The no free lunch theorems: complexity and security

One of the main challenges for decision scientists in the 21st century will be managing systems of ever increasing complexity. As systems like electrical power grids, computer networks, and the software that controls it all grow increasingly complex, fragility, bugs, and security flaws are becoming increasingly prevalent and problematic. It is natural then to ask what consequences this growing complexity has on our ability to manage these systems. In this paper, we take a first step toward addressing this question with the development of the fundamental matrix, a framework for analyzing the broad qualitative nature of decision making. With the fundamental matrix we explain in a qualitative way many theorems and known results about optimization, complexity, and security. The simplicity of the explanations leads to new insights toward potential research directions. Like other "theories" dealing with broad fundamental properties, however, the fundamental matrix has certain limitations that make it largely descriptive. Thus, instead of claiming the last words our goal is to stimulate a dialog and debate that may one day lead to a prescriptive science of complexity.