Animal cooperation in the wild often involves multiple individuals that must tolerate each other in close proximity. However, most cooperation experiments in the lab are done with two animals, that are often also physically separated. Such experiments are useful for answering some pertinent questions, for example about the understanding of the role of the partner and strategies of partner control, but say little about factors determining successful cooperation with multiple partners in group settings. We explored the influence of dominance, rank distance, tolerance, affiliation, and coordination by testing kea parrots with a box requiring two, three, or four chains to be pulled simultaneously to access food rewards. The reward could be divided unevenly, but not monopolized completely. Eventually dyadic, triadic, and tetradic cooperation tasks were solved, showing that non-human animals are capable of tetradic cooperation in an experimental setup. Starting with two chains, we found that in a dyad monopolization of the box by the highest-ranking bird was the largest obstacle preventing successful cooperation. High-ranking birds learned to restrain themselves from monopolizing the box during a single session in which monopolization was hindered by the presence of a large number of birds. Thereafter, restraint by dominants remained the strongest factor determining success in the first trial in dyadic, triadic, and tetradic setups. The probability of success increased with the degree of restraint shown by all dominant subjects present. Previous experience with the task contributed to success in subsequent sessions, while increasing rank distance reduced success notably in the four-chain setup. Supplementary Information The online version contains supplementary material available at 10.3758/s13420-021-00462-9.