•Tourism management requires urgent measures for increasing sustainability.•We propose an asymmetric spatial game for tourists and stakeholders with migration.•Adaptive and fixed rewarding incentives to both populations are applied to the game.•In dense populations and low migration, the adaptive rewarding is the best option.•If density is low, rewarding must be focus on tourists instead of stakeholders. Tourism is a growing sector worldwide, but many popular destinations are facing sustainability problems due to excessive tourist flows and inappropriate behavior. In these areas, there is an urgent need to apply mechanisms to stimulate sustainable practices. This paper studies the most efficient strategy to incentivize sustainable tourism by using an asymmetric evolutionary game. We analyze the application of rewarding policies to the asymmetric game where tourists and stakeholders interact in a spatial lattice, and where tourists can also migrate. The incentives of the rewarding policies have an economic budget which can be allocated to tourists, to stakeholders, or to both sub-populations. The results show that an adaptive rewarding strategy, where the incentive budget changes over time to one or the other sub-population, is more effective than simple rewarding strategies that are exclusively focused on one sub-population. However, when the population density in the game decreases, rewarding just tourists becomes the most effective strategy.