Mohammad Malekraeisi 1 ,* , Ali Akbar Kiani 1 , Hamid Saadatfar 2 1 Department of Computer Engineering, Bi.C., Islamic Azad University, Birjand, Iran ; mohammad.malekraeisi@iau.ac.ir . 2 Department of Computer Engineering, Faculty of Electrical and Computer Engineering, University of Birjand, Birjand, Iran ; saadatfar@birjand.ac.ir . Recei ved: 20 July 202 5 Malekraeisi , M ., Kiani , A. A., & Saadatfar , H . ( 2025 ).

Game - Revised: 18 Septembe r 202 5 Accepted: 16 November 202 5 . Research Annals of Industrial and Systems Engineering , 2 ( 4 ), 306 - 318 .

As computer networks expand and cyberattacks grow more sophisticated, Building Intrusion Detection Systems (IDS) that can withstand intelligent, adaptive adversaries has become a core problem in network security. Much of the existing literature assumes attacker behavior that is static or fully specified in advance, ignoring the uncertainty inherent to any real detection process, including false alarms and missed detections. Such simplifications make it difficult to capture how a defensive system and a learning attacker actually behave against each other over time. We address this gap with a game - theoretic frame work, built on repeated games, that models the ongoing interaction between an IDS and an intelligent attacker. Both players are represented as autonomous agents that follow memory - one strategies and adjust their behavior through gradual learning. To make t he model more realistic, detection uncertainty enters the security payoffs directly through the True Positive Rate (TPR) and the False Positive Rate (FPR), and a separate control parameter captures how quickly the attacker adapts relative to the defender, so we can study fast - adapting attacker scenarios. Numerical simulations produce a phase diagram over the space of attacker learning speed and IDS detection capability, revealing three distinct regimes: attacker dominance, adaptive competition, and a secure state. Faster - learning attackers push the security boundary outward, so the IDS needs stronger detection capability to stay in the secure regime. These findings offer practical guidance for designing and tuning adaptive IDS in dynamic, hostile network e nvironments .

d system , Network s ecurity, Adaptive at tacker, Repeated games , Multi - a gent le arning .

The rapid growth of network infrastructures, cloud platforms, and Internet - based services has made network security central to stability and trust in modern information systems. Intrusion Detection Systems (IDS) sit at the front line of this defense, taske d with flagging malicious activity, cyberattacks, and unauthorized Corresponding Author : mohammad.malekraeisi@iau.ac.ir https://doi.org/10.22105/raise.vi.105 Licensee System Analytics . This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0).

Malekraeisi et al. | Res. Ann. Ind. Syst. Eng . 2 ( 4 ) ( 2025 ) 306 - 3018 behavior [1] . Yet traditional IDSs are increasingly strained by more sophisticated attacks, the rise of intelligent malware, and attackers who rely more and mor e on adaptive, learning - based techniques [2] . Much of the IDS literature still trea ts attacker behavior as static: fixed, predictable, or drawn from previously observed patterns [3] . Real - world attackers, however, increasingly u se machine learning, watch how the defense responds, and gradually refine their strategy, probing IDS behavior and exploiting its blind spots [4] . In effect, they act as intelligent, adaptive agents continually optimizing their decisions inside a hostile environment. At the same time, no real IDS operates without uncertainty: false alarms and missed detections add operationa l cost and can themselves shape what an attacker does next [5] .

Despite this, many analytical and simulation - based models either ignore this uncertainty or treat it too simply, pulling their results away from what actually happens on real networks [6] . Game theory has become a natural tool for capturing this strategic back - and - f orth between defenders and attackers [7] . Rather than analyzing each side's decisions in isolation, security games treat them as part of one interactive process, with each move shaped by what the other player is doing [8] . Pairing game theory with multi - agent learning extends this further, giving researchers a framework for studying how strategies evolve, converge, and compete over the long run among intelligent agents [9] .

Even so, three gaps stand out in the literature. Many game - theoretic security models still assume perfect, error - free detection. The role of the attacker's learning speed relative to the defender has rarely been studied systematically. And most work reports isolated, point - wise results rather than a regime analysis mapping out which regions of the para meter space are secure and which are not [10] .

