Political Complexity: Nonlinear Models of Politics

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This collection illustrates how nonlinear methods can provide new insight into existing political questions. Politics is often characterized by unexpected consequences, sensitivity to small changes, non-equilibrium dynamics, the emergence of patterns, and sudden changes in outcomes. These are all attributes of nonlinear processes. Bringing together a variety of recent nonlinear modeling approaches, Political Complexity explores what happens when political actors operate in a dynamic and complex social environment.

The contributions to this collection are organized in terms of three branches within non-linear theory: spatial nonlinearity, temporal nonlinearity, and functional nonlinearity. The chapters advance beyond analogy towards developing rigorous nonlinear models capable of empirical verification.

Contributions to this volume cover the areas of landscape theory, computational modeling, time series analysis, cross-sectional analysis, dynamic game theory, duration models, neural networks, and hidden Markov models. They address such questions as: Is international cooperation necessary for effective economic sanctions? Is it possible to predict alliance configurations in the international system? Is a bureaucratic agency harder to remove as time goes on? Is it possible to predict which international crises will result in war and which will avoid conflict? Is decentralization in a federal system always beneficial?

The contributors are David Bearce, Scott Bennett, Chris Brooks, Daniel Carpenter, Melvin Hinich, Ken Kollman, Susanne Lohmann, Walter Mebane, John Miller, Robert E. Molyneaux, Scott Page, Philip Schrodt, and Langche Zeng.

This book will be of interest to a broad group of political scientists, ranging from those who employ nonlinear methods to those curious to see what it is about. Scholars in other social science disciplines will find the new methodologies insightful for their own substantive work.

Diana Richards is Associate Professor of Political Science, University of Minnesota.

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Product Details

  • ISBN-13: 9780472109647
  • Publisher: University of Michigan Press
  • Publication date: 5/22/2000
  • Pages: 352
  • Product dimensions: 6.10 (w) x 9.44 (h) x 1.18 (d)

Read an Excerpt

Political Complexity: Nonlinear Models of Politics

By Diana Richards

University of Michigan Press

Copyright © 2000 Diana Richards
All right reserved.

ISBN: 0472109642

Nonlinear Modeling: All Things Suffer Change

Diana Richards

Nonlinear modeling may be summed up by the Latin phrase attributed to Terence, Omnium rerum vicissitudo est or "all things suffer change." At the simplest level, the statement that all things change seems trivial. Contemporary social science always formulates theories in terms of change captured by the concept of a dependent variable. Yet the concept of change is typically linear, namely, changes in variables occur, but the effect is constant. For any unit change in the independent variable, there is a corresponding change in the dependent variable regardless of the magnitude or characteristics of the variables. Yet to say that things suffer change implies that change is not something inconsequential, not something that occurs as smooth identical events, but something that entails the possibility of a radical qualitative effect. In linear processes, things simply change; in nonlinear processes, they can be more accurately described as suffering change.

But what is meant by nonlinear? This is obvious in mathematics but less so in a social science context. In mathematics, a nonlinear function is obviously one that has nonlinear--that is, polynomial or exponential-- terms. But what does nonlinearity mean when one is removed from any mathematical equation, as in the topics and approaches explored in this book? Nonlinear relationships imply that an independent variable does not have a constant effect on the dependent variable. Furthermore, even the direction of change need not be the same across all cases, and there may be qualitatively different phenomena observed from a small change in a variable. The consequences of a nonconstant effect of the independent variables cannot be overemphasized; it is what creates the potential manifestations of nonlinear relations, including the lack of closed-form solutions, complex and varying dynamics, sensitivity to initial conditions, and the possibility of multiple outcomes or dynamic patterns.

But what difference does a linear versus a nonlinear perspective make? As a thought experiment, imagine vision when one only sees things that hold still. Outside of my office window, "the world" consists of a eucalyptus grove, an alley with a row of parked cars, and footpaths through lawn areas. With my hypothetical fixed-point vision, I would only see the heavy trunks of the trees (but not the swaying eucalyptus branches or leaves), the few parked cars and the parking signs (but not the cars jockeying in the alley), and the walkways and lawns (but not the students walking or the bikers dodging the pedestrians). Although this illustration is highly stylized, the point is that if the world involves higher order change but we only look for what is constant we miss a lot. Let's assume for a minute that the social world is largely nonlinear. (I do not take this as a given but consider it an empirical or theoretical question and one that each author is asked to address in his or her essay.) No wonder very few clear empirical relationships have been found over decades of political science. If it is a nonlinear world and we are looking with "linear vision," then we can only catch a small portion. Furthermore, our models of constant effects will miss something fundamental about what we are studying; as the saying goes, it's like throwing a dead bird to model the flight of a live bird.

