Bi-level Cyber Warfare Game

• The paper introduces a bi-level Stackelberg model where a defender commits to cyber-security investments and the attacker best responds to maximize control degradation.
• It employs a networked control system framework with H2 cost functions, binary node attack success probabilities, and a cost-based Stackelberg equilibrium to resolve optimal strategies.
• The work integrates cyber actions with physical control consequences, using robust defense strategies and multi-resolution game architectures to manage uncertainty.

Searching arXiv for recent and foundational papers related to bi-level cyber warfare games, Stackelberg cyber models, and hierarchical attacker–defender formulations. Bi-level cyber warfare game denotes a hierarchical attacker–defender formulation in which one decision-maker acts at an upper level and anticipates a lower-level adversarial response, with cyber actions evaluated through their induced operational consequences rather than as purely abstract network events. In the literature represented here, the concept appears most explicitly in networked control systems as a defender–leader, attacker–follower Stackelberg game tied to closed-loop control degradation (Shukla et al., 2021), while related work broadens the picture toward simultaneous critical-node games (Dragotto et al., 2023), dynamic partially observable Stackelberg games (Zheng et al., 2021), insurance-embedded bi-level cyber conflict (Zhang et al., 2019), and cross-echelon multi-resolution game architectures (Yang et al., 2 Jul 2025). Taken together, these works define a family of hierarchical cyber conflict models in which upper-level commitment, lower-level response, uncertainty, and physics- or mission-based consequence models are jointly represented.

1. Definitional scope and conceptual boundaries

A bi-level cyber warfare game is most precisely instantiated as a leader–follower resource-allocation problem in which the leader commits to a defensive or offensive posture and the follower then optimizes against that commitment. The clearest instance in the supplied literature is a leader–follower bi-level game for cyber-security investment in a networked control system, where the defender is the leader and the attacker the follower (Shukla et al., 2021). In generic bilevel terms given there, the upper level (leader/defender) chooses a defense allocation to minimize expected control degradation induced by the attacker’s lower-level best response, while the lower level (follower/attacker) chooses an attack allocation to maximize expected control degradation under the chosen defense (Shukla et al., 2021).

This differentiates bi-level cyber warfare games from simultaneous attacker–defender models. "The Critical Node Game" is explicitly a 2-player, simultaneous, non-cooperative, complete-information, pure-strategy integer programming game and is therefore not itself bilevel, even though each player solves an optimization problem parameterized by the other player’s variables (Dragotto et al., 2023). Likewise, the earlier networked-control paper formulates a two-player general-sum mixed-strategy game rather than a Stackelberg problem, but already exhibits a nested structure because each realized attack/protection pattern induces a lower-level structured LQR synthesis problem (Shukla et al., 2018).

The broader literature represented here also shows that “bi-level” need not mean a static single-leader/single-follower optimization only. One-sided zero-sum partially observable stochastic games introduce sequential commitment under asymmetric information and public actions (Zheng et al., 2021). Cross-echelon cyber warfare models replace a single upper/lower split with coupled tactical and strategic layers linked by abstraction and refinement operations (Yang et al., 2 Jul 2025). Cyber-insurance formulations embed an attacker–defender cyber conflict as the lower-level game inside an upper-level principal-agent problem of moral-hazard type (Zhang et al., 2019). This suggests that the phrase “bi-level cyber warfare game” is best understood as a family of hierarchical cyber conflict models rather than a single canonical formalism.

2. Canonical Stackelberg form in networked control systems

The most explicit mathematical realization is the cyber-security investment game for a networked control system operating under state-feedback $\mathcal H_2$ control (Shukla et al., 2021). The plant is modeled as a continuous-time linear system

$$\dot{\bm x}(t) = \bm A \bm x(t) + \bm B \bm u(t) + \bm D \bm w(t),$$

with linear state feedback

$\bm u(t) = -\bm K \bm x(t).$

The control objective is the $\mathcal H_2$ cost

$$J(\bm K)= \operatorname{trace}(\bm D^T \bm P \bm D)$$

subject to the Lyapunov equation

$$(\bm A-\bm B\bm K)^T \bm P + \bm P(\bm A-\bm B\bm K) =-(\bm Q+\bm K^T \bm R \bm K).$$

This control layer gives physical meaning to the cyber game because the attacker’s objective is to increase $J$ by disabling communications, not merely to remove nodes abstractly (Shukla et al., 2021).

