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Richardson Revisited: An Analysis of ‘Action-Reaction’ Conflict Models
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- RICHARDSON REVISITED: AN ANALYSIS OF ‘ACTION-REACTION’ CONFLICT MODELS
Richardson Revisited: An Analysis of ‘Action-Reaction’ Conflict ModelsAuthor: Giorgio Gallo Title: Professor, Retired
Location: Pisa, Italy E-mail: gallo@di.unipi.it
RICHARDSON REVISITED: AN ANALYSIS OF ‘ACTION-REACTION’ CONFLICT MODELSAbstract The remarkable pioneering work of Lewis Fry Richardson has started the theoretical analysis of the action-reaction processes leading to the onset of violent conflicts. His work, developed between the two world wars, has started receiving recognition only with the birth of Peace and Conflict Studies in the aftermath of WWII. Here the main model proposed by Richardson will be presented and discussed, together with some of its extensions and derivations. Particular attention shall be dedicated to the epistemological analysis of these models. IntroductionFor Lewis Fry Richardson, a mathematician and physicist, WWI was a fundamental turning point, both from a personal and from a scientific point of view. A dedicated pacifist, as a conscientious objector, he did not serve in the army during the war, and, immediately after the end of the war, he started applying the mathematical tools he had developed for the study of meteorological phenomena to investigate the conditions that make likely the onset of violent conflicts, and also the conditions that make peace possible. With his work, he became a pioneer of the area of peace and conflict studies which were to become relevant as a scientific discipline only after the end of WWII. Talking of the theoretical analysis of the action-reaction processes leading to conflicts, Kenneth Boulding, one of the founders of peace studies, writes: “By far the most extensive theoretical treatment of these processes has been made by Richardson, whose remarkable pioneering work is only now receiving recognition. It would only be just to name these processes Richardson processes in his honor" (Boulding, 1962, p. 25). In what is apparently his first scholarly paper on peace issues, Lewis Fry Richardson addresses the problem of finding a viable mathematical formula to determine the relative voting power to be granted to the participants in the congresses which most likely were to be convened after the end of the Great War’s hostilities. The idea was that in such meetings unanimity was not viable, giving a veto power also to minor States, but also that assigning one vote to each state was not reasonable considering the huge differences in strength among the States. It was quite clear to Richardson that mathematics was not intended to provide a “solution” to what was actually a political problem, but rather to be a decision-aiding tool: “The proper voting strength is, and must always remain, a matter of judgment, of opinion. But a well-chosen mathematical formula may help it to become a matter of considered and agreed opinion. In the choice of the formula we may be guided by a good old rule in applied mathematics, that of taking the simplest formula which makes sense” (Richardson, 1918, p. 195). The problem of voting strength in international assemblies has been addressed again in an unpublished report of 1953 with reference to the UN assembly (Richardson, 1993c). One year later, in 1919, in an essay dedicated to his comrades of the motor ambulance convoy “S. S. Anglaise 13,” in which he had worked during the war, and published in the volume of his collected papers (Richardson, 1993a), Richardson analyzes at length all the different motivation which lay behind the willingness of a state to engage in a war against another state, and defines a set of differential equations which can help in studying the dynamics of the aggressiveness in the relations between two states or two coalitions of states. The dynamics of aggressiveness has been later further presented in a paper and in a letter to the editor published in Nature (Richardson, 1935a, 1935b), and are analyzed at length in a long essay published in 1939, on the eve of World War II (Richardson, 1939). In these writings it is clear that to Richardson mathematical models, more than describing reality, are decision aid tools. His insistence on Ockham’s Razor goes in the same direction: models should be characterized by the maximum simplicity compatible with their capacity to provide decision makers with the right information. The use of mathematical models for decision aid is exactly the aim of Operations Research, which can be defined as “a discipline that deals with the application of advanced analytical methods to help make better decisions” (Informs, The American Operations Research Society). In this sense we can say that Richardson can be considered as an ante-litteram Operations Researcher. Richardson not only had a clear understanding that, in systems of human activities, mathematical models are decision aid tools, but he also knew well that he was not modeling the behavior of single entities, be they states or individuals, but rather that he was modeling the interaction of such entities. Moreover such entities were themselves aggregates of other lower-level entities. He uses the example of gases: “Just as in meteorology it has been found necessary to endow the air with an eddy viscosity to compensate for the motion which we cannot study in detail, so in the social sciences it is to be expected that there will be a need to regard the social groups as endowed with properties, analogous to viscosity, arising from the lack of agreement in the purposes of the individuals which compose the group” (Richardson, 1939, p. 3). Later in the same book (p. 9) he talks of social viscosity with reference to the slow processes in a community that eventually lead to a decision. He was well aware of the systemic structure of the reality he was dealing with. In the following section I present in some detail Richardson’s action-reaction linear conflict model, then, in sections 3, I discuss from an epistemological point of view its meaning, purposes and uses. Finally, in sections 4 and 5, I describe two different nonlinear models which can be derived from it. Some real life applications of these nonlinear models are discussed. Richardson’s ModelThe main idea behind Richardson’s model is that, in a situation in which there are two parties who are potentially foes, each of them responds to the actions of the other by increasing or decreasing its level of aggressiveness or, as he says, “preparedness for war.” In Richardson’s view the two parties are either states or coalitions of states, but we may think also of social groups or armed organizations in contemporary intrastate conflicts. The behavior of the parties is the result of the complex pattern of interactions between the many different individuals constituting them, and that is why Richardson warns against applying his model to the interaction between single individuals. Let’s call X and Y respectively the two parties, and x and y the respective levels of aggressiveness or of preparedness for war. Variables x and y are usually positive, but, at least in principle, they can be negative, representing solidarity or cooperation between the two countries instead of aggressiveness. I call  the function which, for any given level y of action taken by Y, gives the value x of the action decided in response by X. Similarly, Y responds to X according to response function  .
