Affinity maturation of antibodies during immune responses is achieved by multiple rounds of somatic hypermutation and subsequent preferential selection of those B cells that express B cell receptors with improved binding characteristics for the antigen. The mechanism underlying B cell selection has not yet been defined. By employing an agent-based model, we show that for physiologically reasonable parameter values affinity maturation can be driven by competition for neither binding sites nor antigen--even in the presence of competing secreted antibodies. Within the tested mechanisms, only clonal competition for T cell help or a refractory time for the interaction of centrocytes with follicular dendritic cells is found to enable affinity maturation while generating the experimentally observed germinal centre characteristics and tolerating large variations in the initial antigen density.