Left–handers occur at unexpectedly high frequencies at top levels of many interactive sports. This may occur either because left–handed contestants are innately superior or because they enjoy a negatively frequency–dependent strategic advantage when rare relative to right–handers. We analysed the batting records from the 2003 cricket World Cup and showed that left–handed batsmen were more successful than right–handers, and that the most successful teams had close to 50% left–handed batsmen. We demonstrate that this was because left–handed batsmen have a strategic advantage over bowlers, and that this advantage is greatest over bowlers that are unaccustomed to bowling to left–handers. This provides a clear mechanism for negative frequency–dependent success of left–handed batsmen. Our results may also support a historical role for negative frequency–dependent success in fights and other contests in the maintenance of left–handedness by natural selection.