Abstract
Combat between governmental forces and insurgents is modelled in an asymmetric Lanchester-type setting. Since the authorities often have little and unreliable information about the insurgents, ‘shots in the dark’ have undesirable side-effects, and the governmental forces have to identify the location and the strength of the insurgents. In a simplified version in which the effort to gather intelligence is the only control variable and its interaction with the insurgents based on information is modelled in a non-linear way, it can be shown that persistent oscillations (stable limit cycles) may be an optimal solution. We also present a more general model in which, additionally, the recruitment of governmental troops as well as the attrition rate of the insurgents caused by the regime’s forces, i.e. the ‘fist’, are considered as control variables.
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Notes
- 1.
Note that the time argument t will be dropped to simplify notation whenever appropriate.
- 2.
This term models the double-edged sword effect as described in Sect. 1.
- 3.
The attrition rate γ is assumed to be non-negative, but there are no restrictions on β because it can be seen as hiring/firing rate and negative values correspond to dismissing soldiers.
- 4.
A ≤ 4 assures that μ ≤ 1 as the maximum value of I(1 − I) is 1∕4 and \(\left (1 - \frac{1} {1+\epsilon }\right )\) is bounded by 1.
- 5.
Actually one only has to distinguish between the normal case \(\lambda _{0} = 1\) which follows for strictly positive values of \(\lambda _{0}\) by rescaling, or the abnormal case \(\lambda _{0} = 0.\)
- 6.
In economic theory this means that the elasticity of \(\theta\) is smaller than 1.
- 7.
Note that these values do not represent any real situation and are purely fictitious to illustrate the occurrence of periodic solutions in principle.
- 8.
There are, however, two remarkable exceptions. Kaplan et al. (2010) formulate an optimal force allocation problem for the government based on Lanchester’s dynamics and develop a knapsack approximation and also model and analyse a sequential force allocation game. Feichtinger et al. (2012) study multiple long-run steady states and complex behaviour and additionally propose a differential game between terrorists and government.
- 9.
Persistent oscillations, more precisely, stable limit cycles, occur in quite few optimal control models with more that one state, see Grass et al. (2008), e.g. particularly in Sect. 6.2
- 10.
E.g. the interaction between marketing price and advertising is a classical example of a synergism of two instruments influencing the stock of customers in the same direction. A further example is illicit drug control as analysed by Behrens et al. (2000), where prevention and treatment are applied but not at the same time.
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Acknowledgements
The authors thank Jon Caulkins, Dieter Grass, Moshe Kress, Andrea Seidl, Stefan Wrzaczek for helpful discussions and two anonymous referees for their comments.
This research was supported by the Austrian Science Fund (FWF) under Grant P25979-N25.
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Novák, A.J., Feichtinger, G., Leitmann, G. (2016). On the Optimal Trade-Off Between Fire Power and Intelligence in a Lanchester Model. In: Dawid, H., Doerner, K., Feichtinger, G., Kort, P., Seidl, A. (eds) Dynamic Perspectives on Managerial Decision Making. Dynamic Modeling and Econometrics in Economics and Finance, vol 22. Springer, Cham. https://doi.org/10.1007/978-3-319-39120-5_13
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