{"id":"1e717ada-8ab9-433a-9832-c0ced27380ea","arxiv_id":"2507.10621","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":2,"one_line_summary":"A conceptual chapter proposing LLM-Nash and LLM-Stackelberg games that shift equilibrium analysis from action spaces to prompt spaces for AI-driven cyber defense.","lead":"This chapter argues that combining game theory with LLM-powered agents could replace reactive, manual cyber defense with proactive, strategic responses. It introduces theoretical models in which agents choose reasoning prompts rather than raw actions, and it surveys multi-agent workflows for security.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The paper's sole numerical demonstration contradicts its own LLM-Nash definition: with the stated distributions, (x5,y3) is not an equilibrium, so the central claim currently lacks a valid example.","rationale":"The paper is a position piece; its central novel claim is that equilibria over prompt spaces are meaningful for LLM agents. For that claim to hold, at least one internally valid instantiation must be shown. The only quantitative instantiation is the RPS example. The arithmetic in the example is not a harmless typo: after correcting numbers, the claimed equilibrium fails Definition 1, because a unilateral deviation by the defender to y5 strictly improves her payoff. The reader's weakest assumption correctly targeted the fixed-distribution/common-knowledge structure and the unsupported empirical claim; I push slightly further by showing that, even accepting those assumptions and prompt distributions, the specific equilibrium assertion is false. This does not refute the conceptual framework—one could choose different prompts or add mixed strategies over prompts—but it strips the paper of its only evidence that the concept is operational. Given that the chapter is explicitly a vision/program chapter rather than a validated technical result, the appropriate disposition remains conditional: accept a revised version that repairs the example and provides, at minimum, a verified computational instance or real LLM evaluation. I would not escalate to reject because the definition itself is coherent and the chapter's survey content is useful.","tokens_in":26562,"tokens_out":4632,"duration_ms":45783,"concrete_test":"Recompute the full 5×5 expected-utility matrix Uij = µA(xi)ᵀ UA µD(yj) for Section 4.1 exactly (a 10-line script). Then check the two Definition 1 conditions: (i) x5 ∈ arg max_i Ui3; (ii) y3 ∈ arg min_j U5j (defender minimizes in zero-sum). Direct arithmetic gives U53 = 0 and U55 = −0.02, so y5 beats y3 and the pair is not an equilibrium. If instead the authors intend a demonstration with real LLMs, the test is to run the stated five attacker and five defender prompts on a fixed LLM with seed/temperature control, report empirical distributions and utility table, and show the claimed pair survives unilateral deviations; currently no such data exist.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Definition 1 (Section 4.1, Eqs. 15-16) is coherent only if each prompt induces a fixed, known action distribution (µA, µD), and the RPS example is the entire validation of the new solution concept. But direct computation from the published distributions refutes it. For defender y3 = (0.2, 0.6, 0.2), the attacker's pure-strategy expected payoffs under UA are (−0.4, 0, 0.4); hence U53 = µA(x5)ᵀ UA µD(y3) = 0, not 0.02. Against attacker x5 = (0.3, 0.4, 0.3), defender y5 = (0.3, 0.2, 0.5) yields attacker payoff −0.02, strictly less than every other y (y1-y4 give 0); since uD = −uA, the defender strictly prefers y5 over y3. Thus (x5, y3) violates both best-response conditions of Definition 1. The paper's 'empirical evaluations indicate' assertion is also unsupported: no LLM, prompt text, sampling protocol, or data are given, so the claimed distributions are not evidenced. The stated U35 = 0.04 is likewise wrong (correct value is 0.03). The central claim—that LLM-Nash over prompt spaces is a demonstrated solution concept—therefore rests on an invalid example; the framework may be repairable, but as written the argument does not establish it.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript is a chapter-length position paper arguing that game theory and LLM-based agentic AI should be jointly used to model and operate cyber defense. Sections 2–3 review classical security games (static, dynamic, Bayesian, and signaling games with Nash, Stackelberg, SPNE, and PBNE concepts) and argue that LLMs relax classical assumptions of full rationality, common knowledge, and static payoffs. Section 4 introduces the principal original contributions: an LLM-Nash reasoning equilibrium (Definition 1, Eqs. (15)–(16)) in which players choose prompts that induce action distributions through LLMs, and an LLM-Stackelberg equilibrium for signaling with prompt-based message and response policies. Section 5 surveys multi-agent workflow architectures (chain, star, parallel, feedback, and hybrid) and invokes a 'gestalt' games-in-games model for the overall system. The central claimed demonstration is the prompt-space RPS example in §4.1, which asserts that (x5, y3) is