In a linear model, if one wants to account for higher levels of change or dynamics, one must turn to exogenous random shocks. This dominant conceptualization of political phenomena is evident throughout political applications, ranging from describing individual-level preferences to modeling aggregate time-series data. Although no one would argue against the important role linear models and stochastic shocks play in political models, the ramifications of nonlinearity have not been sufficiently realized. In some contexts, interaction or interconnections between variables, rather than a collection of separate variables and a stochastic element, are the main force behind the dependent variable. For example, contemporary ecology, while leaving explanatory space for exogenous random shocks such as volcanic eruptions or meteors, also incorporates sources of the dynamics arising from species interacting with each other in a changing and connected environment. The essays in this volume assert that the same holds in political science, where one can gain explanatory power by considering endogenous interactions and relationships. Therefore, rather than a lineage from physics and its tendency toward linear approximations, the appropriate sister sciences to political science may be ecology and cognitive science, with their traditions of considering complex aggregate behavior arising from the interaction of species or neurons.

Methodologists also have been known to "suffer change." We are so comfortable with linear approaches. Shortly after World War I, Lewis Richardson outlined several mathematical models of arms races. In these models, he described each nation's level of arms expenditure using coupled equations with variables such as budget constraints and hostility levels. The natural specification for some of those models was nonlinear in that the threat from an opponent's level of military spending varied depending on the proportionality with one's own level of spending. Although Richardson's models continue to be the basis for many theories of arms races, the Richardson models we are familiar with today are linear rather than nonlinear. Given the computational limitations of his time, Richardson linearized the models even when his theory suggested a nonlinear specification. This is ironic since Richardson is now seen as a pioneer in early nonlinear theory as a result of his work with fractals and cartography.

Yet Richardson's approach of simplifying his nonlinear theory of interaction to a linear model is part of a long tradition in political science that continues into the present. Nonlinearity implies new theoretical terrain, in that analysis must move beyond solving for a single "solution," and nonlinearity implies new empirical terrain, in that new statistical techniques must be used, many of which are still in the process of developing a rigorous theory of statistical inference. However, a linear approach is not the only approach, and, as the essays in this volume assert, it may not always be the best approach. A linear approach ignores the interaction among variables and, in particular, ignores the complexity arising from interaction effects. This has been one of the biggest contributions of agent-based modeling approaches (e.g., Schelling 1978; Axelrod 1984; Epstein and Axtell 1996): to emphasize that complexity in social outcomes can arise from a few variables in simple but nonlinear relations. Too often refinements of linear models consist of adding increasing numbers of variables rather than stepping back to consider a smaller set of variables in a nonlinear relationship. This is vividly illustrated in David Bearce's essay in this volume, in which his neural network model of the success of economic sanctions outperforms the competing linear model while using less than half the number of explanatory variables. The essays in this collection attempt to show, through a diversity of approaches, that nonlinear modeling can be a constructive enterprise that yields interesting hypotheses about a wide range of political topics.

A Typology of Nonlinearity

Recently there has been a surge in nonlinear modeling in economics (see, e.g., Day 1994; Benhabib 1992; and Day and Chen 1993). Articles in economics journals using only one particular nonlinear approach (that of chaotic dynamics) numbered 168 over the past 10 years, with an average of 28 published academic journal articles annually for the past five years. In addition, there have been several recent special issues of respected journals, including Journal of Economic Theory, Economic Theory, Journal of Economic Behavior and Organization, and Journal of Applied Econometrics, as well as a journal specifically focused on adaptive computational models, Computational Economics.

