Applying Game Theory to Teach the PERT Scheduling Method

Abstract

This research focuses on applying Game Theory to teach the PERT scheduling method to civil engineering students. The work was carried out with students from the Construction Planning and Scheduling course, who were given face-to-face classes on Game Theory applied to teaching the PERT method. This proposal considers that gamification plays a role in user engagement, while Game Theory contributes to data analysis and decision strategies. It began with an initial pre-test evaluation consisting of questions classified by items and evaluation criteria to teach the PERT method using Game Theory, followed by a post-test and a quantitative evaluation that allowed to know the perception and interests of the students. Using Game Theory principles in teaching the PERT method demonstrated improvements in students. In addition, the results presented statistically significant differences in favor of using Game Theory, which would allow it to be proposed as a viable option for teaching PERT and other scheduling methods.

1. Introduction

1.1. Context

Despite advances in active learning techniques and their dissemination, the predominant method of instruction remains the transmission model, where information is a quantifiable resource, and the role of the instructor is to transmit that information to learners who must store it [1]. While the transmission model can be effective for basic understanding, it is often less efficient for developing practical skills such as communication, teamwork, and creativity [2].

Today, there is a recognized need for a learning-centered didactic approach that allows students to acquire knowledge and develop skills, form values, and acquire strategies to act independently and creatively in their professional future [3]. In this sense, game-based learning is recognized as a practical pedagogical approach that provides students with the motivation and opportunity to practice skills that traditional classrooms cannot offer [4].

However, the transmission-based teaching model—still predominant in engineering education—limits opportunities for students to engage in active problem-solving and decision-making [5]. This limitation underlines the need to explore active learning strategies that foster critical thinking and skill development [6]. To address this gap, Game Theory emerges as a promising pedagogical tool emphasizing essential competencies such as decision-making and strategic thinking, which are increasingly demanded in modern engineering contexts [6].

Table 1. Comparison of traditional teaching methods and active learning techniques (adapted from Prince [7], Bonwell and Eison [8], and Michael [9]).

	Traditional Teaching Methods	Active Learning Techniques	Role of Game Theory
Focus	Emphasis on memorization and theoretical understanding	Focus on practical application and skill development	Encourages strategic decision-making and critical thinking
Student Engagement	Passive learning (listening and note-taking)	Active participation in problem-solving and collaboration	Enhances engagement through interactive decision scenarios
Skill Development	Limited development of teamwork, creativity, and communication skills	Strong focus on developing communication and teamwork skills	Simulates real-world scenarios requiring cooperative skills
Motivation	Often low, with minimal interaction	High, due to engaging and hands-on learning environments	Adds excitement and competition to learning activities
Applicability to Real-World	Indirect, often disconnected from practical challenges	Direct application to real-world scenarios	Provides tools for analyzing and solving complex problems

below compares traditional teaching methods with active learning techniques, highlighting the potential of Game Theory to address the gaps in traditional engineering education.

On the other hand, training highly qualified professionals requires implementing learning techniques that promote active teaching, critical thinking, and reflective analysis [10]. Thus, incorporating Game Theory into engineering education provides an opportunity to address these needs by bridging the gap between theoretical knowledge and practical application [11]. A pedagogical approach introducing innovative teaching techniques like Game Theory could provide students with a motivating and stimulating environment.

1.2. Research Problem and Objectives

Game Theory is expected to improve students’ engineering knowledge since this technique can improve problem conceptual formulation [12,13]. To verify whether Game Theory can significantly impact the teaching of project scheduling techniques to future civil engineers, the research considered the teaching of the PERT method (Program Evaluation and Review Technique) as a case study. Its probabilistic nature makes it suitable for applying techniques that improve decision-making, where Game Theory plays an important role.

Despite its potential, there is limited evidence of academic research that has explored the application of Game Theory as a teaching methodology in civil engineering [14]. Current studies lack depth in addressing its integration with project management techniques such as PERT, particularly for fostering strategic decision-making skills in students, which is essential for equipping future engineers for professional success.

A methodology was developed that allowed the application of Game Theory to the PERT method to help students understand the problem content and encourage decision-making in work scheduling. This approach may help improve pedagogical strategies and train students to meet the expectations of the working world [15].

Thus, the general objective of this research is to apply Game Theory to teach the PERT project scheduling method. The specific objectives are the following: (a) To develop a methodology for applying Game Theory to the teaching of the PERT method; (b) To apply the didactic technique of Game Theory to the teaching of the PERT method; (c) To quantitatively and qualitatively evaluate this methodology in comparison with traditional teaching methods; and (d) To analyze the feasibility of applying Game Theory to the teaching of the PERT method to civil engineering students participating in the engineering works scheduling course.

2. Literature Review

2.1. Gaming, Gamification, and Game Theory

For years, gaming, gamification, and Game Theory have been studied. First, Gaming and Game Theory are two very different yet highly related disciplines, where the crucial relationship between them lies in the analytical conclusions drawn from Game Theory, which asserts that even with a particularly rationalistic number of assumptions regarding human behavior, there is no straightforward way to extend the idea of individual rationality to groups of individuals [16]. In this sense, Roungas et al. [17] suggested a framework that applies game concepts belonging to Game Theory in the abstraction of real systems and game design decisions, contributing to developing gaming simulations.

On the other hand, technological advancement, innovative approaches to teaching, and the role of the educator within the teaching-learning process have taken on a relevant role in achieving the expected learning outcome [18]. As shown by various studies and shared practices in different educational settings, using game elements in non-playful contexts can positively affect increasing student engagement and motivation [19]. This approach is especially relevant today, where traditional teaching methods often fail to capture the attention of modern students [20]. In this sense, gamification offers a framework that combines play with the structure of educational objectives, becoming an attractive approach to achieving results [21,22].

Accordingly, Game Theory provides a theoretical approach to addressing strategic decision-making in situations of conflict or cooperation [23]. In a gamified environment based on Game Theory, students face situations that involve decision-making where they must consider their interests, those of their peers and “opponents”, which stimulates critical thinking and collaborative work. In addition, the development of crucial skills, such as the analysis of complex scenarios, work planning, and adaptation to changing contexts, is promoted [24,25]. These elements are essential in the training of engineers capable of facing the challenges of the real world, where planning, resource optimization, and decision-making under uncertainty are crucial components for achieving success [26].

