In this paper we consider dynamic game-theoretic models, where boundedly rational players use simple decision rules to determine their actions over time. The adaptive process which captures the interaction of the players' decisions is the main objective of our study. In many situations this process is characterized by multistability, where e.g. multiple stable (Nash) equilibria emerge as possible long run outcomes. When such coexistence occurs, the selected equilibrium becomes path-dependent, and a thorough knowledge of the basins and their structure becomes crucial for the researcher to be able to predict which one of the multiple equilibria is more likely to be observed in situations described by the game. We demonstrate that, despite the fact that the long run dynamics of the adaptive process might be rather simple, the basins of the attracting sets might have quite complicated structure. In this paper we show that the complexity of basins can be explained on the basis of the global properties of the dynamical system, and we introduce the main tools - critical sets and basin boundaries - which enable the model builder to analyze the extent of the basins and their changes as structural parameters of the model are changed. The main point is that one has to study the global properties of the system, and not restrict the investigation to the local dynamics around the attracting sets.