Increasing Returns and Path Dependence in the Economy

3. Path-Dependent Processes and the Emergence of Macrostructure

CHAPTER 3 Path-Dependent Processes and the Emergence of Macrostructure W Brian Arthur, Yuri M. Ermoliev, and Yuri M. Kaniovski Many situations dominated by increasing returns are most usefully modeled as dynamic processes with random events and natural positive feedbacks (or nonlinearities). In this paper, Ermoliev, Kaniovski, and I introduce a very general class of such stochastic processes. We call these nonlinear Polya processes and show that they can model a wide variety of increasing returns and positive-feedback problems. In the presence of increasing returns or self-reinforcement, a nonlinear Polya process typically displays a multiplicity of possible asymptotic outcomes . These long-run outcomes or equilibria, we show, correspond to the stable fixed points of an associated "urn function," and can easily be identified . Early random fluctuations cumulate and are magnified or attenuated by the inherent nonlinearities of the process. By studying how these build up as the dynamics of the process unfold over time, we can observe how an asymptotic outcome becomes "selected" over time. The paper appeared in the European Journal of Operational Research 30 (1987): 294-303. It is a less technical version of our 1983 Kibernetika paper "A Generalized Urn Problem and its Applications," (which is cited at the end of this paper). For a more rigorous account of the processes described here, see chapter 10 of this volume. Yuri Ermoliev and Yuri Kaniovski are with the Glushkov Institute of Cybernetics, Kiev, Ukraine. Of recent fascination to physical chemists, biologists, and economists are nonlinear dynamical systems of the "dissipative" or "autocatalytic" or "selforganizing " type, where positive feedbacks may cause certain patterns or structures that emerge to be self-reinforcing. Such systems tend to be sensitive to early dynamical fluctuations. Often there is a multiplicity of patterns that are candidates for long-term self-reinforcement; the cumulation of small events early on "pushes" the dynamics into the orbit of one of these and thus "selects" the structure that the system eventually locks into. "Order-through-fluctuation" dynamics of this type are usually modelled by nonlinear differential equations with Markovian perturbations [1,2]. Show33 34 Increasing Returns and Path Dependence in the Economy ing the emergence of structure in the sense of long-run pattern or limiting behavior then amounts to analyzing the asymptotic properties of particular classes of stochastic differential equations. But while these continuous-time formulations work well, their asymptotic properties must often be specially studied and are not always easy to derive. Moreover, for discrete events, continuous-time formulations involve approximations. In this paper we introduce an alternative class of models (developed in previous articles [3,4,5]) that we call nonlinear Polya processes. These processes have long-run behavior easy to analyze, and for discrete applications they are exact. Within this class of stochastic processes we can investigate the emergence of structure by deriving theorems on long-run limiting behavior. In this short paper we survey our recent work on the theory of nonlinear Polya processes. We avoid technicalities as far as possible, and present applications in industrial location theory, chemical kinetics, and the evolution of technological structure in the economy. The limit theorems we present generalize the strong law of large numbers to a wide class of path-dependent stochastic processes. Structure and the Strong Law If a fair coin is tossed indefinitely, the proportion heads tends to vary considerably at the start, but settles down more and more closely to 50 percent. We could say, trivially perhaps, that a structure-a long-run fixed pattern in the proportion of heads and tails-gradually emerges. Of course, it is perfectly possible that some other proportion might emerge-two heads followed by one tail repeated indefinitely, for example, would yield 2/3 and is just as possible as any other sequence. But such an outcome is unlikely. Borel's strong law of large numbers tells us that repeated random variables (drawn from the same distribution) that are independent of previous ones have longterm averages that much approach their expected values. While other outcomes might in principle be possible, they have probability zero. Thus in coin-tossing, where the event "heads" is certainly independent of previous tosses, the average of heads in the total-the proportion heads-must settle down to 0.5, the expectation of each toss being a head. The emergence of a 50 percent proportion has probability one. Further, the "strong" part of the strong law tells us that once the proportion settles down it persists. If we were to repeat...