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Article

AI-Assisted Game Theory Approaches to Bid Pricing Under Uncertainty in Construction

Department for Engineering and Built Environment, Birmingham City University, Birmingham B4 7RJ, UK
AppliedMath 2025, 5(2), 39; https://doi.org/10.3390/appliedmath5020039
Submission received: 28 February 2025 / Revised: 25 March 2025 / Accepted: 27 March 2025 / Published: 3 April 2025

Abstract

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The construction industry is inherently marked by high uncertainty levels driven by its complex processes. These relate to the bidding environment, resource availability, and complex project requirements. Accurate bid pricing under such uncertainty remains a critical challenge for contractors seeking a competitive advantage while managing risk exposure. This exploratory study integrates artificial intelligence (AI) into game theory models in an AI-assisted framework for bid pricing in construction. The proposed model addresses uncertainties from external market factors and adversarial behaviours in competitive bidding scenarios by leveraging AI’s predictive capabilities and game theory’s strategic decision-making principles; integrating extreme gradient boosting (XGBOOST) + hyperparameter tuning and Random Forest classifiers. The key findings show an increase of 5–10% in high-inflation periods with a high model accuracy of 87% and precision of 88.4%. AI can classify conservative (70%) and aggressive (30%) bidders through analysis, demonstrating the potential of this integrated approach to improve bid accuracy (cost estimates are generally within 10% of actual bid prices), optimise risk-sharing strategies, and enhance decision making in dynamic and competitive environments. The research extends the current body of knowledge with its potential to reshape bid-pricing strategies in construction in an integrated AI–game-theoretic model under uncertainty.

1. Introduction

Bid pricing is a critical aspect of the construction industry, where firms compete to secure contracts by submitting proposals that balance competitiveness with profitability [1,2]. The process requires contractors to estimate project costs [3], incorporate risk factors [4], and anticipate competitor behaviour, all while adhering to tight deadlines and limited information [1,5]. Bid pricing is not merely a financial exercise but a strategic endeavour shaped by the inherent uncertainty of the bidding environment, resource availability, and project-specific complexities [4,6,7]. The challenge lies in setting prices that are low enough to win bids but high enough to ensure a sustainable profit margin [2].
Uncertainty plays a central role in bid pricing [8]. Factors such as fluctuating material costs, labour availability, unforeseen project delays, and evolving regulatory requirements can significantly impact cost estimates and profitability [8,9,10]. Furthermore, the competitive nature of the construction industry exacerbates the difficulty, as contractors must consider not only their cost structures but also the strategic actions of their competitors [2,10]. This creates a dynamic and adversarial environment where decision making is often riddled with risk and incomplete information [11,12].
Traditional approaches to bid pricing, such as historical cost analysis and heuristic-based estimations, have proven inadequate in managing these complexities [12,13,14]. One key driver of this trend is that while these methods offer a foundation, they lack the adaptability and predictive capabilities to navigate modern market dynamics. Game theory has emerged as a promising tool for addressing these challenges, offering a structured framework to model and analyse competitive interactions under uncertainty [11,12,13]. However, existing game theory applications in construction bidding often rely on static assumptions and simplistic models, limiting their practical utility [15].
Recent advancements in artificial intelligence (AI) present new opportunities to enhance decision making in bid pricing [16,17]. AI’s ability to process large datasets, identify patterns, and its predictive capacity can, therefore, complement game theory’s strategic framework [18,19]. Integrating AI into game theory models results in adaptive, data-driven approaches that better account for the possible uncertainty and competitive dynamics [18]. This integration can improve bid accuracy, optimise risk-sharing strategies, and enable contractors to make more informed decisions [20,21].
This paper explores an AI-assisted game theory approach to bid pricing under uncertainty, focusing on construction processes. The study develops an exploratory framework that combines AI’s predictive capabilities with game theory’s strategic analysis, applying it to a real UK case study. By examining the model’s performance and potential, the research aims for exploration into how advanced methodologies can transform bid-pricing practices in construction.
This research contributes to academic and practical domains by advancing construction bid-pricing methodologies for contracting processes using game theory for strategic bidding and AI for integrating of wider extant parameters in the modelling with higher accuracy and prediction capacity. The proposed AI-assisted game theory framework has the potential to reshape how bid pricing is approached in the construction industry, paving the way for more resilient and data-driven decision making in an increasingly competitive bidding environment.
Integrating AI and game theory promises to address the multifaceted challenges of bid pricing in construction. While game theory provides a robust foundation for strategic decision making, its traditional applications often assume static conditions and limited data. Conversely, AI allows for handling large datasets, identifying patterns, and generating adaptive predictions. Combining these two methodologies, this paper presents a dynamic and data-driven model that enables better quantification, prediction, management, and optimisation of risks associated with uncertain project environments, including competition.

Research Objectives

  • Develop an AI-powered framework integrating machine learning (ML), risk modelling, and game theory to optimise bid pricing and strategies;
  • Model bidding process for improved success rates by leveraging historical data, competitive dynamics, and uncertainty quantification for strategic bid optimisation;
  • Evaluate the model’s effectiveness through a UK case study in the construction sector.
The findings of this research contribute to the development of an adaptive bidding system that leverages AI-driven analytics to support enhanced, effective and competitive bid-pricing strategies in the construction industry.