The dynamics of the aggressiveness between the two parties X and Y can be described by the following system of differential equations  (1)
 (2)
where D(z,w) is the distance between z and w according to some criterion, and and are positive constants. The idea is that the rate of change in the aggressiveness is proportional to the distance between the optimal level, that is the one given by the response function, and the actual one. If  is positive ( ), then X will try to reach the equilibrium increasing its aggressiveness. The opposite happens when the distance is negative. In the following, for the sake of simplicity, I will assume the distance functions given by:
The response functions chosen by Richardson for his model are linear: 
Richardson calls k and l the defence coefficients, and the fatigue-and-expense coefficients, and p and q are measures of the past grievances between the two parties. In fact, the response increases in force with k, l, p and q, and decreases with and . Thus, the differential equations governing the conflict escalation in Richardson’s model are: 
Richardson starts his analysis from these differential equations, and from them derives the response functions 3 and 4, which he calls equilibrium lines. Solving the linear equilibrium system kyx+p = 0 lxy+q = 0 we get the equilibrium point  such that  and  :
When the system reaches such equilibrium point, there is no further incentive for changes. The equilibrium point is stable when for small displacement from it the system returns to the equilibrium. It is possible to prove that an equilibrium point is stable if  , that is when the product of the slopes of the two curves at is less than 1. In the linear case studied by Richardson, the stability condition becomes:
> kl. (9) Since the product can be considered a measure of the overall cost/fatigue of the system and the product kl a measure of the overall defense/aggressiveness of the system, inequality 9 states that for the stability to be granted it is necessary that in the system the cost considerations bear a higher weight than those for defense. From this analysis two important observations can be derived. The first concerns Richardson’s intuition that, in order to analyze the chances of the onset of a violent conflict, a systemic approach is needed. It is not sufficient to study the nature of the parties involved. It is also essential to analyze the dynamics of their interaction, which, possibly independently from their real intentions and objectives, may lead to a situation in which the outbreak of a violent conflict is unavoidable. The second concerns the interpretation of inequality 9. The cost/fatigue coefficient of a state cannot be considered independent of the internal characteristics of the state itself. An autocratic or dictatorial government may impose sacrifices to its citizens without any risk of losing legitimacy, at least up to a point. In contrast, a democratic government cannot lose the support of its constituency, lest it risks being overthrown at the next elections. In this sense inequality 9 is in accord with the well-known Liberal Peace theory, which states that democratic states very seldom fight wars against each other. The liberal peace theory is actually much more complex than this and also widely disputed (Doyle, 2005; Mearsheimer, 1990). Although in most of his papers on the topic he treats the interactions between two nations, Richardson has also tried to extend the model to the case of three or more nations. In the case of three nations he has shown that: (i) “if each of the three pairs of nations be separately unstable then the triplet is necessarily unstable”, and (ii) “If each of the three pairs of nations be separately stable then remains the possibility that the triplet of nations may be unstable” (Richardson, 1939, pp. 54–55). Richardson was well aware that his model was a quite rough approximation of the reality. But actually he was mainly interested in making a theory of stability and instability of peace. For that a linear approximation is much more useful than complex nonlinear models: “Because, if we had accurate non-linear equations, it is likely that the only possible formulae of solution would be attained by local linear approximations […]. So linear theory is a necessary preliminary” (Richardson, 1939, p. 48). Directory: wp-content -> uploads -> 2017 2017 -> Leadership ohio 2017 -> Ascension Lutheran Church Counter’s Schedule January to December 2017 2017 -> Board of directors juanita Gibbons-Delaney, mha, rn president 390 Stone Castle Pass Atlanta, ga 30331 2017 -> Military History Anniversaries 16 thru 31 January Events in History over the next 15 day period that had U. S. military involvement or impacted in some way on U. S military operations or American interests 2017 -> The Or Shalom Cemetery Community Teaching on related issues of Integral 2017 -> Ford onthult samenwerking met Amazon Alexa en introduceert nieuwe navigatiemogelijkheden van Ford sync® 3 met Applink 2017 -> Start Learn and Increase gk. Question (1) Name the term used for talking on internet with the help of text messege? 2017 -> Press release from 24. 03. 2017 From a Charleston Car to a Mafia Sedan 2017 -> Tage Participants 2017 -> Citi Chicago Debate Championship Varsity and jv previews Download 1.8 Mb. Share with your friends:
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