How do territorial animals gain ownership of an area? Early modelling has considered the evolution of fighting when the winner can claim the right to the resource. Recently, alternative hypotheses have been offered where repeated interactions lead to division of space through 'nagging' instead of one decisive fight. However, these models assume that animals avoid areas in which they have taken part in aggressive interactions, but do not consider whether avoidance itself is adaptive. We aim to bridge this gap between mechanistic and adaptive explanations, by presenting a game-theory model where individuals choose whether to return to an area after a fight with a specific outcome (win, loss, draw). We show that avoidance of areas where fights have occurred can be adaptive, but only if benefits of access to the area are low compared to costs of fighting. Otherwise, one individual (typically the winner) responds by returning to the area, and the other (loser) avoids it. In such cases, space is gained by winning fights. We also consider the role of conventions. If responses to fights were purely conventional, paradoxical strategies where losers of fights gain ownership would be equally logical as common-sense ones where winners claim ownership. Paradoxical solutions can be stable but only when there is little difference in fighting ability between the competitors, when individuals adhere very strictly to a behavioural rule without much random variation, and when the population in its ancient state used a paradoxical strategy.