This paper takes these gaps as its startin g point and proposes a repeated - game framework for the interaction between an adaptive IDS and an intelligent attacker. Both players are modeled as learning agents that follow memory - one strategies, and detection uncertainty is built directly into the payo ff structure through the True Positive Rate (TPR) and the False Positive Rate (FPR) . A separate parameter captures the attacker's learning speed relative to the defender, letting us explore a range of fast - adapting attacker scenarios. Extensive numerical s imulations then yield a phase diagram over the key parameters, revealing three distinct regimes: attacker dominance, adaptive competition, and a secure state. Beyond the theoretical contribution, this regime analysis offers practical guidance for designing and tuning adaptive IDSs in dynamic network environments and for building defenses that hold up against intelligent attackers.

The rest of the paper is organized as follows. Section 2 surveys related work on IDS , security games, and learning - based approac hes. Section 3 formulates the problem and lays out the game - theoretic framework connecting the IDS and the adaptive attacker, including the payoff structure that captures detection uncertainty. Section 4 turns to the learning dynamics, the memory - one strat egies, and the role of the attacker's relative learning speed. Section 5 covers the experimental setup and evaluation procedure. Section 6 presents and discusses the results: strategy convergence, the regime phase diagram, and the influence of key paramete rs.

Section 7 closes the paper with conclusions and directions for future work.

Prior work relevant to this study falls broadly into three strands: IDS and network - security modeling, game - theoretic approaches to cybersecurity, and adaptive m ulti - agent learning in attacker ² defender settings.

IDSs are conventionally grouped into two families: signature - based and anomaly - based. Signature - based systems detect known attacks reli ably but struggle against new or variant ones [1] . Anomaly - based systems take the opposite approach, modeling normal traffic so that deviations flag unknown attacks, at the cost of a higher false - alarm rate [11] . Machine learning has pushed detection accuracy considerably further, with neural networks, support vector machines, and deep - learning models all applied to this problem [3] . Even so, these approaches generally treat the IDS as a passive observer and largely overlook how an attacker's adaptive behavior affects the defense's long - term performance [12] . Many of them also focus narrowly on detec tion metrics, giving only limited attention to the operational costs that false alarms and detection errors actually impose [13] . Game theory occupies a distinct place in cybersecurity resear ch because it models defenders and Game - « attackers strategically rather than in isolation: each side's behavior is analyzed as a response to the other's strategy [6] . Security games of this kind have found wide application in critical - infrastructure protection, network securit y, and cyber defense more broadly [7] . Much of this work relies on static, complete - information games, where both players are assumed to know the full game structure and payoffs [14] , an assumption that rarely survives conta ct with real networks. Repeated games have been proposed to capture longer - term interactions, but many still assume perfect, error - free detection [15] , whi ch again leaves little room to study how false alarms and detection errors actually shape attacker behavior.

Combining game theory with reinforcement learning and multi - agent learning to model dynamic securit y games is an active research area . Here, the defender and the attacker are both cast as learning agents that update their strategies based on environmental feedback [6] , opening the door to studying convergence, instability, and adaptive competition directly. A number o f studies look at adaptive attackers that alter their attack patterns to evade detection [2] , but few examine systematically how the attacker's learning speed shapes the security boundary. As before, most of this work reports point - wise numerical results rather than a regime that maps secure, competitive, and insecure regions across the parameter space [7] .

Taken together, this body of work has advanced the use of game theory and adaptive learning in network security, but gaps remain: detection uncertainty is rarely built explicitly into game - theoretic models, the attacker's relative learning speed is seldom examined for its effect on security stability, and the lack of a comprehensive regime analysis makes it hard to pin down exactly when an IDS can hold a secure state against an intelligent attacker. The framework proposed here is aimed at closing these gaps, offering a systematic, multi - agent - learning - based analysis of the IDS - attacker interaction un der realistic detection uncertainty.

This section lays out the game - theoretic framework used to model the strategic interaction between the IDS and the adaptive attacker. The goal is to capture how both pa rties behave over the long run in a hostile, dynamic environment, accounting explicitly for detection uncertainty and each agent's capacity to learn.

We model the IDS ² attacker interaction as a two - player repeated gam e in which each side behaves as an autonomous, rational agent. Player one, the IDS, monitors the network and decides whether to activate its defensive mechanisms; player two, the attacker, chooses strategies to extract maximum benefit from its attacks. At every round of the game, the IDS chooses between: I. Normal: R outine network monitoring, with no active defense engaged .

II. Active d efense: E ngaging advanced detection and defense mechanisms at some operational cost. The attacker, in turn, chooses between: I. No a ttack: H olding off . II. Attack: Launching an attack on the network. The game repeats over time, with each agent updating its decision based on the system's previous state and the feedback it receives; this is what allows the model to capture long - term, adaptive interaction [14] . The payoffs mirror th e real costs and benefits faced by each side. A successful attack earns the attacker a benefit G and costs the defender a loss L. Engaging active defense costs the IDS an operational amount C ?