Part of the explanation for the abundance and rapid rise of nonlinear modeling in economics is due to the relatively straightforward application of nonlinear theory to existing economic theory. As anyone who has taken an introductory course knows, economics is based on a multitude of functional relationships such as supply and demand curves. The advantage this gives economics with regards to nonlinear modeling is not simply that economics is more formalized but that their formalizations are functions, that is, one-dimensional mappings. The relevance of this fact is that important theorems from nonlinear theory, such as the Sarkovski or Li-Yorke theorems, only hold for one-dimensional mappings. The importance of these theorems is that they demonstrate when a function will have the possibility for dynamic behavior beyond simple convergence to a static equilibrium. Any situation in which one has a combination of an amplification and a discouragement effect is a candidate, which includes a wide range of applications such as capital accumulation, overlapping generations of finite-lived consumers, business cycles, price adjustments, decision theory, economic growth models, and international trade, just to name a few (see Day 1994 or Benhabib 1992 for an overview).

However, unlike economics, political science is largely unable to piggyback its nonlinear theories on existing results such as the Li-Yorke theorem. Formal political theory is largely based on set-valued mappings such as voting theory or game dynamics. Therefore, although nonlinear feedback relations are common, they seldom take the form of one-dimensional functions as in economics. For example, even in the highly formalized topic of multidimensional voting, it makes little sense to describe the mapping from proposal to proposal in terms of a function. Instead, one has set-valued mapping in that for each status quo point a set of subsequent proposals are feasible: the familiar petal-shaped "win set." Instead, different approaches must be used (e.g., Schofield 1980; Richards 1994). Similarly, strategic behavior in political science, often represented in terms of distributions of strategies among a population of players (e.g., Axelrod 1984; Kollman, Miller, and Page 1992, 1997; Epstein and Axtell 1996; Cederman 1997), cannot be described by a one-to-one nonlinear function, despite its underlying nonlinearity.

As a result, nonlinear modeling in political science consists of disparate works, each inventing a method unique to the problem at hand rather than quickly building on a set of existing theoretical results, as is occurring in economics. Rather than the obvious nonlinear iterated functions of the mathematical economics literature,, nonlinear modeling in political science takes many guises. For example, many political scientists are surprised to learn that Robert Axelrod's acclaimed 1986 article "An Evolutionary Approach to Norms," wherein the complexity of dynamic behavior required a graphical presentation and simulations rather than solving for a single solution, is an example of nonlinear modeling.

Thus, nonlinear approaches to political science may be in terms of nonlinear dynamic systems (e.g., Saperstein and Mayer-Kress 1988; Muncaster and Zinnes 1990; Huckfeldt 1990; Brown 1991, 1994; Kiel and Elliott 1996), but they are just as likely to take the form of neural networks (e.g., Schrodt 1991), n-person game theory (e.g., Axelrod 1984, 1986; Nowak and May 1992; Glance and Huberman 1994; Lohmann 1994), symbolic dynamics (e.g., Saari 1989; Richards 1994), or spatial models of agent-based interaction (e.g., Axelrod and Bennett 1993; Axelrod 1997; Kollman, Miller, and Page 1992, 1997, 1998; Epstein and Axtell 1996; Cederman 1997). The specific form of nonlinear approaches has much more flexibility than most people realize.

The diversity of approaches suggests that the term nonlinear methods is a bit of a misnomer, since the incorporation of nonlinear elements cuts across a wide array of methods, including dynamical systems, game theory, spatial voting models, time-series analysis, nonparametric estimation, and logit/probit models. By deliberating sampling from such a wide array of methodologies, the message is that nonlinear methods cut across formal and quantitative approaches rather than comprising a separate methodology that is disjointed from existing approaches. The essays in this collection also include a wide variety of substantive topics, including questions such as:

Is decentralization in a federal system always beneficial? Is it possible to predict alliance configurations in the international system?

Can qualitatively different "political epochs" be identified in the international political economy?

What is the joint effect of campaign contributions, district service, and challenger quality on election outcomes?

What explains the dynamics of collective action toward changing a status quo?

Why are some international environmental regimes formed quickly while others languish, and why are some stable over time while others are fragile?

Is it indeed true (as in the public choice hypotheses) that the longer a bureaucratic agency is in existence the harder it is to remove?

Is international cooperation necessary for effective economic sanctions?

Is it possible to predict which international crises will result in war and which will not involve armed conflict?