Thus, from an educational perspective, this proposal considers that gamification plays a role in user engagement, while Game Theory plays a role in data analysis and decision strategies [27].

2.2. Origins of Game Theory

The first theoretical approach that outlined the concept of Game Theory was the Cournot Competition Model, carried out by Antoine Cournot in 1838, which defined imperfect competitions where two companies with equal cost functions compete with identical goods in a static environment [28,29]. Later, the mathematician Emile Borel introduced the concept of Strategy Games in 1921, and this was promoted by the mathematician John von Neumann in 1928, who anticipated the basis of Game Theory. This theory was developed in collaboration with the economist Oskar Morgenstern in 1939 to analyze negotiation processes. However, it was not until the publication of their book Theory of Games and Economic Behavior in 1944 that the community understood the potential of this tool for decision-making [30].

Based on the principles established by Von Neumann and Morgenstern, the study of non-cooperative scenarios was carried out by John Nash in the 1950s, who proposed a solution for non-collaborative strategy games called the Nash equilibrium [31]. This equilibrium states that no player has incentives to change strategy if the other players do not change theirs, providing a basis for conflict analysis and strategic decisions in non-cooperative scenarios [32].

2.3. Game Theory

A game can be defined as a conflictual activity in which decisions must be made, knowing that the other players also make decisions, determining the outcome of the game based on all the decisions made [30]. According to Aumann and Schelling [33], Game Theory is a situation in which two or more players must decide to achieve a beneficial objective by following specific strategies. This mathematical theory attempts to describe the socioeconomic phenomenon of human relationships in terms of cooperation and conflict between people who must make decisions, called players.

Decision-makers often face conflicts of interest, which drive them to adopt different strategies to determine other individuals’ actions, seeking the most significant possible results and benefits [34]. Game Theory evaluates the problem of decision-making in a group of people under uncertainty and lack of information about the environment, interpersonal decision processes, and the skills and incentives of other players, generating a probabilistic environment [35].

In general, game theoretical models can be classified into (1) static games, in which players make decisions and take actions simultaneously, without knowing the decisions chosen by other players, and (2) dynamic games, in which players make decisions and take actions sequentially, observing the actions of other players [36]. On the other hand, there are two main branches of Game Theory: (1) Cooperative Game Theory, in which players cooperate mutually to obtain more benefits and allocate the gains equitably, and (2) Non-cooperative Game Theory, in which players independently select strategies to maximize the benefit, without collusion between them [37]. In the case of Nash equilibrium, the solution for non-cooperative games is considered under the assumption that all players are rational.

Non-cooperative Game Theory can facilitate predicting possible outcomes and behaviors of decision-makers or players who prioritize their goals and make strategic decisions based on individual rationality [38]. Cooperative Game Theory provides valuable insights into resource-sharing games, in which parties may adopt diverse strategies for utilizing the shared resource.

Thus, the present study builds on these theoretical foundations by demonstrating how Game Theory can enhance decision-making and strategic thinking in educational settings. Specifically, its application to the PERT method leverages the probabilistic nature of Game Theory to simulate real-world scenarios involving uncertainty and dependencies among activities [31,34]. This approach highlights its suitability for teaching project scheduling techniques, bridging the gap between theoretical knowledge and practical applications [39,40].

For an activity to be considered a game, it must have some characteristic elements, although they are not necessarily always involved:

• Player: This is the participant who decides within the game, trying to obtain the best result. The player seeks to maximize their utility under the rules that the game imposes and considers that the decisions made by the other players affect their results. More than one player can participate in games, such as a person, a company, the government, and others, assuming that the player is rational and informed [41].
• Actions: An action is a decision each player can make when acting within the possible alternatives that a player can adopt within the game, using strategies to maximize utility [42].
• Information: It is the degree of knowledge the player has of the variables involved in the game at a given time. This information can be provided by other participants or acquired by the same player. The information can be classified as complete, incomplete, perfect, or imperfect [43].
• Strategy: It plans a player’s action when starting and during the game [31].
• Results: Different scenarios in which the game can end, and are associated with different consequences for each player [30].
• Payment: It depends directly on the result and can be defined as the income or reward that each player receives at the end of the game [44].
• Equilibrium: Corresponds to the strategy profile, comprising the best possible options for each player. There are the Nash and dominant strategies equilibrium [45].

2.4. Applications of Game Theory

Game theory has been applied to different aspects of human life. In the 1950s and 1960s, it was applied, for example, to decisions regarding battles and political problems. In the 1970s, Game Theory revolutionized economic studies and was applied to sociology, psychology, and biology. Its most significant recognition came from the 1994 Nobel Prize award to John Nash, John Harsanyi, and Reinhard Selten.

This theory has also been widely used in various fields, from wireless networks to applications in defense, Internet security, cybernetics, image processing and coding, design of electromagnetic devices for electric vehicles, and target tracking, among others [46]. For example, recent studies have demonstrated the applicability of tools such as Evolutionary Game Theory to create a hypothetical model of limited rationality for the behavior of key stakeholders in mutual aid for safety risk prevention and control [47].

Moreover, game-theoretic approaches have been integrated into optimization techniques to address resource-constrained project scheduling problems. For example, some researchers have worked on integrating the planning and scheduling of multiple manufacturing projects under resource constraints using the raccoon family optimization algorithm [48], demonstrating how innovative optimization frameworks can improve scheduling efficiency and decision-making. Consequently, these creative methodologies could inspire new applications in educational settings, mainly through Game Theory, to enhance students’ understanding of complex scheduling dynamics [20].

2.5. Adaptation of Game Theory to Engineering and Construction

Several researchers have applied Game Theory models to explain and predict outcomes in engineering. Ho [49] used Game Theory to analyze the build-operate-transfer processes in the presence of asymmetric information and its effect on government funding and policies. Drew and Skitmore [50] analyzed bidding schemes in the construction industry using Auction Theory, a subdiscipline of Game Theory. Drew and Skitmore [50] also analyzed the dynamics between contractors and owners in construction claims through a game-theoretic model. In addition, Game Theory has been used to evaluate strategies for selecting subcontractors and to analyze the effect of bid compensation on the competitive bidding process.

