Stability of cross impact matrices

Abstract

The relationship between drivers and outcomes in technological and social systems is usually difficult to interpret. Pairwise direct impacts between variables can be defined by cross-impact matrices. In research, direct impact matrices' row and column sums are primarily utilized to identify critical, influential, dependent, neuter, and inert variables. In this paper, all existing direct and indirect impact paths between variables are considered, with the aims of finding conditions under which the rank order remains stable (i.e., stable equilibrium state exists) and showing the difference between direct impact matrix and stable state matrix. It is shown numerically that the rank order and thus the categorization of variables can change significantly. A stable state does not always exist, but more direct impacts between variables reduce the risk of instability. The proposed method can support management in strategic planning and decision-making: Management should strive for a stable state that serves to determine the rank order on which informed decisions are based. Caution should be exercised when no stable state exists because the variables cannot be categorized, making the system uncontrollable. Management is advised to incorporate additional impact paths between variables into the system—assuming they were incomplete—which ideally establish a stable state.