a reasoning-level equilibrium.","tokens_in":26955,"tokens_out":10081,"duration_ms":103401,"significance":"If the proposed equilibrium concepts were rigorously established, they would provide a useful bridge between the mature security-game literature and the rapidly growing area of LLM-based agents, enabling system designers to reason about strategy at the level of prompts rather than actions. The paper is readable, covers a wide literature, and its workflow taxonomy is a practical contribution. However, the new solution concepts are not backed by any theorem, experiment, or valid numerical example: the single illustrative equilibrium pair is refuted by the paper's own numbers, and the 'empirical evaluations' are not described. The paper ships no code, data, or machine-checked proofs, so those verifiable strengths are absent. The significance is therefore conditional on a corrected or replaced demonstration and on an explicit statement of the epistemic assumptions of Definition 1.","major_comments":[{"comment":"The paper's only numerical demonstration of Definition 1 is invalid. Direct computation from the stated distributions gives U53 = 0, not 0.02, and U35 = 0.03, not 0.04. More importantly, against attacker x5 = (0.3, 0.4, 0.3), the attacker's expected payoff against defender prompts is 0 for y1 through y4 and −0.02 for y5 = (0.3, 0.2, 0.5); since uD = −uA, the defender strictly prefers y5 over y3 = (0.2, 0.6, 0.2), so inequality (16) fails for (x5, y3). The sentence beginning 'Assume empirical evaluations indicate' does not repair this, because no evaluation data are provided and the numerical contradiction is intrinsic to the stated distributions. This is load-bearing: Section 6's claim that the chapter shows strategic behavior emerging from prompt-space equilibria rests entirely on this example.","section":"§4.1, RPS example"},{"comment":"The empirical assertion is unsupported. The manuscript provides no LLM, no prompt texts, no inference protocol, no sample size, and no results; it simply assumes that x5 and y3 are best responses. Because Definition 1 presupposes that the distributions µA(x_i) and µD(y_j) are known to the players, the claimed best responses are not derived from the model but imported as an unexamined assumption. The example should be replaced by a pair verified arithmetically from the stated distributions, or explicitly labeled as a hypothetical illustration; in the latter case it cannot serve as validation of the new concept.","section":"§4.1, 'Assume empirical evaluations indicate...'"},{"comment":"The epistemic assumptions of Definition 1 are not reconciled with the paper's own critique of classical game theory. Equations (15)–(16) are well defined only if each prompt induces a fixed action distribution and both players know these distributions with common knowledge; otherwise the expected utilities and the unilateral-deviation test are not well posed. Yet Section 3.1 motivates LLMs precisely by rejecting common knowledge and perfect rationality. The paper should state explicitly whether LLM-Nash equilibrium is a normative concept over known distributions, a positive description of LLM behavior, or an equilibrium of a game in which the distributions are learned, and it should specify the information available to each player when comparing prompts.","section":"§4.1, Definition 1 vs. §3.1"},{"comment":"No existence, uniqueness, or tractability result is given for either LLM-Nash or LLM-Stackelberg equilibrium, and no experiment is reported. For a chapter that announces new solution concepts, the absence of any positive formal or empirical result—together with the invalid RPS example—means the central claim is currently unsupported. At minimum, a correct finite-space equilibrium example and a statement of conditions under which an LLM-Nash equilibrium exists (or a counterexample to existence) are needed before the concepts can be used as advertised in Section 6.","section":"§4 (overall)"}],"minor_comments":[{"comment":"The phrase 'The LLM-Stackelberg Stackelberg game' contains a duplicated word; also, 'we introduce LLM-Stackelberg game' needs an article.","section":"§4.2"},{"comment":"The text refers to 'Figure 5.1.6', but the surrounding figures are numbered Figure 5, so the cross-reference is incorrect.","section":"§5.1.6"},{"comment":"The phrase 'we call a view Gestalt game-theoretic' is awkward; it should be reworded as 'a gestalt game-theoretic view'.","section":"§5.4"},{"comment":"Several arXiv identifiers are malformed or missing their declaration numbers (e.g., [1] appears as 'arXiv:240518718' and [31] as 'arXiv:250201587'); these should be checked against the journal's reference style.","section":"References"},{"comment":"The notation 'I= {1, 2, . . . , N}' has a spacing issue, and the mindset parameters θ, δ in §4.1 are introduced without a clear domain; both should be cleaned up.","section":"§2.1 and §4.1"}],"recommendation":"major_revision","confidential_remarks":"To the editor: the manuscript is a broad, single-author survey/vision chapter whose main original content is Section 4, and that section's only numerical example is internally inconsistent. I do not recommend rejection because the framework is potentially repairable: the authors can correct or replace the example, state the epistemic assumptions of Definition 1, and temper the claims in Section 6. There is also a question of fit: the paper reads more like an invited book chapter than a self-contained research article, so if the journal expects original results, the authors should be directed to substantiate Section 4 with formal statements or empirical evaluation. The reliance on the author's own games-in-games framework [4] for the gestalt layer is acceptable but remains descriptive rather than independently validated in this manuscript."