2. Literature Review

2.1. Overview of Traditional Bid-Pricing Methods in Construction

Traditional bid-pricing methods in construction have long served as the foundation for contractors to estimate project costs and prepare competitive bids [1,21,22]. These methods typically rely on historical data and expert judgment, including which is largely subjective, as highlighted by Fayek, Ghoshal and AbouRizk [23], and rule-based approaches, such as those related to specific performance metrics, like sustainability performance, as seen in El-Sayegh, AbdRaboh, Elian, ElJarad and Ahmad [22], to determine the total project cost and the markup necessary for profitability. While these approaches provide a baseline for bid preparation, they are often limited in their ability to address the complexities of modern construction projects, particularly under conditions of uncertainty and intense competition.
Cost-based estimation is the most widely used traditional approach in construction bidding [24,25,26]. It involves calculating the direct and indirect costs associated with a project, including labour, materials, equipment, subcontractor fees, and overhead costs that, according to Takano, Ishii and Muraki [25], often require extensive resourcing of these processes for accuracy. Contractors typically add a markup for profit and contingencies to arrive at the final bid price, such as is highlighted in Zaqout, Islam, Hadidi and Skitmore [27]. While straightforward, this method assumes a stable bidding environment for determining the markup and decision to bid, as well as a considered expert regime and other critical factors, which are static to support bidding decision making [27]. This often overlooks dynamic factors such as fluctuating material prices, labour shortages, or changes in regulatory requirements.
Many contractors rely on historical project data to inform bid pricing, whether through data from competitors or other allowable sources, as Lan Oo, Lo and Teck-Heng Lim [1] highlight. By analysing completed projects with similar scopes and scales, they derive cost benchmarks and use these as a reference for pricing new bids. Input-versus-output benchmarking is, according to Shrestha, Migliaccio, O’Connor and Gibson Jr [28], used in the bidding process of large design-and-build highway projects in the United States, while competitive strategic performance benchmarking (CSPB) is proposed by Ercan and Koksal [29]. First, the key strategic performance parameters are defined and a scale using binary logistic regression is applied. Although these approaches can be useful tools for formulating strategy during bidding, they heavily depend on the accuracy and relevance of past data. Aboseif and Hanna [30], for example, argue that the process of benchmarking, among others, remains subjective, relying on qualitative techniques, while a multiplicity of these approaches is highlighted in Palaneeswaran and Kumaraswamy [31]. This approach to benchmarking does not allow for bidding processes that address the dynamism of processes, which could improve bidding and project performance. Moreover, historical analyses often fail to capture the unique risks or uncertainties specific to the current project.
Managerial superiority and similar approaches to organisational competencies still dominate practice in construction and are vital to competitive advantage and strategic performance, according to Ercan and Koksal [29]. Ercan and Koksal [32] also highlight that management strategies, such as cost leadership and differentiation reinforce many of the experiences and intuitions that underpin bid processes. In cases where data are limited or time constraints are pressing, bidding processes often rely on these heuristic or intuitive approaches from the experienced management. These methods rely on the estimators’ expertise and intuition to judge costs and pricing quickly. While this can be effective for seasoned professionals, it introduces subjectivity and can lead to inconsistencies in bid preparation. Heuristic methods also struggle to account for the complex interdependencies or competitive dynamics in the bidding environment.
Hosny and Elhakeem [33] introduced the concept of winning markup (WM), a bid evaluation strategy that determines the optimum markup that maximises both the probability of winning and a profit value. Takano, Ishii and Muraki [25], on the other hand, proposed an optimisation model for simultaneous bid markup and resource allocation. A Mamdani-type fuzzy inference model for bid markup, on the other hand, is proposed by Zaqout, Islam, Hadidi and Skitmore [27]. Such markup models involve applying a fixed percentage or formula-based markup to the base project cost to account for profit, risk, and competitive pressures. These models are easy to implement and are commonly used in the industry. However, they are simplistic and often fail to account for project-specific risks, bid environment, or dynamic competitor strategies even when extended to utilitarian modelling, as used in Dozzi, AbouRizk and Schroeder [34], in an attempt to resolve such difficulties. The rigidity of markup models can lead to underpricing or overpricing in competitive bidding scenarios.
Last are the simulation and probabilistic models. These models are presented as useful in bid-price decision making such as when faced with time-limited bidding environments [24]. Similarly, in situations where there is limited information, probabilistic approaches, such as that proposed by Jang, Lee, Lee and Han [35] using a naïve Bayesian classifier applied to identify good outcomes in bidding processes, have been helpful. Nguyen [36] uses probabilistic approaches to determine the probability of winning a bid contract, balancing the expected value and standard deviation of the bid price. Simulation models for forecasting in bidding processes have been explored in such research as Lee [37] for bid-rigging in design–bid–build contracts. In other studies, such as Elsayegh, Dagli and El-Adaway [38], simulation models have been used in agent-based modelling in competitive construction bidding environments. These advanced traditional methods, such as Monte Carlo simulations or related probabilistic models, have been useful in accounting for the uncertainty in bidding environments. The methods model different scenarios and assess the likelihood of various outcomes, enabling contractors to quantify risks more effectively. However, they are computationally intensive and require specialised expertise, which can limit their adoption in practice.

2.2. Key Concepts in Game Theory for Construction Bidding

The players in the context of construction bidding include the contractors competing for the project [11,12,13]. Each player aims to maximise their payoff (i.e., profit) while minimising their risk [39]. Conversely, from a contractor’s perspective, strategies involve selecting a bid price that balances competitiveness with profitability [4,37]. After accounting for project costs and risks, the payoff is the profit earned by the winning contractor [25]. Information in bidding scenarios can be classified as complete (all players have full information about competitors) or incomplete (players have limited or uncertain information about competitors), the latter of which Ahmed, El-Adaway, Coatney and Eid [11] and Elsayegh, Dagli and El-Adaway [38] highlight can lead to adverse bidding.
The Von Neumann and Morgenstern [40] game theory conceptual understanding, construction bidding utilises similar mathematical models to analyse and optimise bidding strategies. A Nash equilibrium, for example, occurs when no player can improve their payoff by unilaterally changing their strategy, relying on their rationality according to Ahmed, El-Adaway, Coatney and Eid [11]. For a bidding game with n contractors, the Nash equilibrium is represented as follows:
U i s i * , s i * U i s i , s i * , s i s i *
where U i is the payoff of contractor i , s i is the strategy (i.e., bid price) of contractor i , s i are the strategies of all of the other contractors, and s i * and s i * are the equilibrium strategies for contractor i and the other contractors, respectively. The equilibrium ensures that any unilateral deviation from a strategy does not improve outcomes from the bidding [41]. Such concepts have been widely used in construction bidding practices, such as in single-stage bidding and multistage bidding [11], and in electricity market bidding, such as by Song, Liu and Lawarrée [41].

Utility Function for Payoff Calculation

Dozzi, AbouRizk and Schroeder [34] present an alternative bidding environment that varies, and players’ preferences are subjective. Utilitarian models have been used when faced with this dynamism in the bidding environment, coupled with gaps in information underpinning any decision making [34,42]. The payoff or utility function in construction bidding typically accounts for the profit earned if a contractor wins the bid minus the risks and costs associated with the project. It can be expressed as U i = P i · B i C i R i , where P i is the probability of contractor i winning the bid, B i is the bid price submitted by contractor i , C i is the estimated cost of completing the project for contractor i , and R i is the risk cost or penalty associated with uncertainties.
The probability of winning depends on the bid prices submitted by competitors and increases with knowledge of competitors’ behaviours [42]. This can be modelled using a probability density function, f ( x ) , where x is the bid price. Assuming independent and uniformly distributed bids as presented by Friedman [43], the probability of winning for contractor i can be approximated as follows:
P i = j i P r ( B i < B j )
For normally distributed bids with a mean, μ , and standard deviation, σ , this can be expressed as follows:
P i = Φ B i μ σ
where Φ is the cumulative distribution function of the normal distribution.
Game theory models often incorporate risk-adjusted pricing to account for uncertainties in costs and revenues [44]. The risk adjustment, according to Ioannou [44], represents a level of indifference given the risk regime and bidder’s aversion. A risk-adjusted bid price, B i R , is worked out as B i R = C i + α R i , where α represents the risk premium factor reflecting the bidder’s risk tolerance. In terms of the expected payoff for contractor i , E [ U i ] = P i · B i C i R i , and the goal is to maximise E [ U i ] by selecting an optimal B i .
Ahmed, El-adaway, Coatney and Eid [45] highlight bidding with incomplete information or when bidders have varied private information. In these cases, game theory in construction bidding is used in auction models, often modelled as a first-price sealed-bid auction. Contractors submit bids without knowing competitors’ prices, and the lowest bidder wins. Game theory helps contractors determine optimal bid prices by analysing the competitiveness and profitability trade-offs [4]. On the other hand, game theory is seen in modelling collaborative vs. competitive bidding scenarios, such as joint ventures, as well as cooperative games for the former. At the same time, for the latter, modelling is based on non-cooperative games. In bid markup optimisation, contractors use game theory to determine optimal markup rates based on competitors’ expected strategies and bidding environment. This involves balancing the likelihood of winning the bid with the project’s profitability [11,45]. Finally, in competitor behavioural analysis, incorporating Bayesian game theory allows contractors to model competitors’ behaviour based on incomplete information, such as estimating competitors’ cost structures or risk preferences.

