Adaptive behavior has been observed in almost all aspects of real-world. One of the main advantages of acting adaptively is its stabilizing effect on dynamic equilibrium, associated with which are three favorable features: (a) non-destabilizing characteristics, (b) low-speed effectiveness and (c) the convexity of the stabilization regime in terms of the adaptive parameter. It is shown either in theory or by counter-examples that these advantages may not be preserved if the adaptive mechanism is applied to multi-dimensional processes. The necessary and sufficient conditions for the relevant phenomena are provided for two-dimensional dynamic processes with application to duopolistic dynamics. Our findings not only help to clarify hidden misconceptions and prevent potential abuse of adaptive mechanisms, but also illustrate the possible pitfalls arising from generalizing well-known characteristics of low dimensional and/or homogeneous agent models to high-dimensional and heterogenous agent models.