It is, therefore, evident that the applications of Game Theory and Negotiation Theory to construction management have increased in recent decades [51]. Specifically, a Game Theory study was applied to explain the influence of the degree of reliability of the planned work schedule on the behavior of subcontractors and project managers under a traditional contracting scheme of unit prices [52].

On the other hand, Game Theory provides an appropriate framework for studying the joint management of resources through association during construction projects [53]. Furthermore, this theory has been identified as a helpful framework for investigating various aspects of construction projects [54]. However, there is limited evidence that this theory has been used in civil engineering teaching or construction management.

2.6. Game Theory as a Teaching Tool

Regarding active learning techniques, such as simulations and role-playing, these have been successfully applied in engineering education and project management training [40]. Simulations, for example, allow students to engage in realistic scenarios to practice decision-making and resource management in a controlled environment [55]. Role-playing, on the other hand, encourages collaboration and creativity by assigning students specific roles within a team to tackle complex problems [56].

Game Theory offers distinct advantages compared to these methods, particularly its ability to model strategic interactions and incorporate probabilistic decision-making [57]. These features align closely with the characteristics of the PERT method, which involves uncertainty and dependency among activities. While simulations and role-playing are valuable for experiential learning, Game Theory solidly supports the development of analytical and strategic thinking skills, making it an effective tool for teaching scheduling techniques like PERT.

In learning, winning or losing focuses on how an individual can acquire knowledge and how this knowledge is applied in performing his or her activities. In other words, “gaining” new knowledge or “losing” opportunities to learn something new. In this sense, Game Theory has been used mainly to teach economists, market strategists, social and political scientists, and to some extent by practitioners of mathematical optimization [58].

Game Theory has been found helpful in teaching undergraduate and graduate students in engineering, concluding that it improves students’ knowledge by providing them with new tools and techniques to meet challenges in industry with novel and practical approaches. A working knowledge of Game Theory can also improve an engineer’s ability to formulate conceptual problems. In addition to learning the mathematics involved, students are exposed to applications of these concepts in several engineering disciplines, such as electrical and computer engineering, through cutting-edge research [59].

On the other hand, while the application of innovative methodologies to teaching scheduling methods has been explored previously [39], the specific use of Game Theory as a pedagogical tool is underrepresented within the spectrum of teaching project scheduling methods. In this sense, the present study emphasizes the integration of Game Theory within a structured learning environment to bridge the gap between theoretical concepts and practical applications—an area where traditional pedagogies often fall short. Unlike existing approaches, the methodology presented here demonstrates how Game Theory can enhance decision-making, strategic thinking, and analytical skills in a probabilistic context, aligning closely with the inherent nature of the PERT method [60].

2.7. Challenges for Game Theory and Engineering

According to Vygotsky [61], games help to build a vast network of devices that allow the total assimilation of reality, incorporating it to relive it, master it, understand it, and compensate for it. In this way, the game is essentially a means of assimilating reality.

In engineering, logical-mathematical tools have expanded, but the factors to be considered in solutions have increased similarly. Developments in knowledge theory, artificial intelligence, Game Theory, fuzzy logic, genetic algorithms, neural networks, and other fields of combinatorial optimization offer tools to address complex problems beyond classical mathematics, probability calculation, or statistics. These new theories provide tools to tackle problems of previously unattainable complexity, offering acceptable solutions for many modern engineering applications, such as finding critical paths in project completion [62].

The most significant challenges of Game Theory in engineering include the ability to work collaboratively, follow instructions, manage information, and understand the behavior of others. These skills make Game Theory contribute to cognitive and social development. Many organizations seek new professionals with cognitive and human relations skills that allow them to express their knowledge transversally at all organizational levels. Game theory provides comprehensive and potential development in various aspects. Learning based on Game Theory can help one understand concepts, strengthen knowledge already acquired, master algorithms, and reinforce content [63].

Managing Game Theory as a tool associated with scheduling would impact time management from the point of view of management skills by recognizing the need to set goals, work proactively to achieve them, establish priorities, and plan and schedule. In addition, it reduces the adverse effects of misuse of time and loss [63].

Complementing the trajectory of Game Theory, this research expands its application to the scheduling of engineering projects using the PERT method, finding limited bibliographic evidence indicating that Game Theory has been previously incorporated into teaching the PERT scheduling method [52]. This fact is corroborated by Kline and Ayer [64], who provided a comprehensive review of the academic literature from the last 20 years, examining educators’ diverse methods of teaching construction scheduling. The authors state that teaching methods such as virtual simulations, game-based approaches, hands-on experiential learning, and virtual and in-person field trips have been primarily used; however, no Game Theory experiences are mentioned. In addition, the literature shows other articles discussing teaching approaches for construction scheduling. Ilbeigi et al. [65] discuss a gamified pedagogical method for teaching construction scheduling through active exploration. In Sami Ur Rehman et al. [66], the effectiveness of immersive virtual reality for project scheduling in construction education is explored. Thus, the present research seeks to propose a teaching method for the PERT scheduling technique based on Game Theory, a branch of mathematics that focuses on how participants strategize and make decisions, which are crucial in training civil engineering students.

Regarding the applicability of the methodology proposed in this research to other areas, it is possible to emphasize some cases where Game Theory has been used as a basis for teaching diverse disciplines. In Lazo et al. [13], the authors use Game Theory to facilitate the learning process in engineering students, concluding that the design of the assessment process based on Game Theory positively influenced the learning of concepts, skills, and attitudes. Similar results can be observed in Huang et al. [67], where some findings are derived from teaching a class of engineering students organized into cooperative learning groups using a Game Theory-based approach. Another interesting case from the field of physical education is presented by Wen et al. [68], who showed how the use of Game Theory facilitates learning based on the idea that educators and students are willing to reach the best result through an iterative process, where the cooperative relationship gradually evolved into a process of Nash equilibrium.

3. Use of the PERT Method in Construction Planning and Scheduling

Today, the construction industry faces significant challenges due to projects’ increasing complexity and dynamism, including resource procurement, cost overruns, conflicts, and disputes. The involvement of a wide range of stakeholders in large-scale projects inevitably causes conflicts [69].