, and mounting an attack costs the attacker C ? . When an attack is caught, the attacker pays a penalty P , and the IDS collects a matching reward. Together, these terms balance deterrence, operational cost, and potential attack payoff, a structure common throughout the security - game literature [6] . Malekraeisi et al. | Res. Ann. Ind. Syst. Eng . 2 ( 4 ) ( 2025 ) 306 - 3018

Unlike much of the prior work, which treats detection as complete and error - free, this study builds the IDS's inherent uncertainty directly into the model through two parameters: I. TPR: T he probability that an attack is correc tly identified when defense is active . II. FPR: T he probability of a false alarm when no attack has actually occurred. When active defense is engaged , but no attack actually takes place, the IDS bears an extra cost, with probability equal to the FPR, standing in for wasted resources, service disruption, or the need for human intervention.

When an attack does occur while defense is active, it is detected and blocked with probability TPR and VXFFHHGVZLWKSUREDELOLW\ î735 )UDPLQJWKHSD\RIIVWKLVZD\LQH[ pectation, makes it possible to study the system's long - run behavior under realistically imperfect detection [11] . Both agents follow memory - one strategies: each decision depends only on what happened in the previous round, so the game's state space reduces to the four combinations of the ID S's and attacker's prior actions.

Such a structure keeps the model tractable while still letting it capture reactive, adaptive behavior, an assumption widely used in the repeated - games literature [16] .

Both agents update their behavior through gradual learning to maximize expected long - term payoff. To capture asymmetries in adaptation speed, we int URGXFHDFRQWUROSDUDPHWHU?GHILQHGDVWKHUDWLRRIWKH DWWDFNHU VOHDUQLQJVSHHGWRWKH,'6 V/DUJH?YDOXHVGHVFULEHDWWDFNHUVWKDWRXWSDFHWKH,'6LQUHVSRQGLQJ to environmental change, while small values reflect a defender with the learning advantag e [17] . Put together, this study addresses the long - term behavior of a repeated security game played by learning agents, with payoffs shaped by detection uncertainty and asymmetric learning speeds: specifically, under what conditions the IDS can hold a secure state against an adaptive attacker, and how the security boundary moves across the pa rameter space.

This section describes how the IDS and the attacker gradually update their strategies. The framework rests on repeated games with memory - one strategies, in which each agent adjusts its policy parameters over time to maximize expected long - term payoff, a standard formulation in the repeated - games and multi - agent - learning literature for studying strategic, adaptive interaction [16] , [17] .

As noted above, the system's state at step t i s set by the combination of actions taken in the previous step, giving four possible states: ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? (1) Here N and D denote normal monitoring and active defense, respectively, while NA and A denote no attack and attack. The IDS's memory - one policy is given by the probability vector ’ ? ? ’ ? ? ’ ?

? ’ ? ? ’ ? ? ? ( 2 ) where p ? = Pr (D | s ? ) is the probability of activating defense in state s ? , and the attacker's policy is defined analogously as “ ? ? “ ? ? “ ? ? “ ? ? “ ? ? ? ( 3 ) with q ? = Pr (A | s ? ). To keep dimensionality manageable while still reflecting realistic security behavior, we restrict the attacker's policy class: in the state offering a favorable, near - guaranteed opportunity, the attacker Game - « attacks almost always, and in the high - risk state it al most never does. Concretely, two components of q are held fixed , and two are learnable, and symmetrically, only a subset of the IDS's parameters is learnable as well.

Such a design keeps the learning process stable and makes the resulting analysis easier t o interpret. Because the policies are memory - one, the game's dynamics reduce to a first - order Markov chain over the state space 6JLYHQSDQGTWKHWUDQVLWLRQPDWUL[0 ST IROORZVGLUHFWO\DQGWKHVWDWLRQDU\GLVWULEXWLRQ?VDWLVILHV ? ? ? ? ? ? ? ? ? ? ( 4 ) ? ? ? ? We compute ? numerically by power iteration until it converges, and then obtain the expected long - term payoffs of the IDS and the attacker as expectations over the state distribution: ?

