https://doi.org/10.1140/epjb/s10051-025-01104-z
Research -Statistical and Nonlinear Physics
Finite population dynamics resolve the central paradox of the Inspection Game
Instituto de Física de São Carlos, Universidade de São Paulo, 13566-590, São Carlos, São Paulo, Brazil
Received:
28
October
2025
Accepted:
28
November
2025
Published online:
8
December
2025
The Inspection Game is the canonical model for the strategic conflict between law enforcement (inspectors) and citizens (potential criminals). Its classical Mixed-Strategy Nash Equilibrium (MSNE) is afflicted by a paradox: the equilibrium crime rate is independent of both the penalty size (p) and the crime gain (g), undermining the efficacy of deterrence policy. We re-examine this challenge using evolutionary game theory, focusing on the long-term fixation probabilities of strategies in finite, asymmetric population sizes subject to demographic noise. The deterministic limit of our model exhibits stable limit cycles around the MSNE, which coincides with the neutral fixed point of the equilibrium analysis. Crucially, in finite populations, demographic noise drives the system away from this cycle and toward absorbing states. Our results demonstrate that high absolute penalties p are highly effective at suppressing crime by influencing the geometry of the deterministic dynamics, which in turn biases the fixation probability toward the criminal extinction absorbing state, thereby restoring the intuitive role of p. Furthermore, we reveal a U-shaped policy landscape where both high penalties and light penalties (where 
) are successful suppressors, maximizing criminal risk at intermediate penalty levels. Most critically, we analyze the realistic asymptotic limit of extreme population sizes asymmetry, where inspectors are exceedingly rare. In this limit, the system’s dynamic outcome is entirely decoupled from the citizen payoff parameters p and g, and is instead determined by the initial frequency of crime relative to the deterrence threshold (the ratio of inspection cost to reward for catching a criminal). This highlights that effective crime suppression requires managing the interaction between deterministic dynamics, demographic noise, and initial conditions.
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© The Author(s), under exclusive licence to EDP Sciences, SIF and Springer-Verlag GmbH Germany, part of Springer Nature 2025
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