In this collection, the approaches are organized into three perspectives within nonlinear theory: spatial, temporal, and functional nonlinearity. Although the three categories blend and merge, with many essays using elements from more than one perspective, they are useful distinctions in considering the primary source and manifestations of nonlinearity in social behavior.

Spatial Nonlinearity

Approaches in the category of spatial nonlinearity tend to begin with agent-based models and to use questions of optimization and search to examine the implications for distributions of variables over space. This approach has a long history in political science, especially in its use by Schelling (1978) and Axelrod (1984), who are pioneers in this field. In spatial nonlinearity, the focus is on the dynamics over some space, whether it is the policy differentiation of electoral districts (e.g., Kollman, Miller, and Page 1992, 1997), the emergence of nation-states (e.g., Cederman 1997), alliance distributions (e.g., Axelrod and Bennett 1993), or distributions of cultural attributes (e.g., Schelling 1978; Epstein and Axtell 1996; Axelrod 1997). These approaches tend to share a focus on the dynamics of adaptive agents, which, because of the ensuing complexity, are usually analyzed using the methodology of adaptive computational modeling, where inferences are drawn from "computational experiments" as a method analogous to empirical experiments or an exclusively formal deductive approach (see the discussion in Epstein and Axtell 1996).

Spatial nonlinearity is represented in this collection by the essay by Ken Kollman, John H. Miller, and Scott E. Page and that by D. Scott Bennett. Kollman, Miller, and Page present a model with two-party competition wherein citizens move among towns depending on the public policies of each town. Their approach is computational in that the inferences are drawn from numerical experiments rather than analytic solutions. Similarly, Bennett's essay takes a spatial perspective in considering agents' actions over a political "landscape"--a nonlinear multidimensional spatial representation of the key variables over which agents optimize. This nonlinear spatial approach allows one to explain agents' aggregated actions such as how they group into coalitions or alliances. Like Kollman, Miller, and Page's essay, Bennett's is based on large sets of agents interacting among themselves and an environment and is modeled using simulations.

Temporal Nonlinearity

Another approach to nonlinear modeling is to focus on dynamics over time; I label this strain of nonlinear thinking as temporal nonlinearity. This approach probably comes to most social scientists' minds first when they are speaking of nonlinear dynamics. Often temporal nonlinear models take the form of dynamical systems, as in Saperstein and Mayer-Kress 1988; Huckfeldt 1990; Muncaster and Zinnes 1990; Wolfson and Martelli 1992; Richards 1993; and Brown 1991, 1994, 1995. In many of these cases, the nonlinearity is fairly explicit in that the modeling consists of dynamic equations with nonlinear relationships between variables. The focus is on the dynamics over time in the nonlinear system, represented by phase portraits or vector fields (e.g., Brown 1994), which are then used to deduce hypotheses. Temporal nonlinearity also has an inductive branch, represented by research that focuses on identifying nonlinearity in empirical time-series data (e.g., Brock 1986; Brockett, Hinich, and Patterson 1988; Hinich 1982; Richards 1992; McBurnett 1996). Most of this work remains largely linked to empirical economics (for obvious data quality and motivational reasons). The approach in this case is the reverse: to glean empirical clues from a time-series as to the existence and characteristics of the nonlinear relations in order to develop more accurate models.

In the collection of essays presented here, temporal nonlinearity takes a broad and evolving form that differs from the early works on dynamic systems and empirical time-series cited previously. The essay by Chris Brooks, Melvin J. Hinich, and Robert Molyneux takes the most explicit time perspective by examining the presence of political epochs in the dynamics of exchange rates. Walter R. Mebane Jr.'s essay presents an innovative way to overcome problems with the lack of long-term time-series data, as is common in political science. His approach is to test a nonlinear time-series process using cross-sectional data by looking at the dynamics only in a particular region of values, in particular the dynamics near the equilibrium value of his game model. The essays by Susanne Lohmann and Diana Richards link temporal nonlinearity to game theory but with a specific focus on relaxing the common knowledge assumption and exploring the dynamics of learning or information transmission in games.


Excerpted from Political Complexity: Nonlinear Models of Politics by Diana Richards Copyright © 2000 by Diana Richards. Excerpted by permission.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

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