3.1. Project Planning

Effective decision-making in planning seeks to obtain future results based on various desired activities. The planner finds the best options through an integrated decision analysis [70]. Planning an engineering project involves decisions related to manpower, simultaneity of progress, sequence of activities, deadlines, subcontracting, and facility layout. These decisions are defined as project planning.

The current difficulty in controlling work has led to the adoption of more effective scheduling techniques to improve resource allocation and project management. Being efficient implies using a given budget within the necessary time. Planning is a set of decisions to achieve a final product on a desired date, considering the current situation and the internal and external factors that may affect the fulfillment of these achievements [71].

3.2. Scheduling in Construction

According to Alias et al. [72], the successful performance of a project depends on proper organization, scheduling, and control. The project schedule results from project planning and describes all the tasks necessary to complete the project within the planned timeframe, including durations, start and end times for each activity, and the resources and costs assigned. In project scheduling, the critical path is the set of activities that determines the project execution time. A delay in an activity on the critical path will delay the project timeframe, so these activities must receive special attention [73].

Construction planning and scheduling are essential in any construction process since, without proper scheduling, it is impossible to make precise budgets or exact control of the activities. Scheduling must include exhaustive control of costs and deadlines and manage resources efficiently [74]. Several scheduling methods are found in construction, such as the Gantt Chart, the CPM (Critical Path Method), the Fondhal-LPU method (Linear Point Union), and the PERT method (Program Evaluation and Review Technique). These methods are used to control the timing and sequence of activities in a project [75].

In this sense, technological advances have significantly improved the teaching of construction scheduling, making computer knowledge necessary to reduce time in data interpretation and variable control [76]. Scheduling methods are generally based on network systems that allow planning and controlling projects [77]. However, construction scheduling is only the beginning of the construction process; the biggest challenge is to control the schedule by constantly updating the information to make the best decisions and use resources efficiently [78], where the use of Game Theory for teaching a scheduling method such as PERT can bring multiple benefits.

3.3. PERT Method (Program Evaluation and Review Technique)

For non-repetitive activities, the PERT method assumes that the time required is not known in advance, including probabilities in the time analysis and the concept of expected value to obtain the project’s total duration [79]. This method represents the logical sequence and the interrelation of all project activities through a network flow chart, with initiation nodes and branches representing different tasks, which can be successive, simultaneous, or convergent. The PERT method uses probabilistic estimates of the durations of each activity, known as optimistic, pessimistic, and most probable. In this way, it is possible to calculate the probability of completing the project within a given time frame and find a critical path [80].

Like tools such as the Gantt chart and the CPM Critical Path, PERT helps get closer to the desired optimal point in projects [81]. From an administrative perspective and the benefit provided to the decision-maker, PERT specifies and details how planning should be done. Among its advantages, the following stand out: Reduction of time and costs (it helps achieve the objective or complete a project with the minimum expenditure of time and cost) and optimization of resources (it minimizes the costs associated with the project and, in this way, achieves a more significant final utility and attractive returns) [79].

In organizing project activities in an orderly and prioritized manner, tools and methods emerge that simplify planning. The PERT method (Program Evaluation and Review Technique) helps with scheduling control. It allows identifying which activities determine the duration of a project and which ones need to be controlled more to try to shorten their duration.

The PERT method is a set of methods and techniques for effectively scheduling goal-oriented activities, establishing a baseline for scheduling, evaluating costs, controlling, and re-scheduling. It consists of arranging different activities in a network, which, due to their dependencies and links, contribute to the result of the project [82].

In developing a project, there is often uncertainty about the timing of critical activities. These, being random variables, can be associated with a probability distribution. The PERT method takes this uncertainty into account, assuming that the estimated time for each activity is based on three different values: optimistic time (T_o) or the minimum period that elapses when the activity occurs in an ideal manner; pessimistic time (T_p) or the maximum period that elapses when probable but infrequent risks cause long delays; most probable time (T_m) or the most frequent duration, and finally, expected time (T_e) or the expected time for an activity.

The most likely estimate Tm does not have to coincide with the midpoint of (T_o + T_p)/2 because the duration of each activity is assumed to follow a Beta probability distribution.

The standard deviation is represented by Equation (1), and the time variance will be represented by Equation (2):

$${\sigma }_t^2={\displaystyle \frac{T_p-T_o}{6}}$$

(1)

$$v_t={\sigma }_t^2$$

(2)

The midpoint weighs half of the most probable point T_m; that is, the expected value T_e is the average of $(T_o+T_p)/2$ and $2T_m$, as can be seen in Equation (3) and is developed in Equation (4):

$$T_e={\displaystyle \frac{1}{3}}\left[2T_m+{\displaystyle \frac{1}{2}}\left(T_o+T_p\right)\right]$$

(3)

$$T_e={\displaystyle \frac{4{T }_m+T_o+T_p }{6}}$$

(4)

It can also be assumed that the project’s expected duration is a random variable approximating the Gaussian distribution, and probabilities can be calculated using a normal distribution table.

On the other hand, Equation (5) represents the probability that the project is carried out in the desired duration (D):

$$Z={\displaystyle \frac{D-T}{{\sigma }_t}}$$

(5)

where

• $Z$: normalized variable N(0,1)
• $T$: critical path time
• ${\sigma }_t$: standard deviation of critical activities

3.3.1. Elements of the PERT Diagram

Like CPM (Critical Path Method), the PERT method allows for graphically drawing the diagram of the relationships between each activity and their interdependencies [81]. It consists of the following elements: (a) Arrows: These are used to represent an activity to be executed. The start of the arrow represents the beginning of the activity and is associated with the node, while the arrowhead at the end of it represents the direction, from right to left; (b) Nodes: An event or milestone that has no duration, where node i is the start of one or more activities, and node j is the end of an activity. The diagram can only have one start node and one end node. Activities are linked in the nodes; (c) Dummy Activities: Represented by an arrow with a dotted line, they indicate the dependencies between activities. These do not have a duration; (d) Float: The length of time that an activity can be delayed without causing a delay in the completion of the project.