2.3. Game Theory Applications in Construction Bidding

Game theory has significant applications in construction bidding as a computational framework for modelling strategic interactions among rational decisionmakers. In multistage bid environments, contractors have used game theory as a tool to facilitate bidding processes when faced with complex cost estimation [45]. The authors found that game theory helped bidders avoid negative profits and ultimately gain a competitive advantage, even in joint bids. In related research, it was found that game theory, when complemented with reinforcement learning (RL) algorithms, helped bidders avoid the winner’s curse [46].
In the study by Kembłowski, Grzyl and Siemaszko [12] game theory was examined for its contribution to strategy building during bidding processes. The authors found that it provided opportunities for optimal strategy selection even when faced with the dilemma of incomplete information, helping transform such information into imperfect information instead. The studies by Xue [47] and Teo, Bridge and Love [48] similarly reinforced the optimisation potential of game theory for projects with a composite base price, the latter focusing on public–private partnerships (PPPs). In a simulation of a large-value PPP bidding process, when faced with a markup and an investment decision at the same time, De Clerck and Demeulemeester [49] applied game theory to different approaches to two- and three-bidder situations, reinforcing the need for losing bidders to be compensated in oligopolistic bidding environments.
In all these studies, game theory takes a structured approach to understanding how bidders compete for projects under uncertainty, with incomplete information, or with competing objectives. This enables them to make informed decisions about bid pricing. Game theory allows contractors to predict competitor behaviour, assess risk, and optimise bid strategies to balance competitiveness and profitability through competitive dynamics analysis.
While game theory in these cases demonstrates applicability for improving outcomes, in practical applications in construction bidding, it still faces several challenges, not least the complexity of modelling multi-player scenarios with incomplete information, the difficulty of obtaining accurate data on competitors’ strategies and risk preferences and simplistic assumptions in traditional game theory models, such as static conditions, which may not reflect real-world dynamics.
Game theory represents the foundational elements for strategic bidding in construction, but its potential can be further enhanced through integration with AI. AI’s ability to process large datasets and generate predictive analytics can address many of the limitations of traditional game theory applications, providing a more adaptive and dynamic approach to bid pricing. This paper builds on these foundational premises to propose an AI-based game theory framework to improve decision making under uncertainty and competitive dynamics.

2.4. The Role of AI in Addressing Uncertainty and Predictive Decision Making

Uncertainty is a defining characteristic of the construction industry, affecting project costs, bidding environment, and competitor behaviour [9,10,44]. AI has emerged as a transformative approach for addressing these uncertainties by enabling data-driven and predictive decision making [50,51]. It extends the capability by looking between variables and their associated uncertainty [50]. AI can dynamically adapt during runtime with new information about the environment and context [51]. From computing real-time advertisement to housing design optimisation, AI’s ability to process large datasets, identify patterns, and adapt to dynamic environments makes it an invaluable tool for enhancing the precision and robustness of decision making, particularly inherent in bid pricing strategies [50,52,53].

Key AI Techniques for Addressing Uncertainty

ML algorithms can predict project costs, competitor behaviour, and market trends by learning from historical data [54,55]. Understanding accurate manufacturing costs is important in early product development. Hennebold, Klöpfer, Lettenbauer and Huber [55] use ML cost prediction with limited information about the final product with high accuracy. In their study, Sharma, Zaki, Jha and Krishnan [54] demonstrated that the ensemble methods, such as gradient-boosted trees, improved construction cost prediction. In addition, the multi-objective optimisation capabilities presented as a Pareto front for two competing variables ultimately contributed to improved decision making. XGboost (Extreme Gradient Boosting), deep neural network (DNN), and random forest (RF) supported an automation decision support for green construction cost prediction by Alshboul, Shehadeh, Almasabha and Almuflih [56]. Lately, Serugga [52] uses XGBoost and Random Forest classifiers to predict cost performance in offsite construction.
Techniques like regression, classification, and clustering are commonly used to estimate costs, assess risks, and segment competitive landscapes [52,56].
DNNs composed of multiple levels of nonlinear operations are particularly effective for capturing complex, nonlinear relationships in data [57]. For example, convolutional neural networks can process spatial data such as site layouts, while recurrent neural networks (RNNs) can handle sequential data like time-series forecasts of material prices.
Probabilistic models, such as Bayesian networks and Monte Carlo simulations enhanced by AI, enable the modelling of uncertainties and the computation of likelihoods for various outcomes [58]. Their ability to structure knowledge representation among many variables encoding their interdependencies has gained wide appeal in many AI modelling applications, such as in healthcare [59].
Deep reinforcement learning (DRL) and RL bring capacity for a higher-level understanding of the visual world to AI [60]. RL’s capability to scale in complex emergent situations that previously would have been intractable has gained wide appeal in areas such as video gaming by extracting directly from pixels. DRL algorithms, on the other hand, have been applied to robotics, learning directly from camera inputs in the real world [60]. Another related capability, multi-agent reinforcement learning (MARL), is used widely because it provides capacity for sets of agents through their collective interactions with their environment. Applications for MARL include AI-based wireless network settings that solve wireless communication problems [61]. These techniques can be extended to allow contractors to develop adaptive bidding strategies by learning optimal actions (bid prices) based on feedback from competitive environments. Lastly, when faced with large datasets of unstructured data, natural language processing (NLP) brings the capacity to extract patterns and insights to support decision making [62]. NLP tools can extract insights from textual data, such as project specifications or competitor bidding histories, to improve decision making.

2.5. Foundations of AI Applications

AI techniques are underpinned by mathematical models that facilitate learning, prediction, and decision making. Below are the key equations and their applications in addressing uncertainty and predictive decision making in construction bidding.

2.5.1. Cost Estimation Using Regression Models

AI leverages regression techniques to predict project costs ( C ) based on input variables ( X ) , such as project size, material prices, and labour rates. The general form of a regression model is as follows:
C = β o + i = 1 n ( β i X i ) + ε  
where β o is the intercept, β i are the coefficients for the variables, and ϵ is the error term. ML algorithms, such as linear regression, support vector regression (SVR), and gradient boosting, optimise the coefficients ( β ) to minimise prediction error.

