The Mathematics of Deception: A Unified Framework for AI-Driven Honeypot Systems

Source abstract

Honeypots occupy a peculiar place in the security landscape: they are among the few mechanisms whose value is defined by their failure. A firewall that is never tested, an intrusion detection system that never alarms—these are successes. A honeypot that is never attacked is useless. This inversion of the usual security logic gives honeypots an unusually clean signal, but it has also kept their theoretical foundations fragmented. Game theorists model deception in isolation; information theorists treat intrusion detection generically; queuing theorists model traffic separately; and the recent wave of AI-driven honeypots—powered by Large Language Models and Reinforcement Learning—has proceeded largely without formal mathematical treatment.This paper attempts a synthesis. We develop a unified mathematical framework for AI-driven honeypot systems, building on six pillars: stochastic processes (the honeypot as a continuous time Markov chain), game theory (the honeypot as a Bayesian signaling device), detection theory (the honeypot as a near-ideal hypothesis test), information theory (the honeypot as a channel for intelligence), optimization (the honeypot as a constrained POMDP), and AI integration (LLM as observation kernel, RL as policy optimization).Along the way, we derive equilibrium conditions, detection thresholds, optimal deployment fractions, information-theoretic bounds, and LLM deception limits. We prove that the mixed strategy equilibrium is unique and Lyapunov stable, solve the POMDP for a reduced state space to obtain a threshold policy, and show that in the myopic case this threshold coincides exactly with the game-theoretic equilibrium under consistent payoff specifications. We generalize the Poisson detection assumption to a Weibull hazard model and derive a bound on the mutual information achievable under LLM-generated observations via the data processing inequality.We accompany the analytical development with numerical Monte Carlo experiments based on exact Gillespie simulation of the CTMC, replicator dynamics for the game-theoretic component, and parametric sampling for the detection and information-theoretic components. These experiments validate the closed-form predictions and expose the qualitative behavior of the system in transient and steady state. All quantitative claims are analytical; the simulations serve as numerical illustrations, not as empirical validation of real-world deployments.