The PERT method, like CPM, was designed to provide valuable information for project managers and decision-makers. If the critical path is delayed in both tools, the same amount delays the project, and the float is the same in both cases.

3.3.2. Simple PERT Example Using Game Theory

The elaboration of a PERT diagram will be based on the data shown in

Table 2. Precedence matrix.

Activity	Description	Dependence	To	Tm	Tp
A	Mobilization	-	2	3	4
B	Debris removal	A
C	Facilities	A	4	5	6
D	Soil improvement	B
E	Pavement	C, D	3	5	10

, where the times are measured in days.

Activities A, C, and E in Table 2 will be solved using the PERT method with known times, and Equation (4) will be used to find the expected time for each activity. However, activities B and D in Table 2 will be solved using Game Theory applied to the PERT method. For Activity B (debris removal), the following extensive game will be played, as shown in Figure 1: the best combination of equipment must be found to perform the work at the lowest cost and in the shortest time.

Using the pairs of numbers $(X,Y)$, where “X” represents the cost of the activity and “Y” represents the time in days, the best option, also called the equilibrium of the game in (6, 3.5), can be found. The expected time is 3.5 days, the optimal time is 2 days because it is the lowest “Y”, the pessimistic time is 7 days—the longest time that the activity can take—and the average time is 3 days.

As in the previous activity, for activity $D$ (soil improvement), different options in the game in matrix form are found, where the optimum between cost and time is found in the pair of numbers (5, 6).

Table 3. Activity D: soil improvement (cost $, time in days).

	Vibro Replacement Stone Column	Without Vibro Replacement Stone Column
Soil Replacement	(7, 8)	(5, 6) *
Without Soil Replacement	(6, 7)	(15, 0)

*: The pair of numbers (5,6) represents the optimum between cost and time.

is filled in with the times obtained using both methods, and the PERT network is then constructed.

Thus,

Table 4. Precedence matrix with estimated times.

Activity	Description	Dependence	To	Tm	Tp	Te
A	Mobilization	-	2	3	4	3
B	Debris removal	A	2	3	7	3.5
C	Facilities	A	4	5	6	5
D	Soil improvement	B	0	7	8	6
E	Pavement	C, D	3	5	10	5.5

shows the precedence matrix with the final estimated times, and Figure 2 shows the final network with the times calculated using Game Theory.

Given this data, the process’s duration can be determined, as shown in Figure 2. For this specific example, the critical path will be A-B-D-E, indicating that the process will take at least 18 days.

4. Methodology

4.1. Assessment Material

A pre-test consisting of 26 questions with three different modalities was applied to measure the students’ prior knowledge: order sequences, true or false, and multiple-choice questions on construction scheduling. This set of questions was developed based on the authors’ experience and reviewed by professionals and academics with experience in teaching Construction Planning and Scheduling [83,84].

The 26 questions were classified into four categories of evaluation criteria (with six or seven questions per category):

• Identification of activities.
• Duration of activities.
• Critical path.
• Variability and decision-making.

Both the pre-test and post-test questions addressed topics related to construction scheduling and construction processes. After the PERT method was taught through Game Theory, a post-test containing the same set of 26 questions was administered. The questions could be solved traditionally or using Game Theory, thus ensuring that both assessments were comparable [85].

The pre-test and post-test design was aligned with the criteria established by Burton for constructing items in criterion-referenced tests, ensuring their validity and relevance to the content of the Construction Scheduling Method through PERT and Game Theory [86].

4.2. Study Group

The study group consisted of 33 students taking the Construction Planning and Scheduling course, part of the civil engineering curriculum at a relevant Chilean university. In engineering education, the use of small sample sizes in diverse studies has created tension in fields focused on quantitative approaches. However, in recent years, there has been a growing demand for these methods, presenting exciting opportunities to explore new perspectives on issues in engineering education [87]. Accordingly, several authors have developed didactical applications in engineering education, and their research has allowed interesting and extrapolating conclusions to be drawn from their results [39,88].

All students were instructed to learn the PERT method using Game Theory and were given a pre-test and a post-test. The decision to evaluate the same group at the beginning and end of the course (pre and post-test) was made to control for variability between groups and focus on the intervention’s direct impact. This approach ensured that the differences in scores were attributable to the teaching methodology (i.e., Game Theory) rather than pre-existing differences in knowledge or skills between separate groups [89,90].

It must be noted that although the Construction Planning and Scheduling course is a final-year course, the students did not necessarily belong to the same cohort or have the same academic progression. However, to enroll in the course, students had to have previously passed two courses: “Construction Equipment and Methods” and “Operations Research”, which suggests that they all had a common baseline of prior knowledge [91].

On the other hand, there is a phenomenon in which subjects subjected to a study improve their performance just because they know they are being evaluated. This phenomenon is the Hawthorne effect, inherently linked to the excitement and increased attention of individuals who are aware that they will participate in a study [39], which can distort the results of an experiment. In the current study, to reduce the negative impact of this effect, participants were informed that they would be part of a study; however, neither the group that did not use the Theory Game nor the group that applied the Theory Game were informed about who would be in the Control group or the Study group [92]. As a result, both groups showed the same level of enthusiasm and willingness to be part of the study, which helped avoid potential biases related to this effect.

4.3. Teaching Method

Game Theory, a branch of mathematical theory applied to decision-making in conflict situations, offers valuable tools for teaching. From an educational perspective, Game Theory seeks to achieve several objectives: (a) use heuristic techniques to solve complex problems, (b) encourage attitudes such as self-confidence, self-discipline, and perseverance in the search for solutions, and (c) develop observation and communication skills [93,94].

In the context of teaching construction scheduling, students were taught everyday situations associated with building processes. The objective was for students to be able to schedule a specific project, evaluate the best option considering different variables, define the activities, establish durations using Game Theory, and then create the PERT network to calculate the critical path of the project.

