This paper characterizes long-run outcomes for broad classes of symmetric games, when players select actions on the basis of average historical performance. Received wisdom is that when agent's interests are partially opposed, behavior is excessively competitive: ``keeping up with the Jones' '' lowers everyones' welfare. Here, we study the long-run consequences of imitative behavior when agents have sufficiently long memories --- and the outcome is dramatically different. Imitation robustly leads to cooperative outcomes (with highest symmetric payoffs) in the long run. This provides a rationale, for example, for collusive cartel-like behavior without collusive intent on the part of the agents.