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), but its classical analysis is crippled by a paradox: the equilibrium crime rate is found to be independent of both the penalty size ($p$) and the crime gain ($g$). This result severely undermines the policy relevance of the static model, suggesting fines are futile. To resolve this paradox, we employ evolutionary game theory and analyze the long-term fixation probabilities of strategies using finite population dynamics. Our results fundamentally demonstrate that high absolute penalties $p$ are highly effective at suppressing crime by driving the system 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 deterrence levels. Most critically, we analyze the realistic asymptotic limit of extreme population 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 ($x_0$) relative to the deterrence threshold (the ratio of inspection cost to reward for catching a criminal). We find the highly counter-intuitive result of the dominance of the initially rare strategy: crime becomes fixed if $x_0$ is below this threshold, but goes extinct if $x_0$ is above it. These findings highlight the need to move beyond deterministic predictions and emphasize that effective deterrence requires managing demographic noise and initial conditions.