A successful attack on node $i$ disables all communications associated with that node, so the entire $i$-th block row and block column of $\bm K$ become zero (Shukla et al., 2021). Post-attack outcomes are encoded by binary sparsity patterns

$$\dot{\bm x}(t) = \bm A \bm x(t) + \bm B \bm u(t) + \bm D \bm w(t),$$0

and the induced control-performance loss is

$$\dot{\bm x}(t) = \bm A \bm x(t) + \bm B \bm u(t) + \bm D \bm w(t),$$1

with open-loop loss

$$\dot{\bm x}(t) = \bm A \bm x(t) + \bm B \bm u(t) + \bm D \bm w(t),$$2

The cyber layer thus selects node-disruption patterns, while the control layer translates them into structured $$\dot{\bm x}(t) = \bm A \bm x(t) + \bm B \bm u(t) + \bm D \bm w(t),$$3 degradation (Shukla et al., 2021).

The players choose discrete effort allocations

$$\dot{\bm x}(t) = \bm A \bm x(t) + \bm B \bm u(t) + \bm D \bm w(t),$$4

with

$$\dot{\bm x}(t) = \bm A \bm x(t) + \bm B \bm u(t) + \bm D \bm w(t),$$5

and node-level attack success probability

$$\dot{\bm x}(t) = \bm A \bm x(t) + \bm B \bm u(t) + \bm D \bm w(t),$$6

Budget constraints are normalized as

$$\dot{\bm x}(t) = \bm A \bm x(t) + \bm B \bm u(t) + \bm D \bm w(t),$$7

The attacker’s payoff is expected $$\dot{\bm x}(t) = \bm A \bm x(t) + \bm B \bm u(t) + \bm D \bm w(t),$$8-performance loss,

$$\dot{\bm x}(t) = \bm A \bm x(t) + \bm B \bm u(t) + \bm D \bm w(t),$$9

and the defender’s payoff is

$\bm u(t) = -\bm K \bm x(t).$0

Because the defender commits first and the attacker then best-responds, the game becomes a defender-led Stackelberg bilevel optimization (Shukla et al., 2021).

The lower-level attacker best response is

$\bm u(t) = -\bm K \bm x(t).$1

with least-cost tie-breaking

$\bm u(t) = -\bm K \bm x(t).$2

The defender then solves

$\bm u(t) = -\bm K \bm x(t).$3

Equivalently, because the game is zero-sum,

$\bm u(t) = -\bm K \bm x(t).$4

This is the paper’s exact bi-level cyber warfare formulation (Shukla et al., 2021).

3. Cost-based equilibrium, node criticality, and robust defense

A distinctive feature of the Stackelberg NCS formulation is that it is not satisfied with a payoff-equivalent Stackelberg equilibrium alone. Standard backward induction can produce multiple equilibria with identical payoffs but different investment costs, which is undesirable in a long-term security-investment setting (Shukla et al., 2021). The model therefore defines a cost-based Stackelberg equilibrium (CBSE) through cost-based backward induction.

At the lower level, if multiple attacker best responses exist, the attacker selects the one with smallest cost. At the upper level, if multiple defender actions are optimal, the defender selects the lowest-cost one. Formally, with

$\bm u(t) = -\bm K \bm x(t).$5

the pair $\bm u(t) = -\bm K \bm x(t).$6 is the CBSE (Shukla et al., 2021).

Several comparative-statical properties are established. A CBSE exists because the action spaces are finite; all CBSEs induce the same control-performance loss/payoff; a player’s equilibrium payoff is non-increasing in its own cost-per-node when the opponent’s costs are fixed; if attacker cost is very low and defender cost very high, the attacker can drive the system to open-loop loss $\bm u(t) = -\bm K \bm x(t).$7; if defender cost is sufficiently low, it can protect enough nodes to make attacker payoff zero; and increasing a player’s number of investment levels cannot hurt that player’s equilibrium payoff (Shukla et al., 2021). These results make the model a prescriptive planning tool rather than a purely descriptive game.

Node criticality is encoded through the performance-loss terms $\bm u(t) = -\bm K \bm x(t).$8, especially the one-node losses (Shukla et al., 2021). This creates a control-grounded notion of “important” nodes: more important nodes are attacked and protected first unless attack or protection costs become disproportionately high. This suggests a cyber-warfare interpretation close to resource-constrained critical-node interdiction, but with interdiction value derived from structured control loss rather than graph connectivity alone (Shukla et al., 2021).