2.5.2. Risk Modelling with Bayesian Networks

Bayesian networks model the probabilistic relationships among uncertain variables. For example, let P ( C ) represent the probability of a cost outcome given risk factors R 1 ,   R 2 ,   , R n . The posterior probability is computed using Bayes’ theorem, as follows:
P C | R 1 ,   R 2 ,   , R n = P R 1 ,   R 2 ,   , R n C . P ( C ) P ( R 1 ,   R 2 ,   , R n )
where P C | R 1 ,   R 2 ,   , R n represents the posterior probability of a cost outcome given the risks, P ( R 1 ,   R 2 ,   , R n | C ) is the likelihood of the risks given the cost, and P ( C ) is the prior probability of the cost outcome. This approach enables contractors to assess the likelihood of cost overruns under different risk scenarios.

2.5.3. Time-Series Forecasting for Market Trends

RNNs and long short-term memory networks are AI tools for forecasting time-series data, such as material prices or labour costs. The prediction at time t + 1 , denoted as y ^ t + 1 , is computed based on previous inputs, as follows:
y ^ t + 1 = f ( y t ,   y t 1 ,   , y t n ; θ )
where y t ,   y t 1 ,   , y t n represent historical data points, θ the model parameters optimised during training and f the nonlinear function approximated by the neural network. This enables contractors to predict future cost fluctuations and incorporate them into bid pricing.

2.5.4. Competitor Behaviour Prediction Using Classification Models

AI can classify competitors’ likely bid strategies ( S c ) based on historical bidding data ( X ) . Using logistic regression or classification models, the probability of a competitor adopting a specific strategy is as follows:
P ( S c = k X = e x p ( β o + i = 1 n β i X i ) j = 1 k e x p ( β o + i = 1 n β i j X i )
where P ( S c = k X is the probability of strategy k given features X , K is the total number of possible strategies, and β i are the coefficients optimised by the model. This helps contractors anticipate competitor actions and adjust their bids accordingly.

2.5.5. RL for Adaptive Bidding

In RL, a contractor (i.e., agent) learns an optimal bidding strategy ( a t ) by maximising cumulative rewards ( R ) over time [63]. The reward function is defined as R t = U i = P i · B i C i R i . The agent updates its policy, π ( a s ) , which maps states ( s ) to actions ( a ) , using a value function, as follows:
Q ( s , a ) = R t + γ max a Q ( s , a )
where Q ( s , a ) espouses the expected value of taking an action ( a ) in state s , γ is the discount factor for future rewards, and s   a n d   a are the next state and action. RL enables the contractor to adjust bid strategies dynamically based on feedback from the bidding environment.
Bid pricing is a strategic decision-making problem under uncertainty [9,10]. The contractor balances winning the contract (bidding low enough to outcompete rivals) while ensuring profitability (bidding high enough to cover costs and generate a profit). Traditional aids in this area rely on static pricing strategies, but real-world bidding is dynamic, requiring adaptive learning [64]. RL allows the AI to learn continuously from past bidding experiences [65], adjust bid prices dynamically based on a changing bidding environment, predict competitor responses, and optimise bid amounts accordingly. Thus, RL mimics real-world decision making, where contractors adjust their bidding behaviour over time [20,46].
Accurately estimating costs is essential because underestimating leads to project funding gaps, while overestimating results in lost bids [11,66]. Traditional cost estimation methods, such as heuristics or simple regression models, struggle to handle nonlinear relationships between cost factors (e.g., material prices, labour cost, and project complexity) and high-dimensional data (many interacting cost components). A dynamic bidding environment (inflation, demand shifts, and supply chain issues) also means that static models cannot account for these attributes in the modelling. XGBoost can handle structured data (such as bid-pricing datasets), missing values, and noisy data and is computationally efficient and interpretable [67], making it an ideal choice for construction dataset modelling.
On the other hand, traditional risk assessment methods rely on single-value estimates or simple probability distributions, which are merely focused on preparation, negotiation, and decision making, among others [68]. Such simplistic approaches fail to capture the dynamism in cost variations due to inflation and economic shifts, as well as the interdependencies between project risk attributes and uncertainty in cost estimation models. Bayesian methods overcome these limitations by incorporating historical data, continuously updating beliefs about cost risks, and modelling uncertainty explicitly using probability distributions [35,51,69]. Bayesian risk modelling also provides confidence intervals for cost estimates, helping AI make better bid decisions. Therefore, instead of one fixed-cost estimate, Bayesian modelling offers a range of likely costs with probabilities improving outcomes and their reliability.
Traditional cost estimation methods often fail to account for uncertainty, leading to overly optimistic bid pricing (i.e., underestimating risks), overly conservative bid pricing (i.e., losing competitiveness), and failure to incorporate inflation, economic trends, or market shifts [8,38,42]. Monte Carlo simulation overcomes these issues by generating thousands of potential cost scenarios while providing a probability distribution of total costs [70]. It also incorporates market fluctuations and multitudes of other risk factors [71]. This provides the foundation for AI to determine the optimal bid price while considering worst-case and best-case cost variations.

2.6. Gaps in Current Research and Justification for the Study

2.6.1. Gaps in Current Research

While construction bidding, game theory, and AI have been extensively studied, integrating these disciplines remain limited. The existing literature points to several gaps that hinder the development of comprehensive and adaptive frameworks for bid pricing under uncertainty.
Although game theory has been applied to construction bidding, its models often rely on static assumptions and lack adaptability for real-world applications [12,45]. Conversely, AI techniques have demonstrated significant potential for predictive decision making but are rarely integrated with game theory to model strategic interactions in competitive environments [18,20]. Similarly, current game theory models in construction bidding often use oversimplified methods to represent uncertainty, such as fixed probability distributions or deterministic parameters [12]. These fail to capture the dynamic and multifaceted nature of risks in construction, such as fluctuating material prices, changing regulations, and varying bidding environments.
Moreover, many studies overlook the importance of modelling competitor strategies beyond simplistic assumptions (e.g., competitors aim to minimise costs). There is a lack of appropriate aids to predict and incorporate competitor behaviours, which are critical for strategic bid pricing. Knowing how competitors price their bids in competitive environments provides a strategic advantage [4,49]. Traditional bidding approaches ignore or oversimplify competitor actions. However, real-world competitors adjust their bids based on the bidding environment (economic trends and demand fluctuations), past bid losses/wins (learning from experience), project importance (high-stakes projects lead to aggressive bidding), and firm size and reputation (smaller firms may bid lower to win market share). By modelling these factors using AI, it is possible to predict whether competitors will likely submit low or high bids, helping an AI-powered system adjust its bid price strategically [18].
Current research on game theory and AI in construction remains largely theoretical, with limited real-world case studies or industry validation. Particularly in the UK construction industry, minimal evidence demonstrates the practical utility of integrated AI–game theory frameworks. Additionally, traditional research often focuses solely on cost optimisation while neglecting other critical factors, such as risk sharing, adaptive bidding strategies, and the dynamic interplay between uncertainty and competition.

