Inspection-Corruption Game of Illegal Logging and Other Violations: Generalized Evolutionary Approach

Games of inspection and corruption are well developed in the game-theoretic literature. However, there are only a few publications that approach these problems from the evolutionary point of view. In previous papers of this author, a generalization of the replicator dynamics of the evolutionary game theory was suggested for inspection modeling, namely the pressure and resistance framework, where a large pool of small players plays against a distinguished major player and evolves according to certain myopic rules. In this paper, we develop this approach further in a setting of the two-level hierarchy, where a local inspector can be corrupted and is further controlled by the higher authority (thus combining the modeling of inspection and corruption in a unifying setting). Mathematical novelty arising in this investigation involves the analysis of the generalized replicator dynamics (or kinetic equation) with switching, which occurs on the "efficient frontier of corruption". We try to avoid parameters that are difficult to observe or measure, leading to some clear practical consequences. We prove a result that can be called the "principle of quadratic fines": We show that if the fine for violations (both for criminal businesses and corrupted inspectors) is proportional to the level of violations, the stable rest points of the dynamics support the maximal possible level of both corruption and violation. The situation changes if a convex fine is introduced. In particular, starting from the quadratic growth of the fine function, one can effectively control the level of violations. Concrete settings that we have in mind are illegal logging, the sales of products with substandard quality, and tax evasion.