Finite Population Dynamics Resolve the Central Paradox of the Inspection Game

Abstract

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 $p \approx g$) 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.