The same paper also introduces a robust-defense method for the case where the defender does not know the attacker’s budget/resources. Instead of using a Bayesian prior, the defender assumes the most powerful hypothetical attacker: $\bm u(t) = -\bm K \bm x(t).$9 and solves

$\mathcal H_2$0

The resulting method is explicitly described as a worst-case, sequential, distribution-free robust planning rule rather than a Bayesian Stackelberg defense (Shukla et al., 2021). The model proves that this robust-defense method is conservative relative to the complete-information Stackelberg outcome: $\mathcal H_2$1 or equivalently

$\mathcal H_2$2

4. Related formulations: simultaneous games, dynamic games, and game-of-games architectures

Several adjacent models clarify what a bi-level cyber warfare game is not, while also supplying components that can be repurposed into a bilevel structure.

"The Critical Node Game" defines a cloud-network attacker–defender model on an undirected graph $\mathcal H_2$3, with defender variables $\mathcal H_2$4, attacker variables $\mathcal H_2$5, and knapsack constraints $\mathcal H_2$6, $\mathcal H_2$7 (Dragotto et al., 2023). Its defender and attacker objectives

$\mathcal H_2$8

$\mathcal H_2$9

are parameterized by the opponent’s choice, but the solution concept is pure Nash equilibrium or approximate $$J(\bm K)= \operatorname{trace}(\bm D^T \bm P \bm D)$$0-Nash equilibrium rather than Stackelberg commitment (Dragotto et al., 2023). The paper is explicit that it is not explicitly bilevel, not Stackelberg, and not sequential. Still, the binary attack/defense variables, node criticality parameters $$J(\bm K)= \operatorname{trace}(\bm D^T \bm P \bm D)$$1, and payoff structure with $$J(\bm K)= \operatorname{trace}(\bm D^T \bm P \bm D)$$2 are directly reusable in a bilevel defender-leader reinterpretation (Dragotto et al., 2023).

The earlier networked-control paper formulates a general-sum mixed-strategy Nash game over binary attack patterns $$J(\bm K)= \operatorname{trace}(\bm D^T \bm P \bm D)$$3 and protection patterns $$J(\bm K)= \operatorname{trace}(\bm D^T \bm P \bm D)$$4, with realized communication pattern

$$J(\bm K)= \operatorname{trace}(\bm D^T \bm P \bm D)$$5

and payoffs

$$J(\bm K)= \operatorname{trace}(\bm D^T \bm P \bm D)$$6

(Shukla et al., 2018). Although not Stackelberg, it already has the upper-level/lower-level flavor because every payoff entry is computed from a lower-level structured LQR problem: $$J(\bm K)= \operatorname{trace}(\bm D^T \bm P \bm D)$$7 This suggests that bilevel cyber warfare games often arise when strategic cyber actions induce a constrained control or optimization problem beneath them (Shukla et al., 2018).

Dynamic and partially observable hierarchy is represented by the one-sided two-player zero-sum partially observable stochastic game with public actions (Zheng et al., 2021). There the leader’s policy maps beliefs to mixed actions,

$$J(\bm K)= \operatorname{trace}(\bm D^T \bm P \bm D)$$8

while the follower’s policy maps belief-state pairs to mixed actions,

$$J(\bm K)= \operatorname{trace}(\bm D^T \bm P \bm D)$$9

The Stackelberg equilibrium embeds the follower’s best-response correspondence

$$(\bm A-\bm B\bm K)^T \bm P + \bm P(\bm A-\bm B\bm K) =-(\bm Q+\bm K^T \bm R \bm K).$$0

and the leader solves against that lower-level response (Zheng et al., 2021). This extends the static bilevel idea into a finite-horizon, one-sided-information, sequential cyber-conflict setting.

At a still broader scale, the multi-resolution cross-echelon framework defines tactical Micro Base Games (MBGs) as extensive-form games and a strategic Macro Strategic Game (MSG) as a zero-sum Markov game over abstracted states (Yang et al., 2 Jul 2025). Zoom-in and zoom-out operators couple the two layers: macro continuation values define micro terminal utilities, and micro terminal outcome probabilities update macro transition/action probabilities (Yang et al., 2 Jul 2025). This is not a standard bilevel program, but it supplies a clear hierarchical attacker–defender architecture in which higher-resolution tactical games and lower-resolution strategic games are mathematically linked.