2.6.2. Justification for the Study

This study is motivated by the need to address the identified gaps and provide a comprehensive, adaptive, and practical framework for bid pricing in the construction industry.
This study integrates AI and game theory and provides a framework that combines the strategic modelling of game theory with the predictive capabilities of AI. This approach enables contractors to make data-driven decisions while accounting for the complex dynamics of competition and uncertainty, contributing to bridging the gap between AI and game theory.
There is an increasing need to advance uncertainty modelling in construction processes, which is a major aspect of project complexity. The proposed framework leverages AI to model uncertainty more dynamically and comprehensively. Techniques such as ML and probabilistic models allow for the incorporation of real-time market data, competitor behaviour, and risk factors, overcoming the limitations of static models.
The study enhances the predictive modelling of competitor strategies by employing AI algorithms such as classification models and RL. This ensures that contractors can anticipate and respond to competitor actions more effectively, contributing to the emergent need for improved competitor behaviour modelling. Additionally, by applying the proposed framework to a case study in the UK construction industry, this research bridges the gap between theoretical advancements and practical implementation. It demonstrates the utility of AI-assisted game theory in addressing real-world challenges, bringing practical relevance to the integrated capabilities.
Unlike traditional methods, this study adopts a holistic approach that considers cost optimisation, risk management, and strategic competition. This ensures that the proposed framework is robust and adaptable to the multifaceted nature of construction bidding. This research, finally, advances theory in this domain by integrating AI and game theory during the critical stage of construction projects. It also offers a practical aid that contractors can use to improve their bid-pricing practices, ultimately enhancing competitiveness and profitability in their processes.
This research addresses these gaps, advancing the knowledge of AI and game theory’s integration into construction bidding, providing a practical framework that improves decision making in an increasingly competitive and uncertain industry.

3. Methodology

This study adopts an exploratory research approach to investigate the integration of AI and game theory in bid pricing under uncertainty in the construction industry. Given the complexity and dynamic nature of construction bidding, an exploratory methodology is well-suited to uncovering new insights, supporting the development of theoretical frameworks, and evaluating their practical implications [72,73].
Exploratory research is employed when existing knowledge on a topic is limited or fragmented because it allows for flexibility and adaptability in the research discourse [72]. Since integrating AI-assisted game theory in bid pricing is still an emerging area, a structured yet flexible approach is required to understand how AI can enhance traditional game-theoretic bidding models. It also helps identify key variables influencing bid pricing under uncertainty. Lastly, exploratory research forms the basis for developing and testing an AI-driven game theory framework using real-world data. This approach, therefore, allows for the iterative refinement of models and hypotheses based on empirical findings, ensuring that the proposed framework remains adaptable and applicable to industry needs.
The literature review informs the theoretical framework development by looking at existing bid-pricing models, game theory applications, and AI techniques. Theoretical gaps are identified, forming the basis of an integrated AI-assisted game theory framework. A case study approach focuses on a real-world construction bidding scenario for a low-rise residential construction project in the UK. Data sources include historical bid records, competitor pricing strategies, market trends, and project cost databases. Interviews and expert consultations with industry professionals were conducted to validate key assumptions and inform the feature engineering process.
The game theory model was then developed to simulate competitive bidding dynamics, incorporating AI-driven insights into risk prediction and competitor behaviour. AI models, such as ML regression for cost estimation, Bayesian networks for uncertainty modelling, and RL for dynamic bid strategy optimisation, are integrated into the framework. A simulation environment was developed to test the framework using real and engineered bid data. Scenarios were designed to evaluate how AI-enhanced game theory influences bid outcomes under different bidding environments and uncertainty levels.
The model’s performance was finally evaluated by comparing its predictive accuracy and strategic effectiveness against traditional bid-pricing methods. A sensitivity analysis was conducted to assess the robustness of the framework under varying levels of risk and competitive behaviour through accuracy (MAE and R 2 scores) and reliability testing.
The exploratory research approach allows for exploring the integrated AI and game theory model, requiring a flexible and open-ended research design. By incorporating a case study, the research ensures practical relevance and industry applicability. AI’s predictive capabilities provide an empirical basis for improving bid-pricing strategies, making this approach more robust than purely theoretical models. The framework can be deployed and developed to scale, extended to other construction contexts, and adapted to different competitive environments. By employing an exploratory research approach, this study generates valuable knowledge into how AI-assisted game theory can be brought to bid pricing in construction processes, ultimately bridging the gap between theoretical advancements and practical implementation.