“ ( 5 ) ? ? ’ ? ? ? ? ? ? ? ” ? ? • ? ? ’ ? “ ? ? ? ? ? ? ? ? ’ ? “ ? ? ? ? ? ? ” ? ? • (6) ? ? ’ ? “ ? ? ? ? ? Here r D and r A denote the stage payoffs, which already incorporate imperfect detection through TPR and FPR along with the operational and false - alarm costs introduced in Section 3. The IDS seeks to maximize U D while the attacker seeks to maximize U A , giving a two - objective game in which each agent nudges its policy parameters in the direction that improves its own long - term payoff: ?

? ? ? ? ? ? ? ? ? ? ? ? ( 7 ) ? ? ? ? ? ? ? ? ? ? (8) ? ? ? Differentiating U analytically is awkward once the stationary distribution and the TPR/FPR - dependent payoffs are involved, so we approximate the gradient numerically via finite differences: ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ( 9 ) ? ? ? ?? ZKHUH?LVDVPDOOVWHSVL]HXVHGIRUWKHDSSUR[LPDWLRQ

To capture fast - adapting attackers, we set the attacker's learning rate relative to the IDS's as ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ( 10 ) $V?JURZVWKHDWWDFNHUUHDFWVWRFKDQJHPRUHTXLFNO\ , and the competition intensifies. This single parameter drives the regime, or phase - diagram, analysis presented later. 7RNHHSWKHQXPHULFVVWDEOHDQGWKHSROLFLHVYDOLGSUREDELOLWLHVZHFOLSWKHSDUDPHWHUVWR>?î?@DIWHUHYHU\ update: ’ ?

? …Ž‹’ ? ’ ? ? ? ? ? ? ? ? ? “ ? ? …Ž‹’ ? “ ? ? ? ? ? ? ? ? ? ( 11 ) Clipping keeps probabilities from getting stuck at 0 or 1 and avoids numerical problems when computing the stationary distribution. The fixed parameters within the restricted policy class are similarly re - constrained at every itera tion so that the intended policy structure is preserved. Convergence is tracked by monitoring how much the stationary distribution, or the payoffs, change across successive iterations. In practice, the implementation runs a fixed number of iterations to re ach steady state, computing the long - WHUPSD\RIIIURP?DWHDFKOHDUQLQJVWHS Malekraeisi et al.

| Res. Ann. Ind. Syst. Eng . 2 ( 4 ) ( 2025 ) 306 - 3018

The overall learning process is presented below in algorithmic form. Algorithm 1 . Gradual learning of the IDS and the attacker within a repeated - game framework with imperfect detection . 1: Initialize p and q 2: Enforce probabilistic constraints on p and q 3: for t = 1 to T do 4: Compute stationary distribution ? for Markov process M (p , q) 5: Compute long - term payoffs: 6: U D (p, q) ? IDS payoff 7: U A (p, q)

? Attacker payoff 8: Approximate IDS gradient using finite differences: 9: ? p U D (p, q) 10: Update IDS strategy: 11: p ? p + ? D ? p U D 12: Approximate attacker gradient using finite d ifferences: 13: ? q U A (p, q) 14: Set attacker learning rate: 15: ? A = ? ? D 16: Update attacker strategy: 17: q ? q + ? A ? q U A 18: Apply clipping and policy constraints (if required) 19: end for 20: Set p* ?

p, q* ? q 21: Return p*, q*, U D (p*, q*), U A (p*, q*)

%HFDXVH?DQGWKH735)35SDLUWRJHWKHUJRYHUQWKHV\VWHP VEHKDYLRUWKHJDPHFDQVHWWOHLQWRTXDOLWDWLYHO\ different regimes depending on where these parameters sit. Running the algorithm DERYHRYHUDJULGRI ? Game - « TPR) values produces a regime map delineating the attacker - dominance, adaptive - competition, and secure regions, giving a systematic way to determine when the IDS, despite operational costs and detection errors, can still maintain d eterrence against a learning attacker.

To evaluate the model, we simulated the IDS ² attacker interaction as a two - agent repeated game, with both sides follow ing memory - one strategies so that each round's decision depends only on the previous round's state. Payoffs were computed in expectation, folding in TPR and FPR so that detection uncertainty is built directly into the simulation. The payoff parameters, inc luding the benefit of a successful attack (G), the defender's loss on a breach (L), the cost of active defense (C d ), the cost of mounting an attack (C a ), and the detection penalty (P), were varied over reasonable ranges for the analysis. Adaptive behavior was modeled by updating each strategy's selection probability gradually, based on cumulative payoff, and the control parameter ?JRYHUQHGWKHDWWDFNHU VOHDUQLQJVSHHGUHODWLYHWRWKHGHIHQGHUFRYHULQJVFHQDULRVZKHUHWKHDWWDFNHUOHDUQV more slowly, at t he same pace, or faster than the IDS. Simulations ran over a long time horizon so that behavioral stability and payoff convergence could both be assessed. We evaluated performance using the two agents' expected long - term payoffs together with a regime anal ysis over the parameter space: extracting each agent's average steady - state payoff and mapping out the attacker - dominance, adaptive - competition, and secure regions, which together indicate what conditions are needed for stable security.

