Explaining the emergence of cooperative behaviours in a selfish world remains a major challenge for sociobiology. The iterated prisoner's dilemma offers a well-studied metaphor with which to explore theoretically the evolution of cooperation by reciprocation. Our current understanding is that 'tit-for-tat' should be the very first step (if not the aim) of evolution towards cooperation, but that mobility of the players in space seems to raise a devastating obstacle to the spread of tit-for-tat, by allowing egoists to exploit cooperation and escape retaliation. The second point is based on models that represent mobility only implicitly (in terms of travelling costs) and assume random interactions. Here we develop a more explicit theory of spatial iterated games: individual mobility is represented in terms of a diffusion process and interactions - defined locally - are inherently non-random. Our model reveals the existence of critical levels of individual mobility allowing for the evolution of cooperation. In fact, tit-for-tat can spread and take over among mobile players even when originating from extreme rarity. The dynamics of invasion of tit-for-tat develop as a travelling wave which propagates the cooperative strategy through space. Significant mobility is required to make the pioneering moves of cooperators towards the front of invasion less hazardous; it also contributes to neutralizing those defectors who may intrude the core of a cluster of cooperative players.
