Integrated Resource Allocation and Strategy Synthesis in Safety Games on Graphs with Deception

Deception plays a crucial role in strategic interactions with incomplete information. Motivated by security applications, we study a class of two-player turn-based deterministic games with one-sided incomplete information, in which player 1 (P1) aims to prevent player 2 (P2) from reaching a set of target states. In addition to actions, P1 can place two kinds of deception resources: "traps" and "fake targets" to disinform P2 about the transition dynamics and payoff of the game. Traps "hide the real" by making trap states appear normal, while fake targets "reveal the fiction" by advertising non-target states as targets. We are interested in jointly synthesizing optimal decoy placement and deceptive defense strategies for P1 that exploits P2's misinformation. We introduce a novel hypergame on graph model and two solution concepts: stealthy deceptive sure winning and stealthy deceptive almost-sure winning. These identify states from which P1 can prevent P2 from reaching the target in a finite number of steps or with probability one without allowing P2 to become aware that it is being deceived. Consequently, determining the optimal decoy placement corresponds to maximizing the size of P1's deceptive winning region. Considering the combinatorial complexity of exploring all decoy allocations, we utilize compositional synthesis concepts to show that the objective function for decoy placement is monotone, non-decreasing, and, in certain cases, sub- or super-modular. This leads to a greedy algorithm for decoy placement, achieving a $(1 - 1/e)$-approximation when the objective function is sub- or super-modular. The proposed hypergame model and solution concepts contribute to understanding the optimal deception resource allocation and deception strategies in various security applications.