Case Study and the Assisted Game Theory Model

To evaluate the effectiveness of the AI-assisted game theory bid-pricing model, a low-rise, 54-flat residential redevelopment project in the midlands was used, where multiple contractors competed for the contract under a fluctuating bidding environment. The development is for temporary residential independent living for young persons in the local area.
Data collection and model implementation were based on the principles outlined in seven steps below, including historical and market data and competitor behaviour. Bayesian risk modelling quantified the cost uncertainty, the Monte Carlo simulation generated 10,000 possible cost scenarios, and the RL (PPO) optimised bid-pricing strategy.
This study proposes an AI-assisted game theory approach that models bid pricing as a strategic decision-making process under uncertainty. Figure 1 presents the conceptual framework of the model. This model integrates ML, Bayesian risk modelling, Monte Carlo simulations, and RL to optimise bid pricing under uncertainty in construction projects. Below is a step-by-step breakdown of how each component works and the key steps, as follows:
Step 1. Data Collection and Feature Engineering
Data collection and feature engineering aim to construct a realistic dataset that captures the cost dynamics, competitor strategies, and market uncertainties affecting bid pricing in construction. This first step, therefore, gathers historical bidding data (5 years of past bids that are available and deemed sufficient for the scope of the modelling), market trends, and risk factors for projects with similar scopes, selecting key cost-influencing features and preparing the data for ML models. These data come from historical data bidding sets for project costs, final bid prices submitted, and winning bid amounts. Market and economic data, on the other hand, helps with material price trends (steel, concrete, timber, etc.), inflation and exchange rate data, and labour data from the Office of National Statistics (ONS) and seasonal influences (peak vs. off-peak demand). Competitor behaviour data are important for highlighting behaviours regarding previous bid prices from competitors, market position (SMEs or large firms), and win rate in past bids, which are publicly available. Finally, project-specific data relating to the complexity (low, medium, and high difficulties), lead times, and scope, as well as context-based cost adjustments, exist.
The process then embarks on feature engineering, selecting the most relevant features or engineering new ones to enhance the AI’s performance (Table 1).
Data are then cleaned and transformed before training, including filling in missing values for missing material or labour costs or using industry averages for missing competitor win rates. Similarly, the process uses mean/median and mode imputation for missing numerical continuous and categorical data respectively. The data are then normalised for continuous features (e.g., material and labour costs) using Min–Max scaling or standardised for economic indicators to avoid large numerical differences.
One-hot encoding converts categorical variables (e.g., project complexity: low, medium, and high) into numerical formats (see summary in Table 2).
Step 2. Cost Estimation (XGBoost + Hyperparameter Tuning)
The cost estimation step is one of the most critical components of the AI-assisted bid-pricing model. It determines the baseline cost for a project by leveraging XGBoost, an ML algorithm that predicts costs based on historical project data and market factors. This step predicts realistic project costs based on past data and current trends. XGBoost estimates the total project cost, C t o t a l , using a regression model, C t o t a l = f X + ϵ , where X is the feature set (e.g., material cost, labour cost, and bidding environment), f X the XGBoost prediction model, and ϵ the error term (unexplained variance). XGBoost then optimises its predictions using gradient boosting with a rich feature set (Table 1), which iteratively improves the model by minimising the residual error through equation F m X = F m 1 X + η . h m ( X ) , where F m X is the prediction at iteration m , F m 1 X is the previous iteration’s prediction, η is the learning rate (controls step size), and h m ( X ) is the decision tree model that reduces errors.
The AI model is trained on historical bid data, using XGBoost regression to predict the total project cost. This step involves data preparation, where irrelevant columns in datasets are removed, datasets are one-hot encoded for categorical variables (e.g., small/medium/large firm sizes), and split into training (80%) and testing (20%) subsets. Model parameters are then defined, such as the number of trees and learning rate, and the model is trained on the training data subset. The model’s performance is evaluated for prediction accuracy using the mean absolute error (MAE) for closeness to actual values and the R 2 score for cost variation insights.
Step 3. Risk Modelling (Bayesian Networks)
This stage is used to quantify uncertainty in pricing by incorporating probabilistic cost variations. Bayesian risk modelling estimates the true cost of the project by considering both known costs and uncertainty factors using the relationship C r i s k = C t o t a l + N ( 0 , σ ) , where C t o t a l is the baseline estimated project cost; N ( 0 , θ ) is the risk factor, modelled as a normal distribution; and σ is the standard deviation, representing the cost variability. Using Bayes’ theorem, the AI continuously updates its cost estimates as new bid-pricing data become available.
P C D = P ( D | C ) P ( C ) P ( D )
where P C D is the posterior probability (updated cost estimate given data D ), P ( D | C ) is the likelihood (how well the cost data match the estimated model), P ( C ) is the prior probability (initial belief about project costs), and P ( D ) is the evidence (all observed data used in the estimation). As new bid cost data are collected, Bayesian modelling refines the cost predictions dynamically. The framework uses PyMC3 to model the uncertainty in bid pricing. The first step involves defining the Bayesian model in which the base project cost is assumed to follow a normal distribution based on the historical bid data. The bid environment uncertainty factor is modelled using a half-normal distribution (to ensure positive values). The final cost prediction includes an additional risk factor (Figure 1). This approach uses Bayesian risk modelling to analyse the posterior distribution of risk-adjusted costs, such as the mean expected cost (MEC) and confidence interval (CI).
Step 4. Monte Carlo Simulation
The Monte Carlo simulation step is crucial for estimating potential cost variations in construction bidding. Since real-world project costs fluctuate because of the bidding environment, material price changes, and unexpected risks, this technique quantifies uncertainty by simulating thousands of possible cost outcomes. This step is important in generating thousands of potential cost outcomes to assess market fluctuations by incorporating stochastic uncertainty. Monte Carlo simulation works by repeatedly sampling random values for key cost components and computing the total project cost for each scenario using the relation C m = M . N 1.2 , 0.1 . I + L . N 1.1 , 0.05 . I , where M and L are the materials and labour costs, respectively, N   ( μ , σ ) is the normal distribution representing uncertainty in costs, and I is the market inflation factor, sampled from U ( 0.95 , 1.1 ) (uniform distribution). Using the relationship first ensures that the higher material/labour costs’ volatility leads to larger deviations in cost estimation, and inflation effects are incorporated dynamically.
The Monte Carlo Simulation in the model defines the cost components with uncertainty, where each cost component is modelled using random distributions. A total of 10,000 trials produce a distribution of possible total project costs (Figure 1).
Step 5. Competitor Behaviour Prediction
The competitor behaviour prediction step is essential for accurately modelling rivals’ bidding strategies. By predicting whether competitors will bid aggressively (low bid) or conservatively (i.e., a high bid), the AI can adjust its bid pricing to increase the probability of winning while maintaining profitability. In this step, the model uses ML to classify competitor bidding strategies. Using a supervised Random Forest classifier, the model predicts whether a competitor will bid aggressively or conservatively.
A key feature in this step is feature engineering for competitor prediction that supports building an accurate competitor bidding prediction model based on the following features in Table 1.
The process then classifies competitors into aggressive bidders (lower bid prices and higher competition risk) or conservative bidders (higher bid prices and lower risk-taking). This classification helps the AI anticipate competitor actions and adjust its bid accordingly. Equation (7) is then applied for the probability of a competitor choosing strategy k ( P ( S c = k X ) , X i , the input features (past bid price, market demand and economic conditions), and β i are the coefficients assigned to each feature. If P ( S c = k ) is greater than 0.5, the model classifies the competitor as aggressive; otherwise, it classifies them as conservative.
Training the competitor behaviour prediction component involves data processing by removing irrelevant variables (e.g., unrelated cost factors) and balancing datasets using SMOTE to avoid class imbalance (if aggressive bidders are rare, we generate synthetic data). Lastly, the process one-hot encodes categorical variables (e.g., small/medium/large firms). Random Forest training involves splitting the dataset into 80% training /20% testing.
At this point, the model is evaluated for its performance in accuracy (i.e., how well it classifies bids) and precision–recall (i.e., how well it distinguishes aggressive from conservative bids).
The probability distribution of simulated project costs, against risk-adjusted estimates shows a normal (Gaussian) peaking at around 50,000GBP (Figure 2).
Step 6. Reinforcement Learning (PPO)
The final step in the AI-assisted bid-pricing model leverages RL to optimise bid pricing dynamically. This ensures that the bid maximises profitability while remaining competitive in the market. In this step, the process optimises bid pricing dynamically based on real-time learning from past bids. AI learns optimal bid pricing using PPO.
RL models decision making as an interaction between an agent (i.e., AI) and an environment (i.e., bidding competition). For the agent, the AI model learns the optimal bid price. The environment helps simulate the bidding environment (competitors, project costs, and winning conditions). Following Equation (8), the state, s , represents the current bidding environment (e.g., material costs, demand, and competitor behaviour). An action, a , represents the bid price chosen by the AI, a , while the profit or loss after submitting the bid is the reward, R . The policy, π ( a s ), is the AI’s strategy for choosing bid prices. Its main objective is to maximise the cumulative rewards (i.e., profit) over multiple bids.
To train the AI, the model uses PPO, an RL algorithm. PPO allows for stability in modelling for continuous bid pricing. It also balances exploration vs. exploitation in ensuring that AI tries new strategies, improving its learned bid strategy while simultaneously bringing dynamism to the process, such as where cost factors fluctuate. The reward function (i.e., bid profit maximisation) has the following relationship:
R t = B i C i   i f   b i d   w i n s C i   i f   b i d   l o s e s
where B i is the AI-predicted bid price and C i is the estimated project cost. The objective is to learn a bid price that maximises the cumulative reward over many bids.
The RL agent trains over multiple bid simulations to refine its bidding strategy. Initially, the model defines the state (i.e., market demand, material costs, and past competitor bids) and action (i.e., bid price range) spaces. The PPO algorithm optimises the bid-pricing policy.
Next, the AI submits bids in a simulated market, choosing a bid price based on past learning and determining whether it wins or loses the bid. AI receives a reward (profit/loss) based on the bidding outcome. The AI then adjusts its bidding strategy, updating its policy function to improve bid selection. Over time, it learns the optimal bid range to balance winning contracts and profitability, after which it applies itself to the case scenario. AI continuously refines its strategy as new bid data become available.
Step 7. Final bid decision
In this step, AI recommends a bid price that balances winning probability and profitability.
Advanced feature engineering adds predictive accuracy. This includes data related to project location, which shows that certain locations have higher labour/material costs; seasonality trends, which show that construction costs fluctuate seasonally; and macroeconomic indicators, which show that inflation and interest rates impact costs.
Competitor behaviour prediction with more contextual data uses external market signals. These include previous bidding history to track whether a competitor won/lost similar projects, competitor’s market position to look at how large firms may price differently from small contractors, and sentiment analysis of market reports to extract market confidence from construction news. This helps the model better anticipate aggressive vs. conservative bidding.
Deep Bayesian networks can refine risk calculations for more robust uncertainty estimation than basic probabilistic models. Correlations between cost fluctuations and economic cycles help capture market-induced cost fluctuations, leading to more realistic bid pricing. The PPO is chosen for its stability as it is better suited for continuous action spaces (bid price adjustments in real time). The PPO provides better long-term strategy adaptation.