The PERT method was taught to students according to the following steps:

• Theoretical Presentation: The professor introduced Game Theory, including its elements, types of games, and equilibria, using support material and interactive examples.
• Teaching the PERT Method: Students received information about the PERT method’s definition and methodology. Then, practical exercises were carried out to build the PERT network, calculating the estimated times using the traditional method and Game Theory to finally identify each problem’s critical path [79].
• Practical Training: The students were given two real construction scheduling problems. They solved these problems by applying the mathematical tools learned in class. Doubts were resolved, and aspects regarding the construction of the PERT network and the decisions to be made concerning the Game Theory taught were clarified, which allowed for more effective training [95].

4.4. Assessment of the Teaching Process

Once the class sessions were over, the post-test was administered individually to the participants. The questionnaire consisted of 17 questions, scored from 1 to 7, where 1 was disagreement, 7 was agreement, and there were three open questions. In addition, the 33 students were asked to complete an opinion survey, where different aspects of the class were evaluated, such as the teaching method used, the difficulty of the instrument applied, and the quality of the classes by the teacher. This survey sought to measure the students’ perceptions to obtain a qualitative assessment of the teaching methodology.

4.4.1. Quantitative Assessment

The pre-test and post-test results were compared to assess the feasibility of incorporating Game Theory into teaching the PERT method. Both tests contained 26 questions focused on the PERT method for construction scheduling. This comparison allowed the impact of the new methodology on student performance to be measured quantitatively [96].

4.4.2. Qualitative Assessment

From a qualitative perspective, the aim was to discover students’ opinions about the new teaching methodology and their perception of the teaching-learning process. To this end, an open-ended questionnaire was applied to collect comments and opinions from students, providing a more complete view of the methodology’s impact [97,98].

5. Analysis of Results

The results of the assessments were processed statistically, where their central tendency and dispersion measures represented the numerical variables. In contrast, the categorical variables were represented by the frequency and percentage of each of their classes. To determine if there were changes in the test results, both by dimension and by the total results, the t-Student test was applied for paired samples, and the Wilcoxon test was applied in cases where normality was not proved through the Shapiro–Wilk test. To determine if the proportion of correct answers increased in the second assessment, the McNemar test was applied since the same subjects took the test on the second occasion. A significance level of 0.05 was used; therefore, each time the p-value associated with a test was less than or equal to 0.05, it was considered statistically significant.

5.1. Qualitative Analysis

As previously mentioned, this research’s qualitative analysis was conducted to understand the students’ perceptions of the experience, the applied methodology, and the professor’s performance. To mitigate potential biases, anonymity was ensured in the feedback collection process, and a diverse set of open-ended and scaled questions was employed. This approach reduced the likelihood of socially desirable responses and gave more authentic insights into the methodology’s practicality and effectiveness.

The study used a 19-question questionnaire, with 33 individuals asked to rate each question with a score from 1 to 7. Then, they had to answer three open questions related to the presentation of the content, the advantages and disadvantages of the method, and its application in scheduling construction projects. These questions measured the students’ perceptions and strengthened the quantitative analysis carried out in a complementary way.

The results obtained for each question are presented in

Table 5. Summary statistics for perception survey questions.

Question	Media	S.D.	Min	Max
Q1	6.24	0.87	4	7
Q2	6.55	0.67	5	7
Q3	6.61	0.61	5	7
Q4	6.67	0.60	5	7
Q5	6.52	0.76	4	7
Q6	6.48	0.67	5	7
Q7	6.45	0.71	5	7
Q8	6.48	0.67	5	7
Q9	6.27	0.88	4	7
Q10	6.58	0.71	4	7
Q11	6.33	0.82	4	7
Q12	6.55	0.71	5	7
Q13	6.42	0.87	4	7
Q14	6.64	0.60	5	7
Q15	6.48	0.71	5	7
Q16	6.33	0.89	4	7
Q17	6.73	0.67	4	7

. The information indicates that no individual rated questions with a score lower than 4 (the minimum threshold considered). Furthermore, the average for each question was higher than 6.24 out of 7, obtained by question 1 (I found the class interesting). In contrast, question 17 (The practical exercises were an excellent complement to the theory) received the highest average score within the questionnaire (6.73).

The open questions contained in the qualitative questionnaire (applied after the last assessment) were as follows (along with a summary of the students’ answers per question):

• Comment on the conceptual clarity and exposition of the contents.

The students expressed clarity regarding the content transmitted, the professor’s mastery of the subject, and the appropriate and rapid understanding of the method implemented. The majority of the opinions reflected a positive evaluation by the students regarding the class presentation, the practical lectures being well received, and having internalized the content in a more didactic way. Most of the opinions agreed that the proposed teaching method (Game Theory), due to its novelty and easy understanding, was crucial for achieving a greater understanding of the PERT scheduling method.

• Comment on applying Game Theory to learning the PERT method (advantages or disadvantages).

According to the responses given by the students, the Game Theory tool was consistent with the bibliographic review carried out for the development of this research. They also believed that Game Theory was a mathematical tool with great potential for decision-making involving time and costs. The advantages most repeated in the students’ responses were that Game Theory helped to better understand decision-making, the simplicity of the method, and the clarity of its use. The disadvantages stated were that decisions in a real-world environment, with more decision variables involved, could make the method more difficult to use.

• Do you consider the equilibrium studied in Game Theory and its application to the PERT method practical in decision-making when scheduling a construction project?

At this point, the responses were unanimous, as the entire universe surveyed leaned towards the idea that the equilibrium studied in Game Theory and its application to the PERT method helped make decisions in the scheduling of construction projects. The students also complemented their responses by indicating that the technique learned was a helpful tool for resolving conflicts when making decisions and better understanding the relationship between costs and benefits.

In summary, the students highlighted three primary attributes of the methodology: (1) its ability to improve strategic decision-making skills, (2) the enhanced comprehension of scheduling concepts through practical application, and (3) its effectiveness in simulating real-world scenarios, which students found highly relevant to their professional development.

5.2. Quantitative Analysis

5.2.1. Preliminary Analysis

To conduct the quantitative analysis, 26 closed questions focused on the PERT method for construction scheduling were asked of the students in a pre- and post-test. To classify the areas of knowledge to be assessed, the 26 questions were grouped into four categories, each fundamental for understanding the PERT method. The first category, “Identification of activities”, grouped 6 questions and sought to measure the students’ ability to identify activities in a construction project. The second category, “Duration of activities”, also included 6 questions to assess the student’s ability to determine the duration of activities in a construction project. The third category, called “Critical Path”, included 7 questions, and its objective was to assess whether the students could determine the critical path of a project. Finally, the fourth category, “Variability and Decision”, which also consisted of 7 questions, sought to measure students’ understanding of the concepts associated with variability (typical of the PERT method) and decision-making (typical of Game Theory).

