The dynamics of asymmetric interactions: Some insights from formal theory and Richardson models

Abstract

Sometimes a large power attempts to influence the actions of one or more smaller agents that may be influenced by other small agents. This situation gives rise to amixed system of symmetric and asymmetric interactions and each case of this kind seems to have unique characteristics. This article asks, can a formal approach bring out some general features of such interactions? We extend a well-known model of the dynamics of arms races in new directions, and produce some counterintuitive results that may have interest for policy debates.

Keywords:

• asymmetric conflict
• insurgency
• political violence
• mathematical modeling

Acknowledgements

The authors are grateful to the editor for extensive help in expanding the arguments in this article and in presenting them in a manner that made them comprehensible to a wider audience than would otherwise have been the case.

Notes

1. Mathew Sussex helped with this, and other, examples and provided helpful insights into the problem of influence in international systems.

2. For a discussion of this problem, see Katzner (2009). See Dunne, García-Alonso, Levine, Ron, and Smith (2006) and Smith, Dunne, and Nikolaidou (2000) for a discussion of this issue and some econometric studies under different assumptions about measures.

3. The exposition is non-rigorous to aid understanding. For a rigorous presentation, see Coram and Noakes (2010).

4. Normally this would be expressed as .

5. See Richardson (1960). Rapoport (1974) is generally credited with rediscovering the Richardson model. For a discussion of its history and an overview of some applications see Nicholson (1991). The empirical work is too extensive to be properly presented in anote and would require a bibliographical treatment. For applications prior to 1989 see Anderton (1989), who identified over 100 of these. For fairly haphazard examples ofrecent applications, see Blank, Enomoto, Gegax, McGuckin, and Simmons (2008), Bolks and Stoll(2000), Dunne, Nikolaidou, and Smith (2005), Intriligator and Brito (1985, 2000), Saperstein (1999), Schrodt (1978), Jun Sik (2003), Smith et al. (2000).

6. This possibility is discussed in Saperstein (1999).

7. Specifically, if the matrix A is invertible it is only necessary to choose u : x = –A ^–1 h and since u is unrestricted this is in the attainable set.