4. Results and Discussion

Through modelling, the AI learns that bid prices increase by approximately 5–10% in high-inflation periods through the analysis of complex relationships among cost factors. The competitor behaviour also suggests aggressive pricing strategies in the UK market, to which the AI adjusts the bid prices based on inflation and competitor data.
XGBoost’s critical role in cost estimation in the model is in reducing errors in the predicted project costs (Table 3 and Table 4), as well as in feature importance analysis, where AI understands what drives cost fluctuations (Figure 3), updating cost predictions dynamically. This ability and its complementary capacity to seamlessly integrate with Monte Carlo simulation and Bayesian risk modelling for more accurate bid pricing makes XGBoost a vital component of the AI model.
Similarly, the AI cost estimates are generally within 10% of actual bid prices, showing that the model is accurate but flexible (Figure 4). This small variation is expected in real-world cost estimation because of small errors. Only 8% of the variability is due to errors, noise, or factors not captured by the model (Table 5).
The Bayesian risk modelling produces the expected costs with confidence intervals by analysing the posterior distribution of risk-adjusted costs. The Monte Carlo simulation results, on the other hand, provide crucial decision-making insights. For example, MEC represents the most likely project cost based on thousands of scenarios, while the cost variance indicates the expected costs’ volatility. A 95% C I falls within the cost range, where the project cost is expected to fall 95% of the time (Figure 5). If 95% of costs fall below the threshold, AI ensures the bid price exceeds this to avoid losses.
Bayesian risk modelling enhances AI bidding by integrating realistic cost uncertainty modelling. The AI does not assume fixed costs but accounts for risk factors. It also brings confidence intervals for smarter bidding to the process, where AI avoids pricing that is too low or too high, increasing the bid win probability. The bidding process can also integrate market-adaptive learning through which Bayesian models update continuously as new bid data are collected. Finally, the integration with RL allows the AI to learn the optimal bid pricing using Bayesian risk modelling as a cost input.
The Monte Carlo simulation, on the other hand, improves bid pricing through uncertainty quantification, where AI gains a probabilistic understanding of the cost variations. In the process, it helps prevent underbidding by ensuring that bid prices account for worst-case cost increases. Overall, the component improves competitiveness, allowing for dynamic AI adjustments to bids based on risk-aware forecasts. The approach, therefore, ultimately contributes to improved strategic pricing, supported by AI to find the optimal bid price where it can outbid competitors while remaining profitable.
A confusion matrix (Figure 6) of the AI-powered competitor predictions shows that after training on past bidding data, the AI classified 70% of competitors as conservative and 30% as aggressive. Since most competitors bid conservatively, the AI adjusted its bid to slightly lower to increase its probability of winning. The AI would have maintained a safer pricing strategy to avoid losses if more competitors were aggressive.
Figure 6 similarly shows the competitor strategy classification indicating a high accuracy score of 87% meaning that the model performed well overall, with only a 13% misclassification rate. A strong precision score of 88.4% reinforces the model’s ability to classify a conservative strategy as aggressive, meaning fewer false positives while a moderate recall (82.6%) suggests that some aggressive strategies are missed, meaning false negatives exist. The model returns an F1 score of 85.4%, which represents a strong balance between precision and recall, making this model reliable for strategic decision making. Therefore, competitor prediction improves AI’s bid strategy by stratifying how conservative bidders bid higher to ensure profit. The AI, in this case, can bid slightly lower while maintaining profitability. Ultimately, a good strategy creates a higher probability of winning without major profit loss. Conversely, when aggressive bidders are in the market, the AI’s responsiveness means it does not need to reduce prices significantly to remain competitive. The AI, therefore, helps to avoid excessive underbidding and protects profitability.
The PPO learning curve shows how the AI improved its bidding strategy over time. It started with random bidding but gradually learned profitable bid pricing. The cumulative rewards increased, meaning the AI won more profitable bids. After 100 training iterations, the AI converged to an optimal strategy.

