Coevolutionary dynamics is widely investigated in chemical catalysis, biological evolution, social and economic systems; often formulated within the unifying framework of evolutionary game theory. However, populations are finite, and the replicator-type deterministic differential equations are to be replaced by a stochastic process, which, if available, must be based on a microscopic description of the interaction process. Often it has been argued that a spatial system is necessary to preserve biodiversity (or coexistence), but we have demonstrated that stabilization is possible even in the well-mixed (mean-field) system [1,2,3]. Here I will discuss how the stability of cyclic coevolutionary dynamics can be reversed, depending on the population size, the interaction process, and the payoffs. The reversal of stability can change, from an average drift towards the boundaries of the phase space (resulting in extinction times polynomial with population size [4]) to an average drift towards the coexistence fixed point (resulting in an exponential scaling of extinction time [4]). \\ [1] A. Traulsen, J.C. Claussen and C. Hauert, PRL 95, 238701 (2005) \\ [2] Jens Christian Claussen EPJB 60, 391 (2007) \\ [3] Jens Christian Claussen and Arne Traulsen PRL 100, 058104 (2008) \\ [4] Markus Sch\"utt and Jens Christian Claussen, to be submitted (2009)