5. Uncertainty, observability, and robustness

Uncertainty is treated in several distinct ways across the literature, and these distinctions materially affect the meaning of “bi-level cyber warfare game.”

In the NCS Stackelberg model, uncertainty appears in two forms. First, the same security investment is evaluated under uncertain future system models $$(\bm A-\bm B\bm K)^T \bm P + \bm P(\bm A-\bm B\bm K) =-(\bm Q+\bm K^T \bm R \bm K).$$1, and the nominal-model game is reported to be fairly robust across large uncertain model sets (Shukla et al., 2021). Second, the defender may not know the attacker’s budget, leading to the distribution-free worst-case robust-defense rule already described (Shukla et al., 2021). This is a robust-planning rather than a Bayesian-information treatment.

The OTZ-POSG framework treats uncertainty as one-sided partial observability. The follower observes the true state $$(\bm A-\bm B\bm K)^T \bm P + \bm P(\bm A-\bm B\bm K) =-(\bm Q+\bm K^T \bm R \bm K).$$2, while the leader acts on a belief

$$(\bm A-\bm B\bm K)^T \bm P + \bm P(\bm A-\bm B\bm K) =-(\bm Q+\bm K^T \bm R \bm K).$$3

updated by the Bayes rule given in the paper (Zheng et al., 2021). Public actions leak information, so the lower-level response problem is itself strategic over time: a follower may sacrifice immediate reward to avoid revealing hidden state. This motivates the paper’s $$(\bm A-\bm B\bm K)^T \bm P + \bm P(\bm A-\bm B\bm K) =-(\bm Q+\bm K^T \bm R \bm K).$$4-Stackelberg equilibrium and $$(\bm A-\bm B\bm K)^T \bm P + \bm P(\bm A-\bm B\bm K) =-(\bm Q+\bm K^T \bm R \bm K).$$5-sacrifice policy notions (Zheng et al., 2021).

Distributional uncertainty is handled differently in the distributionally robust bi-level framework. There the leader solves

$$(\bm A-\bm B\bm K)^T \bm P + \bm P(\bm A-\bm B\bm K) =-(\bm Q+\bm K^T \bm R \bm K).$$6

subject to a lower-level response

$$(\bm A-\bm B\bm K)^T \bm P + \bm P(\bm A-\bm B\bm K) =-(\bm Q+\bm K^T \bm R \bm K).$$7

under a Wasserstein ambiguity set

$$(\bm A-\bm B\bm K)^T \bm P + \bm P(\bm A-\bm B\bm K) =-(\bm Q+\bm K^T \bm R \bm K).$$8

(Shen et al., 7 Nov 2025). This paper is not a cyber application paper, but it is directly relevant as a methodological template for a bi-level cyber warfare game under distribution shift and incomplete information. A plausible implication is that cyber-warfare models with uncertain threat distributions can be embedded in a bi-level DRO structure when continuous convex approximations are acceptable.

The insurance framework introduces yet another uncertainty layer: the insurer cannot observe defender effort directly, so the upper-level contract problem is of moral-hazard type (Zhang et al., 2019). The lower-level cyber conflict remains strategic and covert through FlipIt, while the upper level sees only induced losses and must maintain insurer and defender participation constraints (Zhang et al., 2019). This suggests that some cyber-warfare bilevel models are not only attacker–defender but principal–agent–attacker structures.

6. Algorithms, tractability, and application domains

Bi-level cyber warfare games are computationally difficult because they typically combine combinatorial cyber action spaces, lower-level best-response or control-design problems, and uncertainty. The NCS Stackelberg model shows this explicitly: after precomputing the loss vector $$(\bm A-\bm B\bm K)^T \bm P + \bm P(\bm A-\bm B\bm K) =-(\bm Q+\bm K^T \bm R \bm K).$$9 over all $J$0 sparsity patterns, exhaustive traversal has worst-case complexity

$J$1

(Shukla et al., 2021). To address scale, the paper extends a bidirectional parallel evolutionary genetic algorithm (BPEGA) to a cost-based BPEGA (CB-BPEGA), with complexity

$J$2

and proves convergence in payoff to CBSE as the number of iterations tends to infinity when crossover and mutation rates are positive (Shukla et al., 2021).

