Book Review: Computation and complexity in economic behavior and organization
Bulletin of the American Mathematical Society
More and more, my fellow economists are coming to recognize that all forms of economic activity entail computation. Given the way mathematics has emerged as the preferred language of economic theory, it is perhaps surprising that this aspect of economics has only recently begun to be treated formally. The late emergence of the computational theme in economics may be a consequence of the fact that the notion of computational limits is itself relatively new in mathematics. Gödel and Turing first
... iscovered limits to what can be proved or computed in the 1930s, and formal theories of computational complexity and algorithmic information only began to emerge in the 1960s ,  . Model-building styles in economics have been slow to catch up. Since the beginnings of complexity theory in mathematics, it has been recognized that computational limits have economic implications. When Herbert Simon introduced "bounded rationality" in the 1950s, he had in mind inherent limits to human information-processing capabilities: "Broadly stated, the task is to replace the global rationality of economic man with a kind of rational behavior that is compatible with the access to information and the computational capacities that are actually possessed by organisms, including man, in the kinds of environments in which such organisms exist" (, p. 99). In addition to the kinds of intrinsic human psychological limitations Simon emphasized, computational complexity issues are obviously important for many of the practical problems faced by business firms in their day-to-day activities. Well-known examples include the Traveling Salesman Problem (TSP), the Production Planning Problem, and many others included in Garey and Johnson's long list of NP-complete problems  . More recently, economists have come to realize that any theory of decision-making that is supposed to weigh costs and benefits is incomplete without taking account of the costs of the decision-making process itself. What Mount and Reiter (hereafter MR) have done is to develop a measure of the "complexity" of economic computations relative to a set of elementary or "primitive" functions and set of directed graphs. Any computation that can be expressed as a superposition of the primitive functions can be represented by a corresponding graph structure, and MR show that only a particular type of graph (ordered trees) is required. The measure of complexity is then simply the height of the ordered tree (the length of the longest path in the tree). The model involves some interesting mathematics -results about superpositions of functions are the content of Hilbert's 13th problem, nomography, and Leontief's Theorem. The MR setup provides a natural way to think about computations that are performed by humans and machines in conjunction, by specifying that some of the primitive functions are those that are easy for humans (like handwriting recognition) but difficult for computers.