This section presents and discusses the simulation results for the proposed IDS ² attacker framework, focusing on the convergence of the learning dynamics, the effect of detection quality, the regime analysis in the (TPR, ? VSDFHDQGDMRL nt look at security and the economic efficiency of defense.

We first checked the convergence behavior of both agents' policies and long - term payoffs to rule out any influence from transient learning effects on the final results. Fig . 1 shows that the IDS's long - term payoff, DFURVV GLIIHUHQW ? YDOXHV VHWWOHV LQWR a steady state after an initial transient, evidence that the update mechanism, which relies on maximizing long - term payoff through the Markov chain's stati onary distribution, is numerically stable.

Fig . 2 shows the same pattern for the attacker's long - term payoff, which converges without notable oscillation in the later stages of learning. Together, these results indicate that the two agents' adaptive intera ction settles into a stable equilibrium, making meaningful long - term analysis possible. Fig. 1. Convergence of the IDS's policy parameters over learning iterations . Malekraeisi et al.

| Res. Ann. Ind. Syst. Eng . 2 ( 4 ) ( 2025 ) 306 - 3018 Fig. 2. Convergence of the attacker's policy parameters over learning iterations . We also examined how the IDS's and the attacker's probabilistic policies converge. Figs . 3 and 4 show that the action - selection probabilities across the different game states settle at fixed values, confirming that the learned policies are sta ble, further evidence that the framework consistently captures stable behavior for both learning agents.

Fig. 3. Variation of the IDS's long - term payoff over iterations (convergence of U D ) . Fig. 4. Variation of the attacker's long - term payoff over itera tions (convergence of U A ) .

Here we examine how the IDS's detection quality shapes the game's strategic dynamics, through a sensitivity analysis over the TPR and the learning - VSHHGSDUDPHWHU? Table 1 reports only representative and boundary points, those marking a shift in behavioral regime, to keep the discussion focused. At low detection quality 735 IRULQVWDQFH WKH,'6 VH[SHFWHGSD\RIIVWD\VQHJDWLYHUHJDUGOHVVRI?ZKLOHWKHDWWDFNHUH arns either a positive payoff or only a marginal loss.

Such a pattern is a clear signature of attacker dominance: attacking remains strategically worthwhile, and active defense fails to deter. As the TPR rises further, the system passes through a transitio n region. Around TPR = 0.6 in Table 1 GHSHQGLQJRQ?WKHUHJLPHVKLIWV from attacker dominance into a mixed, arms - race - like state, where both players' payoffs are negative , yet neither has a clear incentive to disengage; strategic updates instead settle into an unstable, competitive equilibrium.

This finding shows that improving detection quality alone does not guarantee security; the attacker's adaptive behavior still plays a decisive role. Once TPR crosses roughly 0.72, Table 1 shows the regime shifting qualitatively into the secure region: the IDS's expected payoff turns positive while the Game - « attacker's payoff drops substantially, the signature of stable deterrence, where attacking is no longer strategically attractive. The learning - VSHHG SDUDPHWHU ? DOVR matters for where these boundaries fall: at intermediate TPR values, Table 1 VKRZVWKDWFKDQJLQJ?DORQHFDQSXVKWKHV\VWHPEHWZHHQWKHPL[HGDQG secure regimes. Security stability, in other words, depends not just on raw detection quality but also on how the IDS's learning process weighs past experience. Overall, these results suggest that network security is not simply a direct function of the TPR but emerges from the interplay between detection quality and adaptive learning dynamics. For IDS design, the practical takeaway is that stable security requires crossing specific thresholds on both the detection and the learning dimensions, not just improving one in isolation.