The simultaneous cloud-network critical-node game takes a different route, using a tailored ZERO Regrets cutting-plane algorithm to compute an equilibrium maximizing a chosen objective or an approximate $J$3-NE (Dragotto et al., 2023). The existence problem for pure NE in integer programming games is noted to be $J$4-complete (Dragotto et al., 2023), underscoring that simultaneous discrete games can be hard in a way different from single bilevel programs.

The partially observable Stackelberg game converts the one-stage problem into a linear-fractional program and then a linear program via a Charnes–Cooper transformation, yielding a leader value that is piece-wise linear and a follower value that is piece-wise constant (Zheng et al., 2021). For multiple stages, the paper proposes a belief-space partition method, but warns that the number of partitions can grow doubly exponentially in horizon (Zheng et al., 2021).

The distributionally robust bi-level model replaces the lower-level problem by KKT conditions

$J$5

$J$6

and dualizes the Wasserstein DRO layer into a finite convex reformulation (Shen et al., 7 Nov 2025). This is computationally attractive when convexity and strong duality hold, but those assumptions are restrictive for many cyber problems involving binaries, graphs, or nonconvex attack surfaces.

The application domains in the supplied literature are correspondingly diverse. The most fully instantiated physical-cyber Stackelberg model is validated on wide-area control of electric power systems, including the IEEE 39-bus and 68-bus systems (Shukla et al., 2021). The critical-node simultaneous game is grounded in cloud networks and evaluated on an anonymized Ericsson cloud-native application network (Dragotto et al., 2023). The insurance framework targets IoT networks under APTs with influence-network risk propagation (Zhang et al., 2019). The cross-echelon framework uses an enterprise-network campaign path with deception via decoy files (Yang et al., 2 Jul 2025). These examples suggest that bi-level cyber warfare games are most mature where one can define a tractable consequence model beneath the strategic layer.

7. Strategic significance, misconceptions, and limitations

A persistent misconception is that a bi-level cyber warfare game is simply any attacker–defender game. The supplied literature makes the distinction explicit. Simultaneous Nash models, even when each player solves an optimization problem parameterized by the other’s variables, are not bilevel unless one player’s decision is embedded as an upper-level commitment and the other’s as a lower-level response (Dragotto et al., 2023). Likewise, mixed-strategy attacker–defender games over attack and protection patterns can contain nested lower-level control design without being Stackelberg in the strategic layer (Shukla et al., 2018).

Another misconception is that cyber value can be measured solely by node compromise or connectivity disruption. The strongest bilevel formulations in this set derive cyber value from mission or control consequences. In the NCS Stackelberg game, the critical quantity is loss in optimal closed-loop $J$7 performance caused by attack-induced communication sparsity (Shukla et al., 2021). In the insurance framework, the key state variable is not one-shot compromise but expected control fraction $J$8 and induced interdependent risk $J$9 over a network (Zhang et al., 2019). In cross-echelon campaign models, tactical outcomes matter because they alter strategic transition opportunities and continuation values (Yang et al., 2 Jul 2025). This suggests that robust bi-level cyber warfare modeling requires a consequence layer beneath the cyber action layer.

The main limitations are equally consistent across the literature. Many models are static or one-shot; many assume complete information or stylized uncertainty structures; many use continuous resource allocations where real cyber actions are binary or graph-constrained; and many depend on convexity or finite action spaces for tractability. The NCS Stackelberg model uses a stylized node-level success function $i$0 and does not model correlated attacks or adaptive multi-stage campaigns (Shukla et al., 2021). The critical-node cloud game is not sequential and does not resolve leader identity (Dragotto et al., 2023). The distributionally robust framework assumes convexity, compactness, continuous differentiability, and strong duality (Shen et al., 7 Nov 2025). The cross-echelon framework lacks a single canonical bilevel optimization and instead offers an abstraction/refinement architecture (Yang et al., 2 Jul 2025).

A plausible implication is that future work on bi-level cyber warfare games will continue to hybridize these strands: Stackelberg resource commitment, dynamic partial observability, network interdependence, robust uncertainty models, and cross-echelon abstraction. The supplied literature already shows the main design pattern. One defines critical cyber assets; maps their compromise into operational degradation, mission loss, or interdependent risk; imposes attacker/defender budget or policy constraints; and then solves a leader–follower or hierarchical game in which the upper level anticipates the lower-level rational response (Shukla et al., 2021). That design pattern, rather than any single equation, is the defining structure of the bi-level cyber warfare game.