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here is the short version: this is a readable survey/position chapter on game theory meeting LLM agents in cybersecurity, but the central claimed contribution—LLM-Nash and LLM-Stackelberg equilibria over prompt spaces—is not demonstrated. The only concrete example is arithmetically wrong, and the phrase 'empirical evaluations indicate' is unsupported by any data.\n\nThe survey parts earn credit. Sections 2 and 3 give a competent tour of static/dynamic/Bayesian/signaling security games, cyber deception, LLM capabilities, and the relaxation of common-knowledge and perfect-rationality assumptions. Section 5's taxonomy of multi-agent workflows (chain, star, parallel, feedback, hybrid) is useful for practitioners. The author clearly knows the literature, and the references are largely appropriate.\n\nThe soft spot is load-bearing and sits in Section 4. Definition 1 is coherent but it is classical Nash/Stackelberg equilibrium with prompts relabeled as strategies: if each prompt induces a fixed action distribution, Eqs. (15)-(16) are exactly the standard best-response inequalities. That is a change of vocabulary, not a new solution concept. More seriously, the RPS example contradicts Definition 1. With the stated distributions, (x5,y3) yields attacker payoff 0, not 0.02 as claimed; the defender can strictly improve by playing y5 (attacker payoff -0.02, so defender +0.02). So (x5,y3) is not an equilibrium. U35 is also 0.03, not 0.04. The 'empirical evaluations' assertion substitutes for missing evidence: no LLM, prompt text, sampling protocol, or data appear anywhere. For a paper whose whole point is new equilibrium concepts, this is a load-bearing flaw in the only demonstration. The gestalt/games-in-games layer leans on the author's own framework without independent validation, which is not wrong by itself but adds to the sense that the piece is programmatic rather than settled.\n\nWho would get value? Readers wanting an accessible map of game-theoretic concepts and LLM-agent architectures in security will find Sections 2, 3, and 5 useful. But as a research contribution, the core claim is unestablished. I would not send this to a research venue in its current form. If the author corrects the arithmetic, adds real LLM experiments showing prompt-to-distribution mappings, and either supplies a theorem that separates LLM-Nash from classical Nash or explicitly positions this as a conceptual framework, it could be a citable chapter. For now, treat it as a draft.","headline":"Readable survey with a flawed core: the LLM-Nash example is arithmetically wrong, and the 'new' equilibrium concepts reduce to classical ones under relabeled prompts.","tokens_in":27397,"tokens_out":5569,"would_cite":false,"duration_ms":52829,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["91A80","91A10"],"pacs":[],"model":"deepseek-v4-flash","headline":"This chapter argues that the arrival of large-language-model agents changes the natural locus of strategy in cybersecurity games: instead of choosing actions, an agent chooses a reasoning prompt, and the model's output distribution over…","keywords":["game theory","large language models","agentic AI","cybersecurity","Nash equilibrium","Stackelberg equilibrium","prompt engineering","multi-agent systems"],"falsifier":"Recompute the 5×5 expected-utility table in the Rock-Paper-Scissors example from the stated action distributions and zero-sum payoff matrix, and check whether (x5,y3) is indeed a mutual best response; the paper states U53=0.02 and U35=0.04, but direct computation yields different values if the arithmetic is in error. A second check is to instrument an actual LLM with the five x-prompts and five y-prompts, measure the induced action frequencies, and test whether the best responses match the equilibrium conditions.","tokens_in":26386,"feed_emoji":"🧠","tokens_out":6971,"duration_ms":74408,"temperature":0.7,"pith_summary":"The chapter argues that the arrival of large-language-model agents changes the natural locus of strategy in cybersecurity games: instead of choosing actions, an agent chooses a reasoning prompt, and the model's output distribution over actions is the resulting mixed strategy. It introduces two equilibrium concepts over prompt spaces—the LLM-Nash equilibrium and the LLM-Stackelberg equilibrium—and claims these capture how bounded, language-conditioned agents would settle into stable strategic patterns. The paper also proposes that multi-agent workflows built from LLM agents can