5. Conclusions and Future Work

AI helps balance cost, competition, and risk dynamically to aid decision making. This research used structured bid-related data to underpin an effective AI predictive model for project costs with high accuracy while optimising bid-pricing strategies dynamically. The AI-assisted bid-pricing model differs from traditional bidding in that it first and foremost offers accurate cost estimations using XGBoost’s hyperparameter tuning for precise cost forecasting. The AI-powered XGBoost cost estimation significantly improved the bid accuracy, increasing the win rates while maintaining profitability. This research contributes to the development of an adaptive bidding system that leverages AI-driven analytics to support enhanced, effective, and competitive bid-pricing strategies in the construction industry.
XGBoost transforms cost estimation from a static prediction task to an adaptive AI-driven process by providing real-time, high-accuracy cost forecasts. This means that bidding processes can be augmented with AI-assisted smarter bidding decisions supporting higher profitability with reduced risk, as well as an accuracy of 87%, even on a small dataset. This also means more competitive and strategic bid pricing because the model is precise (88.4%) in classifying bidding strategies, representing the potential to improve bidding success rates.
Risks are quantified in an integrated manner using Bayesian networks and Monte Carlo simulations to model cost uncertainty. Competitor awareness is brought to the fore, understanding their risk approaches to bidding within the integrated model. The model is effective in predicting competitor bidding patterns and attitudes using ML classification with an 85.4% return on the F1 score, representing a balance between precision and recall. This can form the basis for strategic bidding optimisation, allowing the PPO to adjust pricing strategies dynamically. Integrating these capabilities ensures an adaptive approach to bidding with the potential for generalisability and customisation to different project contexts and scopes in various sectors. The model brings dynamic learning to bidding processes where AI can adapt bidding strategies over multiple projects. The market-aware model allows AI adjustments to bids based on competitor behaviour. This can mean optimised profitability supported by RL that ensures the AI balances profit vs. bid success. This approach allows for scalability and generalisability across various project contexts and sectors. AI and game theory, therefore, bridge the gap between predictive analytics and strategic decision making, enabling contractors to bid smarter and win more projects.
Integrating AI with game theory creates a powerful framework that enhances contractors’ ability to navigate uncertainty, anticipate competitor behaviour, and make more informed bid-pricing decisions. This paper, therefore, forms the basis for a synergistic, novel approach to strategic decision making in bidding practices in the construction industry. It is shown that AI not only adapts bid pricing dynamically based on who it is competing against but also simultaneously balances winning the contract vs. maintaining profit margins in an integrated manner with high accuracy and precision. The AI, therefore, improves the bid efficiency over time by learning from past competitors and their strategies.
The AI-assisted game theory model offers a new approach to bid pricing in construction. It enables contractors to navigate uncertainty in project environments, predict competitor strategies, and optimise their bids. This methodology bridges theoretical advancements with real-world applications, making it a valuable tool for improving competitiveness in the construction industry.

5.1. Limitations of the Study

While the AI-assisted game theory bid pricing model significantly enhances bid prediction accuracy and strategic decision making, several limitations exist, including data availability, quality, and dynamic bidding environment. Poor quality, incomplete or biased datasets (e.g., missing competitor bid data) impacted the model’s predictions. Additionally, the competitor bidding model assumes that firms follow consistent bidding strategies based on historical patterns, which is not always the case, as they often alter strategies unexpectedly, such as collusive bidding or strategic loss-leader bidding. Above all, integrating XGBoost, Bayesian networks, Monte Carlo simulations, and RL represents computational complexity. Real-time bidding simulations and modelling of large-scale projects may require high-performance computing resources. Lastly, optimising the RL training’s efficiency in the model remains an area for potential improvement.

5.2. Future Research

Future work should explore methods to augment datasets with synthetic data or real-time market feeds. Research could also explore the integration of live economic indicators and adaptive learning mechanisms into the model. Further research can enhance the model with game-theoretic adaptive learning to further improve competitor strategy prediction. This could include advanced game theory models integrated with AI, such as Stackelberg competition models where firms react strategically to leader–follower bidding dynamics or auction-based models (e.g., Nash equilibrium strategies in sealed-bid tenders). More importantly, further research could help towards achieving a generalised AI model that can adapt across different regions, project types, and construction sectors, and ultimately, live construction tenders to assess its effectiveness in these various project contexts.

Funding

This research received no external funding.

Data Availability Statement

The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.

Conflicts of Interest

The author declares no conflicts of interest.

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Figure 1. The proposed AI-assisted game theory bidding model.
Figure 1. The proposed AI-assisted game theory bidding model.
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Figure 2. Bayesian risk distribution for project cost (approximate).
Figure 2. Bayesian risk distribution for project cost (approximate).
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Figure 3. Montecarlo simulation results.
Figure 3. Montecarlo simulation results.
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Figure 4. Feature importance in cost estimation (XGBoost).
Figure 4. Feature importance in cost estimation (XGBoost).
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Figure 5. Cost forecast vs. actual bid prices.
Figure 5. Cost forecast vs. actual bid prices.
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Figure 6. Competitor strategy classification (confusion matrix).
Figure 6. Competitor strategy classification (confusion matrix).
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Table 1. Key features used in AI cost estimation.
Table 1. Key features used in AI cost estimation.
Feature NameDescriptionWhy It Matters
Material CostTotal cost of raw materialsMajor component of total project cost
Labour CostTotal worker wages for the projectHighly variable, depends on location and project type
Project ComplexityScale of difficulty (low/medium/high)Complex projects require more time and resources
Market DemandIndicator of project demandHigh demand raises bid prices
Competitor Win Rate% of past bids won by competitorsHelps AI predict aggressive/conservative bidding
Economic IndexCaptures inflation, GDP, and bidding environmentAffects material and labour prices
Seasonality FactorWhether the project occurs during the peak or off-peak seasonCost fluctuations due to supply/demand cycles
Table 2. Feature engineering.
Table 2. Feature engineering.
FeatureMean Value (UK Market)
Material CostGPB 150,000–600,000
Labour CostGPB 100,000–350,000
Project Complexity Score1 (Low)–5 (High)
Economic Index (UK Inflation)105 (Base = 100)
Competitor Aggressiveness Score0.65 (More Aggressive)
Table 3. A rich feature set.
Table 3. A rich feature set.
FeatureDescription
Material CostCost of raw materials (steel, cement, etc.)
Labour CostTotal wages for workers
Project ComplexityA numerical rating of project difficulty
Market DemandIndicator of how much demand exists for the project type
Economic IndexCaptures inflation and market stability
Competitor BehaviourBidding patterns of rival firms
Seasonality FactorCost variations due to seasonal trends
Table 4. Feature engineering for competitor prediction.
Table 4. Feature engineering for competitor prediction.
FeatureDescription
Past Bid PriceThe competitor’s last bid amount
Market DemandEconomic indicators affecting project bidding
Competitor Market PositionWhether the firm is small, medium, or large
Past Win RatePercentage of bids won by the competitor
Project ComplexityHigher complexity may influence bidding behaviour
Economic IndexCaptures inflation, interest rates, and market stability
Table 5. AI model’s results with the XGBoost.
Table 5. AI model’s results with the XGBoost.
StatisticValue (GBP)
Predicted Cost2.85 M
MAE100 K
R² Score0.92
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Serugga, J. AI-Assisted Game Theory Approaches to Bid Pricing Under Uncertainty in Construction. AppliedMath 2025, 5, 39. https://doi.org/10.3390/appliedmath5020039

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Serugga J. AI-Assisted Game Theory Approaches to Bid Pricing Under Uncertainty in Construction. AppliedMath. 2025; 5(2):39. https://doi.org/10.3390/appliedmath5020039

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Serugga, Joas. 2025. "AI-Assisted Game Theory Approaches to Bid Pricing Under Uncertainty in Construction" AppliedMath 5, no. 2: 39. https://doi.org/10.3390/appliedmath5020039

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Serugga, J. (2025). AI-Assisted Game Theory Approaches to Bid Pricing Under Uncertainty in Construction. AppliedMath, 5(2), 39. https://doi.org/10.3390/appliedmath5020039

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