Table 1. Configuration of TPR/FPR test scenarios and summary of key results (average U _D, average U_A, and resulting regime) . Regime Attacker payoff IDS payoff Learning - PHPRU\SDUDPHWHU ? TPR Attacker dominance 0.031 î 0.371 0.01 0.5 Attacker dominance î 0.195 î 0.193 0.01 0.6 Adaptive competition î 0.2 î 0.191 0.42 0.6 Secure î 0.717 0.242 0.01 0.7 Adaptive competition î 0.661 0.19 0.17 0.7 Secure î 0.775 0.263 0.5 0.8 Secure î 1.015 0.476 0.5 1

To build a fuller, more systematic picture of the long - run IDS ² attacker interaction, we carried out a regime analysis over the two - GLPHQVLRQDO 735? VSDFHZKHUH?FDSWXUHVWKHDWWDFNHU VOHDUQLQJVSHHGUHODWLYHWR the IDS , and TPR captures the IDS's intrinsic detection quality. This analysis lets us study the joint effect of the attacker's learning capability and the defense's detection accuracy on network - security stability. The system's behavior falls into three qualitative regimes: attacker - dominant, competitive (arms - race), and secure.

Attacker dominance shows up mainly at low TPR, where the IDS cannot build effective deterrence and attacking still pays off economically; the defense fails against an adaptive attacker even though the IDS itself is learning. As detection quality improves, the system moves through a transit ion region that behaves like an arms race . B oth agents keep updating their strategies ; neither gains a lasting edge, and network security ends up hinging heavily on the learning parameters and operational conditions.

Table 2 quantifies these transition thr HVKROGVDQGVKRZVWKDWWKH735QHHGHGWRHQWHUWKHVHFXUHUHJLPHULVHVZLWK?)RUVPDOO?DVORZO\ DGDSWLQJDWWDFNHUWKH,'6FDQVHFXUHWKHV\VWHPDWDIDLUO\PRGHUDWH735$V?JURZVDQGWKHDWWDFNHUUHDFWV more quickly to past experience, the minimum TPR needed to stay secure climbs substantially. This coupling EHWZHHQ?DQG735LVDNH\GHVLJQLQVLJKWUDLVLQJGHWHFWLRQDFFXUDF\DORQHZLWKRXWUHJDUGIRUWKHDWWDFNHU V learning capability (or vice versa), does not guarantee stable security. Network se curity instead depends on a careful balance between detection quality and both players' learning dynamics. More broadly, the regime analysis here shows that the long - term IDS - attacker interaction can be mapped as distinct qualitative regions LQWKH 735?

) space, a framework that not only explains the underlying behavior but also yields practical thresholds for designing and tuning adaptive IDSs, linking game - theoretic analysis directly to network - security engineering. Table 2. Sample boundary SRLQWVPLQLPXP735UHTXLUHGWRHQWHUWKHVHFXUHUHJLPHIRUVHYHUDOYDOXHVRI? . Regime Above Threshold Regime Below Threshold Minimum TPR For Secure Regime ?

Secure Attacker dominance § 0.01 Secure Attacker dominance § 0.05 Secure Adaptive competition § 0.1 Secure Adaptive competition § 0.3 Secure Adaptive competition § 0.5 Secure Adaptive competition § 0.7 Secure Adaptive competition § 0.9 Malekraeisi et al. | Res. Ann. Ind. Syst. Eng . 2 ( 4 ) ( 2025 ) 306 - 3018

Complementing the regime analysis above, this section looks at the IDS's long - term payoff map over the 735? VSDFHZKLFKOHWVXVZHLJKVHFXULW\VWDELOLW\DJDLQVWWKHHFRQRPLFHIILFLHQF\RIGHIHQVHWRJHWKHU going beyond the p urely regime - based view. The heat map in Fig . 6 shows the IDS's payoff generally improving as TPR rises; at low TPR, the payoff is mostly negative, reflecting an ineffective defense against an adaptive attacker. These low - payoff regions largely coincide wi th the attacker - dominant regime in Fig .

5 ; under these conditions, the IDS is doing poorly both from a security standpoint and a cost - benefit one. At intermediate TPR (matching the mixed/arms - race regime in Fig . 5 ), the IDS's payoff hovers near zero or tur ns slightly positive, a borderline situation in which defense and attack are roughly balanced and small changes in learning parameters or operational cost can tip the advantage either way. Practically speaking, this is where network security is most fragil e and where IDS parameters need the most careful tuning.