be modeled as layered games-in-games, with a 'gestalt' equilibrium characterizing the whole system. If this view is right, cyber defense becomes a matter of designing reasoning environments (prompts, retrieval, fine-tuning) rather than just patching actions.","feed_headline":"LLM agents shift cyber games from actions to prompts","feed_subtitle":"Two new solution concepts—LLM-Nash and LLM-Stackelberg equilibrium—put reasoning itself on the table.","key_machinery":"The load-bearing object is the reasoning prompt as a metastrategy: a structured natural-language string that conditions an LLM's generative policy and thereby induces a probability distribution over the agent's actions. Equilibrium is defined at the level of prompt pairs, with the LLM-Nash definition (Definition 1) requiring expected-utility best responses in prompt space, and the LLM-Stackelberg definition adding a leader's anticipatory maximization over the follower's prompt-based best response. The third mechanism is the 'gestalt' game: a games-in-games composition in which local cooperative and adversarial games among workflow agents are nested so that the whole system exhibits an emergent strategic equilibrium that is not decomposable into isolated pairwise games.","core_discovery":"The central claim is that equilibrium analysis in cybersecurity should move from the action space to the reasoning space. For a two-player LLM game, each agent's mindset includes private information, a prompt space, and an LLM whose parameters generate a probability distribution over actions for each prompt. Definition 1 defines an LLM-Nash equilibrium as a pair of prompts from which neither player can raise expected utility by unilaterally choosing a different prompt. The LLM-Stackelberg equilibrium extends this to a leader-follower setting: the leader chooses a prompt anticipating that the follower will best-respond in its own prompt space for every received message. The paper illustrates the Nash concept with an extended Rock-Paper-Scissors game in which the claimed equilibrium (x5,y3) induces behavioral distributions that differ from the classical uniform random equilibrium; it concludes that stable equilibria under LLM constraints may be rational within the agents' reasoning models yet suboptimal in the classical sense. For multi-agent workflows, the chapter combines these local games into a 'gestalt' game-theoretic description based on the games-in-games framework.","pith_inferences":["The chapter's framing implies that the security of an LLM-based system is the security of its prompt space: adversarial prompt injection becomes an attempt to shift the induced distributions and thereby break the equilibrium, so defense should formally track equilibrium-preserving prompt perturbations.","A direct testable extension is to treat soft prompts as continuous and use gradient-based optimization to solve for LLM-Nash equilibria; whether these optima match the discrete prompt examples in the chapter is an open question.","The same equilibrium language could govern multi-agent alignment: choose system prompts so that the ensemble of models is at an equilibrium, meaning no single model gains by secret deviation—an implicit but unstated design principle for agent swarms.","The gestalt games-in-games view suggests that cross-level coupling (a local agent's prompt change affects the global game) can be analyzed through nested equilibria; this is a natural place to bring in learning dynamics such as regret minimization to test convergence."],"forward_implications":["Defenders can be engineered at the reasoning level: changing prompts, retrieval context, and fine-tuning budgets becomes the main control knob, rather than hard-coded defense policies.","Equilibrium outcomes under LLM constraints may be suboptimal compared with classical Nash equilibria, so defenders should expect and plan for bounded, language-conditioned adversaries that stabilize on non-classical patterns.","The LLM-Stackelberg model gives a formal language for studying deception, phishing, and persuasion in natural language, connecting game theory to message-level interaction rather than symbolic signals.","Multi-agent LLM workflows (chain, star, parallel, feedback) can be analyzed as layered games, so questions about robustness, oscillation, and coordination become equilibrium-selection questions.","Alignment of LLMs to human preferences can be reframed as computing Nash or Pareto equilibria among competing preference models, which is exactly the direction the paper points to via Nash learning from human feedback."],"supporting_citations":[{"why":"Supplies the foundational game-theoretic framing of cybersecurity interactions that the chapter extends.","marker":"[34]"},{"why":"Provides the signaling-game model that the LLM-Stackelberg formulation builds on.","marker":"[45]"},{"why":"Establishes level-k bounded rationality as the behavioral premise behind prompt-level reasoning.","marker":"[2]"},{"why":"Gives the dynamic deception-by-design game that the paper cites for belief manipulation under adversarial signaling.","marker":"[17]"},{"why":"Introduces Nash learning from human feedback, the alignment