External conversions of player strategy in an evolutionary game: A cost-benefit analysis through optimal control

Abstract

We consider an optimal control problem based on the evolutionary game theory model introduced by Shortet al. (Short, M. B., Brantingham, P. J. & D'Orsogna, M. R. (2010) Cooperation and punishment in an adversarial game: How defectors pave the way to a peaceful society.Phys. Rev. E82(6), 066114.1–066114.7) to study societal attitudes in relation to committing and reporting crimes. Since in [26] (Short, M. B., Brantingham, P. J. & D'Orsogna, M. R. Cooperation and punishment in an adversarial game: How defectors pave the way to a peaceful society.Phys. Rev. E82(6), 066114.1–066114.7) it is shown that the presence of criminal informants leads to diminishing crime, in this paper we investigate the active recruitment of informants from the general population via external intervention, albeit at a cost to society. While higher recruitment levels may be the most beneficial in abating crime, these are also more expensive. We thus formulate our optimal control problem to account for finite resources, incurred costs and expected benefits, and determine the most favourable recruitment strategy under given constraints. We consider the cases of targeted and untargeted recruitment, and allow recruitment costs to depend on past cumulative payoffs within a given memory time-frame so that conversion of more successful individuals may be more costly than that of less successful ones. Our optimal control problem is expressed via three control functions subject to a system of delay differential equations, and is numerically solved, analysed and discussed under different settings and in different parameter regimes. We find that the optimal strategy can change drastically and abruptly as parameters and resource constraints vary, and that increased information on individual player strategies leads to only slightly decreased minimal costs.