From the results obtained and shown in

Table 6. Shapiro–Wilk test for each category and the total of responses.

ID	Category	Questions	Correct Answers (%)		Difference	p-Value
			Pre-Test	Post-Test
I	Identification of activities	1, 2, 3, 4, 7, 14	72.7	80.8	8.1	0.0032
II	Duration of activities	5, 15, 18, 19, 20, 22	52.0	68.2	16.2	0.0102
III	Critical Path	9, 10, 11, 12, 13, 17, 21	70.1	86.6	16.5	0.2100 *
IV	Variability and Decision	6, 8, 16, 23, 24, 25, 26	60.2	81.0	20.8	0.0737 *
	Total	1–26	63.9	79.5	15.6	0.4833 *

*: normally distributed.

, it is possible to notice that the correct answers by the students in the post-test were percentage-wise higher than the correct answers obtained by the students in the pre-test in each category. However, to evaluate whether these differences were statistically significant, it was necessary to perform other statistical tests, so the normality of the data was previously verified. To do so, Table 6 shows the results of the Shapiro–Wilk Normality test, where the “Critical Path” and “Variability and Decision” dimensions, together with the total number of responses, were normally distributed (p > 0.05), in which case the t-Student test for paired samples was applied. The other dimensions, not passing the normality test, were analyzed using the Wilcoxon test.

Table 6 shows that all categories had a percentage difference greater than 8% in favor of the post-assessment, where category I “Identification of activities” had the minor difference (8.1%), while category IV had the most significant percentage difference between the pre- and post-assessment (20.8%). The minor difference in category I could be explained because “Identification of activities” is a task that is transversal to any project scheduling method, so it is unnecessary to have used Game Theory to improve. It was different for the category “Variability and Decision”, which, with a difference of more than 20% improvement between the pre- and post-test results, showed that those subjects more closely linked to probabilistic methods (such as PERT) could benefit more from topics associated with decision-making, which was widely addressed through Game Theory during the class. Overall, the total difference was more significant than 15% between the pre- and post-test at the end of the learning process of the PERT method using Game Theory.

5.2.2. Quantitative Analysis Using the Wilcoxon Test and t-Student Test

Figure 3 shows the averages of correctly answered responses for each category. In all categories, a higher percentage of correct responses was obtained in the post-test, confirmed in the analyses below.

Table 7. Differences between pre- and post-test.

ID	Category	Pre-Test		Post-Test		% Improvement	p-Value
		Media (S.D.)	Median (Q1–Q3)	Media (S.D.)	Median (Q1–Q3)
I	Identification of activities	4.36 (0.99)	5.0 (4.0–5.0)	4.84 (0.83)	5.0 (4.0–5.0)	8.1%	0.007 W
II	Duration of activities	3.12 (1.24)	3.0 (2.0–4.0)	4.09 (0.84)	4.0 (4.0–4.0)	16.2%	0.001 W
III	Critical Path	4.91 (1.13)	5.0 (4.0–6.0)	6.06 (0.79)	6.0 (6.0–7.0)	16.5%	0.001 T
IV	Variability and Decision	4.21 (0.99)	4.0 (3.0–5.0)	5.67 (0.99)	6.0 (5.0–6.0)	20.8%	0.001 T
	Total	16.61 (2.74)	17 (15–18)	20.67 (1.83)	21 (20.0–22.0)	15.6%	0.001 T

^W: Wilcoxon test; ^T: t-Student test.

shows the results of the t-Student test for paired samples for the dimensions “Critical Path” and “Variability and Decision” and for the total results, where statistically significant differences were found (p < 0.05) in favor of the post-test group. For the dimensions “Identification of Activities” and “Duration of Activities”, the Wilcoxon test was performed, where statistically significant differences were also found in both dimensions (p < 0.05) in favor of the post-test group.

In other words, the analysis by categories obtained statistically significant differences in the results for each category. According to the above results, the methodology proposed significantly changed the students’ post-test scores for the total number of responses and each category. This finding shows the benefits of using Game Theory to teach the PERT method.

5.2.3. Quantitative Analysis Using the McNemar Test

The analysis was performed question-by-question using the McNemar test to determine if the proportion of individuals who answered the questions correctly was similar before and after implementing Game Theory as a didactical technique for teaching the PERT method for scheduling projects. Analyzing the percentage differences in each question, it was observed that in 20 of 26 questions, there were proportional differences in favor of the post-test. Additionally, in 2 of the 6 remaining questions, the same number of students answered correctly.

Table 8. Questions with statistically significant differences according to the McNemar test.

Question	Category	Correct Answers (%)		Difference	p-Value
		Pre-Test	Post-Test
Q7	Identification of activities	79%	97%	18%	0.031
Q8	Variability and Decision	9%	64%	55%	0.001
Q12	Critical Path	33%	85%	52%	0.001
Q13	Critical Path	70%	94%	24%	0.008
Q15	Duration of activities	52%	97%	45%	0.000
Q16	Variability and Decision	33%	79%	45%	0.001
Q18	Duration of activities	61%	97%	36%	0.002
Q20	Duration of activities	12%	33%	21%	0.039
Q21	Critical Path	70%	91%	21%	0.039
Q26	Variability and Decision	52%	82%	30%	0.031

shows only those questions with statistically significant differences (p < 0.05). It should be noted that these questions had percentage variations in favor of the intervention group (post-test); that is, they achieved better performance after being taught Game Theory to learn the PERT scheduling technique, with question 7 having the lowest variation (18%) and question 8 reaching a higher percentage variation (55%).

Figure 4 shows a portion of those students who faced the pre- and post-test and improved their performance, reaching a difference between the percentages of correct questions of 55%. On the other hand, when comparing the total average scores for the 26 questions of the pre- and post-test groups, a percentage difference of 4.1% is evident in favor of the post-assessment, where in the pre-test, the students answered correctly, on average, 16.6 questions, and in the post-test, 20.7.

