Omniculus — Game-Theoretic Monitoring Model

A design note for the Phase 3 attention allocator. The problem: you have limited analyst/detection attention and an adversary who adapts to where you look. How do you allocate coverage to maximize defense?

Why Stackelberg, not simultaneous Nash

In security the defender effectively commits first (you set up monitoring), and the attacker observes and best-responds (cases the target, routes around gaps). That asymmetry is a Stackelberg Security Game (SSG) — the model behind real deployments like airport patrol and Coast Guard scheduling (ARMOR/PROTECT). The solution concept is the Strong Stackelberg Equilibrium (SSE): the defender's committed (possibly randomized) strategy that maximizes payoff given the attacker will best-respond.

Key insight: the optimal defense is usually randomized coverage, not a static one. A fixed deployment is fully predictable and gets routed around. Randomization is what makes a system genuinely vigilant.

Formalization for Omniculus

Element Meaning here
Targets t ∈ T Assets/paths an adversary could hit: hosts, segments, data stores, ingress points (graph nodes).
Resources r Units of detection/analyst attention (budget-limited — the core tension).
Coverage c_t Probability target t is monitored under the mixed strategy; Σ c_t ≤ r.
Payoffs Defender covered/uncovered utility per target, weighted by asset value; attacker mirror.
Attacker types Distribution over adversary profiles → Bayesian Stackelberg game.

For small T, solve the SSE via the multiple-LP / ERASER formulation. For large/structured target spaces, use compact representations or column generation.

Realism upgrade: bounded rationality

Pure SSG assumes a perfectly rational attacker. Real adversaries aren't. Replace deterministic best-response with SUQR (Subjective Utility Quantal Response): the attacker picks weaker targets more often but not deterministically. SUQR yields markedly more robust deployments and is the current standard.

How it connects to the rest of Omniculus

  • The graph supplies the target set T and asset values.
  • The allocator outputs a coverage policy the detection engine uses to prioritize attention.
  • The attacker's best-responses are the hardest test cases — mine them to retune detections (feeds the feedback/ loop). This is the adversarial curriculum.

Testing the model (lab only)

  1. Equilibrium checks — is coverage on high-value targets? Is attacker expected payoff driven ≤ 0? Sanity-check degenerate cases.
  2. Agent-based simulation — defender executes the policy vs. attacker populations: rational best-responder, quantal/noisy, adaptive/learning (bandit or RL — the real vigilance test), and off-model/zero-day.
  3. Red-team / purple-team — humans find what the model omits; fold back in.
  4. Shadow mode — detect-only comparison before anything influences action.

Metrics

  • Detection rate and time-to-detect vs. baseline.
  • Attacker expected payoff at equilibrium (target ≤ 0).
  • Coverage robustness — detection rate under a learning attacker over time (flat = good; declining = exploitable predictability).
  • False-positive rate (vigilance that floods analysts is useless).
  • Exploitability gap — optimal vs. naive uniform allocation; quantifies what the game-theoretic design bought.

Suggested implementation path

Start tiny (|T| = 5–10) where the equilibrium is hand-checkable. Solve the LP with PuLP/scipy, wrap in an episode-loop simulation, implement the adaptive attacker as a multi-armed bandit before reaching for full RL.