Fig. 5. Phase diagram in the (TPR - ? VSDFHVKRZLQJWKHWKUHHUHJLPHUHJLRQV . Fig. 6. Heat map of the IDS's long - WHUPSD\RIIDVDIXQFWLRQRI?DQGWKHWUXHSRVLWLYHUDWH . Fig . 6 also reveals that being in the secure regime (per Fig . 5 ) does not always mean the IDS's net payoff is PD[LPDO )RU VRPH ? 735 FRPELQDWLRQV VWDEOH GHWHUUHQFH LV DFKLHYHG DQG WKH DWWDFNHU V SD\RII GURSV sharply, yet the costs of active defense and adap tive learning still eat into the IDS's net payoff, a reminder WKDW´VHFXUHµDQG´HFRQRPLFDOO\RSWLPDOµDUHQRWWKHVDPHWKLQJ7KDWVDLGWKHUHJLRQVRIKLJK,'6SD\RII in Fig .

6 line up structurally with the secure regime in Fig . 5 , which supports the reg ime analysis: the IDS's payoff peaks mostly where detection quality is high , and the attacker's adaptation speed is kept in check. Payoff maps, then, work well as a quantitative complement to the qualitative regime picture. In short, designing an adaptive IDS has to look past stable security alone and weigh cost ² benefit trade - offs at the same time. Pairing regime analysis with long - term payoff maps gives engineers a fuller basis for decision - making, one that can guide practical IDS configurations balanced b etween security and economic efficiency.

Game - «

We compared the proposed framework against the classical approaches reported in [6] and [14] ; Table 3 summarizes the comparison. Table 3. Comparison of the features of the proposed method with previous studies . Feature Proposed Method Method [6] Method [14] Modeling ? ? ? Adaptive attacker with tunable learning speed ? ? ? Regime analysis ? ? ? Memory - one strategy ? ? ? Bidirectional gradual learning ?

? ? Modeling false - alarm cost ? ? ? As Table 3 shows, the traditional static - game models in [6] and [14] treat the attacker as passive and predictable, while the framework p roposed here captures the attacker's dynamic, adaptive behavior directly. Conventional approaches also tend to be overly optimistic about the security state; by explicitly accounting for the attacker's UHODWLYHOHDUQLQJVSHHG?WKHSURSRVHGPRGHOGUDZVPR re realistic security boundaries.

This study proposed a repeated - game framework for analyzing how an intrusion detection system and a learning attacker interact under imperfect detection. Its main novelty is modeling the adaptive behavior of both sides at once while folding detection quali ty directly into the payoff structure. The results confirm that the learning mechanism is numerically stable and that the long - term analysis holds up methodologically.

Detection quality, particularly TPR , behaves as a regime - transition parameter, moving th e system from attacker dominance through adaptive competition to stable deterrence. Deterrence, however, does not guarantee economic desirability: the cost of active defense can still erode net payoff. And the faster an attacker adapts, the higher the dete ction quality needed to hold deterrence, which shows that security is a relative notion shaped by the dynamics of the threat itself.

Promising directions for future work include extending the model to multi - attack scenarios, modeling false alarms and delay s more precisely, examining convergence indicators in more depth, generalizing to multi - agent settings, and validating the framework in more realistic scenarios. Overall, combining game theory with adaptive learning offers a quantitative way to derive desi gn thresholds for IDS , and shows that stable security calls for attention to detection quality, defense costs, and attacker adaptability all at once.

The authors declare that no funds, grants, or other support were recei ved during the preparation of this manuscript.

The authors have no relevant financial or non - financial interests to disclose.

The datasets generated and/or analyzed during the current study are available from the corre sponding author on reasonable request. Malekraeisi et al. | Res. Ann. Ind. Syst. Eng . 2 ( 4 ) ( 2025 ) 306 - 3018

Conceptualization, M.M. and H.S.; Methodology, M.M. and A.K.; Software, M.M.; Validation, M.M., A.K., and H.S.; Formal analysis, M.M.; Investigation, M.M. and A.K.; Writing ² original draft, M.M.; Writing ² review and editing, A.K. and H.S.; Supervision, H.S. All authors have read and agreed to the published version of the manuscript.

[1] Liu, H., & Lang, B. (2019). Machine learning and deep learning methods for intrusion detection systems: A survey. Applied sciences , 9 (20), 4396. https://doi.org/10.3390/app9204396 [2] Ferrag, M. A., Maglaras, L., Moschoyiannis, S., & Janicke, H. (2020). Deep learnin g for cyber security intrusion detection: Approaches, datasets, and comparative study. Journal of information security and applications , 50 , 102419. https://doi.org/10.1016/j.jisa.2019.102419 [3] Khraisat, A., Gondal, I., Vamplew, P., & Kamruzzaman, J. (2 019). Survey of intrusion detection systems: techniques, datasets and challenges.

