We consider a simple binary market model containing N competitive agents. The novel feature of our model is that it incorporates the tendency shown by traders to look for patterns in past price movements over multiple time scales, i.e. multiple memory-lengths. In the regime where these memory-lengths are all small, the average winnings per agent exceed those obtained for either (1) a pure population where all agents have equal memory-length, or (2) a mixed population comprising sub-populations of equal-memory agents with each sub-population having a different memory-length. Agents who consistently play strategies of a given memory-length, are found to win more on average -- switching between strategies with different memory lengths incurs an effective penalty, while switching between strategies of equal memory does not. Agents employing short-memory strategies can outperform agents using long-memory strategies, even in the regime where an equal-memory system would have favored the use of long-memory strategies. Using the many-body 'Crowd-Anticrowd' theory, we obtain analytic expressions which are in good agreement with the observed numerical results. In the context of financial markets, our results suggest that multiple-memory agents have a better chance of identifying price patterns of unknown length and hence will typically have higher winnings.