensions of evolutionary game theory. This construction yields a discrete one-parameter family of cellular automata. We show that there is a range of parameter values for which this system exhibits complex dynamics with long range correlations between states in both time and space. In this region the dynamics evolve to a self-organized critical state in which structures exist on all time and length scales, and the relevant statistical measures have power law behaviour.