5.3. Discussion of Results

The main findings from the qualitative and quantitative analyses carried out in this research are summarized below.

From a qualitative point of view, the evaluation to assess the students’ perceptions of the experience showed that they highlighted the clarity of the content taught, the professor’s preparation, and the rapid understanding of the subjects covered. They also mentioned that Game Theory was a powerful tool for decision-making, highlighting its ease of use. However, they considered that its application to more complex problems present in real-world contexts could make its application difficult. Finally, when asked about the use of Game Theory to learn a scheduling method such as PERT, the answers were unanimous in that it was a handy teaching tool, significantly improving their understanding of decision-making, a cornerstone of Game Theory, but also very valuable to address the problem of uncertainty and variability present in construction projects and which PERT addresses.

Based on the quantitative analysis performed, the results obtained showed that in the four categories evaluated (“Identification of Activities”, “Duration of Activities”, “Critical Path”, and “Variability and Decision”), there were statistically significant differences between the pre- and post-test, confirming an evolution in the understanding of the contents after having used Game Theory to learn PERT. An interesting result is that although the differences between the pre- and post-test were significant in all categories, the smallest difference was in “Identification of Activities”, which could be related to the fact that this skill is required by students in any other scheduling method (i.e., not only in the PERT method). As such, students acquire it previously in the same Construction Planning and Scheduling course and related courses in their study plan by studying other scheduling techniques, such as Gantt Chart, CPM (Critical Path Method), and others. On the other hand, the most significant differences in the results were found in the “Variability and Decision” category, which corresponded precisely to all those questions where Game Theory and PERT play a much more critical role. Similarly, the detailed question-by-question analysis corroborated this finding, where the questions that showed the most significant differences were also those related to the “Variability and Decision” category.

5.4. Implications for Educators

This research introduces new insights from a methodological perspective by providing a practical approach that applies Game Theory to teach a scheduling technique commonly used in engineering (PERT or Program Evaluation and Review Technique). Even though the research specifically focuses on civil engineering students, professors from other disciplines can modify the methodology presented here for their subjects. For instance, instead of using Game Theory to teach PERT, they might teach other scheduling techniques, such as CPM (Critical Path Method), Gantt charts, or Lines of Balance. The methodology can also be applied to various engineering courses involving decision-making (e.g., Structural Analysis, Computer Simulation, among many others). In other words, educators can retain the didactical tool of Game Theory while adapting the specific engineering topic to be taught, with special emphasis on those courses where decision-making plays a critical role.

Regarding practical contributions, this study may serve as a valuable resource for engineering professors by (a) providing an introductory understanding of Game Theory and the elements of scheduling courses (e.g., PERT, CPM, Gantt, etc.) necessary for implementing challenging teaching methods that engage students; (b) facilitating the blending of Game Theory with PERT (or other courses where decision-making plays a role) in engineering education; and (c) including minor adjustments to the proposed methodology to accommodate students from different engineering disciplines or academic settings.

In terms of how to scale the proposed approach for larger class sizes, some authors have raised recommendations. To scale a way of teaching without significantly increasing the course-related resources, Petrovic and Pale [99] suggest incorporating (a) an audience response system to make more effortless interactivity during live lectures and (b) peer review for the scalability of evaluations and to relieve professor workloads. Similarly, to improve the scalability of engineering courses, Schefer-Wenzl and Miladinovic [100] propose reducing the load on the main lectures by outsourcing much of the non-scalable tasks to teaching assistants. In this sense, all these recommendations may be incorporated into the present study if the proposed methodology is scaled to larger class sizes.

In the end, this study offers a methodology to complement other teaching techniques, particularly for engineering courses requiring students to face decision-making processes. Utilizing Game Theory to teach a scheduling method like PERT may lead to a more engaging teaching environment compared to traditional engineering teaching techniques, ultimately enhancing academic performance by offering a teaching tool that manages engineering problems related to decision-making.

6. Conclusions

In the context of the Construction Planning and Scheduling course, part of the civil engineering academic program, the present research showed that Game Theory proved to be an appropriate didactical tool for teaching the PERT project scheduling technique. This methodology was explicitly applied to a group of civil engineering students, whose effectiveness was evaluated through pre- and post-written tests composed of closed questions that were statistically analyzed. The results revealed significant improvements in the understanding and analysis of the project scheduling concepts and the main components and use of the PERT method.

The application of Game Theory not only improved students’ understanding of the PERT method but also allowed them to obtain higher scores in subsequent assessments. Such progress was evidenced through multiple statistical analyses, highlighting an increase in the students’ ability to apply these concepts practically when supported by graphical and decision-making tools, both characteristics of Game Theory.

Additionally, the study emphasizes that Game Theory offers a challenging pedagogical approach that may help bridge the gap between theoretical knowledge and practical application. The potential to enhance decision-making skills, critical thinking, and student engagement positions it as an innovative tool in engineering education. This research sets the foundation for how Game Theory may contribute to reshaping learning outcomes across engineering disciplines.

Thus, the findings underscore the potential of Game Theory to transform traditional engineering education by promoting active learning environments. It facilitates the development of critical skills such as problem-solving and strategic decision-making, which are essential in professional practice.

On the other hand, this research offers a replicable framework for widely integrating Game Theory into engineering education. Educators may adopt this methodology to enhance their academic engineering program by emphasizing interactive and strategic decision-making problems. Moreover, this framework is adaptable to other engineering aspects, including resource management and risk analysis, enabling broader application across disciplines. Consequently, educators can further enhance their students’ engagement and learning outcomes by incorporating advanced gamified tools based on decision-making approaches.

Regarding limitations, although evaluating the same group before and after the intervention may not provide a direct comparison with traditional teaching methods, the present study has shown that implementing Game Theory is a dynamic and challenging teaching tool that could be extended to other areas beyond a single academic context (i.e., beyond the civil engineering curriculum). In this sense, future research should include control groups that utilize traditional methodologies compared with Game Theory’s applications in the classroom, not only in civil engineering programs but also in other undergraduate and graduate engineering courses.