Competitive ability is a major determinant of fitness, but why individuals vary so much in their competitiveness remains only partially understood. One increasingly prevalent view is that realized competitive ability varies because it represents alternative strategies that arise because of the costs associated with competitiveness. Here we use a population of great tits (Parus major) to explore whether individual differences in competitive ability when foraging can be explained by two traits that have previously been linked to alternative behavioural strategies: the personality trait 'exploration behaviour' and a simple cognitive trait, 'innovative problem-solving performance'. We assayed these traits under standardized conditions in captivity and then measured competitive ability at feeders with restricted access in the wild. Competitive ability was repeatable within individual males across days and correlated positively with exploration behaviour, representing the first such demonstration of a link between a personality trait and both competitive ability and food intake in the wild. Competitive ability was also simultaneously negatively correlated with problem-solving performance; individuals who were poor competitors were good at problem-solving. Rather than being the result of variation in 'individual quality', our results support the hypothesis that individual variation in competitive ability can be explained by alternative behavioural strategies.