Crisis instability, chaos, and predictability: Saperstein's model of the outbreak of war
In building his predictive model of the outbreak of war Saperstein (1994:150) starts from the following conception of prediction. He represents the present state of the international system by X_{o}. A set of parameters, lambda, is assumed to change that state into a different state X_{n}. Furthermore, he presumes that different sets of parameters will result in different futures, i.e. different values of X_{n}. Prediction, then, implies knowing X_{n}, given X_{o} and a specific set of parameters, lambda. In doing so he introduces a dynamical system that can be expressed mathematically as a function:
where lambda represents the set of parameters that define the form of the function f. Different specifications of lambda, be it sets of policy options or distributions of capabilities, imply different functional relations between present and future states of the international system. Equation (0.4) relates the system state at time n to the next time n+1. Through iterating the function it becomes possible to move into the future. For example, to go from the present, X_{o}, to the future, X_{n}, we iterate Equation (0.4) as follows:
The ultimate outcome of any policy or process is the value of X far ahead in the future: X_{n} = X_{inf} as n approaches infinity. This set of values is called the attractor of Equation (0.4), and varies as the parameter lambda is varied. Thus for Saperstein (1994: 150) the problem of prediction becomes: "given X_{o} and lambda, what is the attractor X_{inf} ?"
As a second assumption comes the notion of crisis instability, which is defined as an extreme sensitivity of system behaviour to small changes in input or system parameters (Saperstein 1994: 156). Small perturbations - such as differential growth of power - occur constantly in anarchical settings as the international political system. What we need to find out is when such small variations lead to crisis instability, a system state that Saperstein assumes to be equivalent to a high probability of the outbreak of war. Illustrative of such instability is again the murder of Archduke Franz Ferdinand in the Balkans in 1914, which was a small perturbation compared to the ensuing mass slaughter that World War I turned out to be. Conversely, the downing of a Korean Boeing 747 by Soviet warplanes did not escalate into a major armed conflict, even if it involved the loss of hundreds of lives. So these examples suggest that the Eurasian security complex was clearly "crisis unstable" at the beginning of the century, whereas it was relatively stable in its latter half with small perturbations leading to small changes in the system output. More generally, it appears that some system states may be "crisis stable", whereas other tend to be crisis unstable. In terms of Equation (0.4), this means that for some ranges of lambda small changes in X_{o} will lead to small changes in X_{n}, whereas for other ranges of lambda small deviations in X_{o} may produce highly disproportionate shifts in X_{n}. The major purpose of the model is to get to know whether a present system state, X_{o}, and/or set of parameters, 8, will lead to the desired end state, X_{inf}, i.e. crisis stability (a system state where the outbreak of war is unlikely) or to an undesirable - that is strange - attractor, , i.e. crisis instability (a system state where the onset of war is to be expected).
Saperstein's notion of crisis instability is clearly inspired by the extreme sensitivity to small changes often displayed by non-linear physical systems, a quality that is now commonly referred to as "chaos" (Waldrop 1994; Brown 1995a; De Vree 1991). Of special interest is that the same non-linear system may show both non-chaotic behaviour and chaotic behaviour. In the first case prediction is possible because the future uncertainty is of the same order of magnitude as present uncertainty. As a result small variations in input, delta(x_{o}) (due to disturbances and measurement errors), lead to commensurately small variations in system output, delta(x_{(t)}) (see Figure 1). Put differently, the paths from closely neighbouring starting points remain close.
Figure 1: Predictability
In the second case prediction it is not possible as the model system is extremely sensitive to small perturbations, delta(x_{o}), resulting in disproportionate shifts in system output, X_{n} (see Figure 2). More specifically, as the paths separate "exponentially", the rate of increase of the separation is itself proportional to the separation, hence increases as the latter increases, leading to an explosively increasing separation. Consequently, the final outcomes over all possibilities allowed in the system. Therefore, prediction becomes impossible, as every future outcome is possible, meaning that we are in the dark about what is to happen. Such a situation is typically chaotic.
Figure 2: Chaos or unpredictability
In Saperstein's model the outbreak of war is ultimately equivalent to the onset of chaos. As a working hypothesis he assumes that the transition from stability to war in an international system can be represented by a transition from non-chaotic to chaotic behaviour in the non-linear model representing the interactions between the competing elements (i.e. states) of the system. In other words, if the onset of chaos is predicted in the model, then crisis instability and therewith the outbreak of war is to be expected in the system being modelled (Saperstein 1994: 157).