In developing mammalian muscle, axon branches of several motor neurons co-innervate the same muscle fibre. Competition among them results in the strengthening of one and the withdrawal of the rest. It is not known why one particular axon branch survives or why some competitions resolve sooner than others. Here we show that the fate of axonal branches is strictly related to the identity of the axons with which they compete. When two neurons co-innervate multiple target cells, the losing axon branches in each contest belong to the same neuron and are at nearly the same stage of withdrawal. The axonal arbor of one neuron engages in multiple sets of competitions simultaneously. Each set proceeds at a different rate and heads towards a common outcome based on the identity of the competitor. Competitive vigour at each of these sets of local competitions depends on a globally distributed resource: neurons with larger arborizations are at a competitive disadvantage when confronting neurons with smaller arborizations. An accompanying paper tests the idea that the amount of neurotransmitter released is this global resource.