Cybersecurity , 2 (1), 1 ? 22. https://doi.org/10.1186/s42400 - 019 - 0038 - 7%0A%0A [4] Ayub, M. A., Johnson, W. A., Talbert, D. A., & Siraj, A. (2020, March). Model evasion attack o n intrusion detection systems using adversarial machine learning. I n 2020 54th annual conference on information sciences and systems (CISS) (pp. 1 - 6). IEEE. https://doi.org/10.1109/CISS48834.2020.1570617116 [5] Ahmed, M., Mahmood, A. N., & Hu, J. (2016).

A survey of network anomaly detection techniques. Journal of network and computer applications , 60 , 19 ? 31. https://doi.org/10.1016/j.jnca.2015.11.016 [6] Š—œ‘ŠŽ’???? ???‘ž?????•™ŒŠ—?????ŠŒóŠ›??????? ž‹Šž¡?? ???????????? Š–Ž?‘Ž˜›¢?–ŽŽœ?—Ž ˜›”? security and privacy. Acm computing su rveys ( CSUR ) , 45 (3), 1 - 39 . https://doi.org/10.1145/2480741.2480742 [7] Zhang, H., Mi, Y., Liu, X., Zhang, Y., Wang, J., & Tan, J. (2023). A differential game approach for real - time security defense decision in scale - free networks.

Computer networks , 224 , 109635. https://doi.org/10.1016/j.comnet.2023.109635 [8] Dehghan, H., & Bigdeli, H. (2026). A Stackelberg signaling game theory model for modeling the defender and attackers under imperfect information situation. OPS EARCH , 1 ? 17. https://doi.org/10.1007/s12597 - 026 - 01152 - 8%0A%0A [9] Chen, J., Xu, Z., Li, Y., Yu, C., Song, J., Yang, H., ... & Wu, Y. (2024, March). Accelerate multi - agent reinforcement learning in zero - sum games with subgame curriculum learning. In proceedings of the AAAI conference on artificial intelligence (Vol. 38, No. 10, pp. 11320 - 11328).

https://doi.org/10.1609/aaai.v38i10.29011 [10] Lan, Y., & Hu, Q. (2023). Intrusion detection and response of network node attacks based on game theory. In proceedings of the 2023 international conference on artificial intelligence, systems and network s ecurity (pp. 386 - 390). https://doi.org/10.1145/3661638.3661711 [11] Estévez - Pereira, J. J., Fernández, D., & Novoa, F. J. (2020 ). Network anomaly detection using machine learning techniques. In Pro ceedings (Vol. 54, No. 1, p. 8). MDPI.

https://doi.org/10.3390/proceedings2020054008 [12] Apruzzese, G., Andreolini, M., Ferretti, L., Marchetti, M., & Colajanni, M. (2022). Modeling realistic adversarial attacks against network intrusion detection system s. Digital threats: research and practice ( DTRAP ) , 3 (3), 1 ? 19. https://doi.org/10.1145/3469659 [13] Ahmad, Z., Shahid Khan, A., Wai Shiang, C., Abdullah, J., & Ahmad, F. (2021). Network intrusion detection system: A systematic study of machine learning an d deep learning approaches.

Transactions on emerging telecommunications technologies , 32 (1), e4150. https://doi.org/10.1002/ett.4150 [14] Ho, E., Rajagopalan, A., Skvortsov, A., Arulampalam, S., & Piraveenan, M. (2022). Game theory in defence applications : A review. Sensors , 22 (3), 1032. https://doi.org/10.3390/s22031032 [15] Wang, Z., Xu, S., Xu, G., Yin, Y., Zhang, M., & Sun, D. (2020). Game Theoretical Method for Anomaly - Based Intrusion Detection.

Security and communication networks , 2020 (1), 8824163. https://doi.org/10.1155/2020/8824163 Game - « [16] FathimaAH, N., Khraisat, A., Ibrahim, S., & Li, G. (2025). Adaptive memory replay for network intrusion detection: Tackling data drift and catastrophic forgetting. Computer networks , 111712. https://doi.org/10.101 6/j.comnet.2025.111712 [17] Malloy, T., & Gonzalez, C. (2023 ). Learning to defend by attacking (and vice - versa): Transfer of learning in cybersecurity games. In 2023 IEEE european symposium on security and privacy wor kshops (EuroS&PW)

(pp. 458 - 464). IEEE. https://doi.org/10.1109/EuroSPW59978.2023.00056