method the paper maps onto game-theoretic equilibria.","marker":"[35]"},{"why":"Documents non-transitive preferences and Condorcet cycles that motivate equilibrium rather than ranking-based alignment.","marker":"[32]"},{"why":"Demonstrates verbalized Bayesian persuasion in LLM prompting, a direct precedent for prompt-driven strategic communication.","marker":"[31]"},{"why":"Defines the games-in-games framework used for the multi-agent gestalt game model.","marker":"[4]"},{"why":"Models coupled signaling and FlipIt games for APT scenarios, an example of the gestalt structure.","marker":"[43]"},{"why":"Formulates a cross-layer Stackelberg game for the timing of cyber deception, supporting the leader-follower prompt model.","marker":"[59]"}],"fun_headline_variants":["LLM games: equilibrium shifts from actions to prompts","Cyber defense equilibria now defined over prompt space","LLM-Nash equilibrium: reasoning replaces action in cybersecurity","Game theory meets LLMs: new equilibria for reasoning agents","From actions to prompts: how LLMs change cyber game theory"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The LLM-Nash equilibrium definition assumes that each prompt reliably induces a known probability distribution over actions and that both players know these distributions well enough to optimize against them; if an LLM's output distribution is unstable, unobservable, or unknown to the other player, the equilibrium concept does not apply.","fun_headline_variants_meta":{"raw":{"variants":["LLM games: equilibrium shifts from actions to prompts","Cyber defense equilibria now defined over prompt space","LLM-Nash equilibrium: reasoning replaces action in cybersecurity","Game theory meets LLMs: new equilibria for reasoning agents","From actions to prompts: how LLMs change cyber game theory"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00065,"raw_usage":{"total_tokens":3041,"prompt_tokens":1063,"completion_tokens":1978,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":679,"completion_tokens_details":{"reasoning_tokens":1898}},"tokens_in":679,"tokens_out":1978,"duration_ms":16937,"temperature":1.0,"reasoning_tokens":1898,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T17:46:52.701916+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the 5×5 expected-utility table in the Rock-Paper-Scissors example from the stated action distributions and zero-sum payoff matrix, and check whether (x5,y3) is indeed a mutual best response; the paper states U53=0.02 and U35=0.04, but direct computation yields different values if the arithmetic is in error. A second check is to instrument an actual LLM with the five x-prompts and five y-prompts, measure the induced action frequencies, and test whether the best responses match the equilibrium conditions.","supporting_citations":[{"cited_title":"ACM Computing Surveys (CSUR) 45(3):1–39","cited_arxiv_id":null,"evidence_quote":"Supplies the foundational game-theoretic framing of cybersecurity interactions that the chapter extends."},{"cited_title":"IEEE Transactions on Information Forensics and Security 14(7):1871–1886","cited_arxiv_id":null,"evidence_quote":"Provides the signaling-game model that the LLM-Stackelberg formulation builds on."},{"cited_title":"In: 2015 IEEE 14th International Conference on Machine Learning and Applications (ICMLA), IEEE, pp 941–947","cited_arxiv_id":null,"evidence_quote":"Establishes level-k bounded rationality as the behavioral premise behind prompt-level reasoning."},{"cited_title":"In: Decision and Game Theory for Security: 8th International Conference, GameSec 2017, Vienna, Austria, October 23-25, 2017, Proceedings, Springer, pp 273–294","cited_arxiv_id":null,"evidence_quote":"Gives the dynamic deception-by-design game that the paper cites for belief manipulation under adversarial signaling."},{"cited_title":"arXiv preprint arXiv:231200886 18","cited_arxiv_id":null,"evidence_quote":"Introduces Nash learning from human feedback, the alignment method the paper maps onto game-theoretic equilibria."},{"cited_title":"arXiv preprint arXiv:250310990","cited_arxiv_id":null,"evidence_quote":"Documents non-transitive preferences and Condorcet cycles that motivate equilibrium rather than ranking-based alignment."},{"cited_title":"arXiv preprint arXiv:250201587","cited_arxiv_id":null,"evidence_quote":"Demonstrates verbalized Bayesian persuasion in LLM prompting, a direct precedent for prompt-driven strategic communication."},{"cited_title":"IEEE Transactions on Control of Network Systems 7(3):1056–1068","cited_arxiv_id":null,"evidence_quote":"Defines the games-in-games framework used for the multi-agent gestalt game model."},{"cited_title":"In: International Conference on Decision and Game Theory for Security (GameSec), Springer, pp 289–308, DOI 10.1007/978-3-319-25594-1 17","cited_arxiv_id":null,"evidence_quote":"Models coupled signaling and FlipIt games for APT scenarios, an example of the gestalt structure."},{"cited_title":"arXiv preprint arXiv:250521244","cited_arxiv_id":null,"evidence_quote":"Formulates a cross-layer Stackelberg game for the timing of cyber deception, supporting the leader-follower prompt model."}],"review_version":1}