This section provides a comprehensive overview of MCDM, game theory, and integrated MCDM and game theory, its proposed methodology, and an illustrative example. Table 1 contains the notation used in the study.

Table 1 Notations Used in the StudyMCDM for sentiment analysis

Multiple Criteria Decision Making (MCDM) is a field within decision science dedicated to addressing decision problems marked by numerous, often conflicting criteria or objectives. Its primary aim is to provide decision-makers with structured methodologies for evaluating and selecting the most suitable alternative from a pool of options that vary across multiple criteria. MCDM involves considering multiple criteria or objectives in the decision-making process, ranging from quantitative metrics to qualitative factors. MCDM offers a structured approach to sentiment analysis by treating sentiment classification as a decision-making process. In this context, sentiment categories such as positive, negative, or neutral are viewed as alternatives, while various factors or features used to evaluate sentiment are treated as criteria. To apply MCDM to sentiment analysis, a decision matrix is constructed, with rows representing sentiment alternatives and columns representing evaluation criteria. We begin by creating a decision matrix X with dimensions m×n, where m represents the number of sentiment alternatives (e.g., positive, negative, neutral) and n represents the number of evaluation criteria (e.g., context score, emotion score, wordcount features). Each row of the matrix corresponds to a sentiment alternative, and each column corresponds to an evaluation criterion. So X looks like this.

\(X=\left[ } }}&}}&&}} \\ }}&}}&&}} \\ \vdots & \vdots & \ddots & \vdots \\ }}&}}&&}} \end} \right]\)

Where \(}\) represents the evaluation score of sentiment alternative i on criterion j. Weights are then assigned to each criterion to reflect their relative importance in sentiment classification. These weights, usually determined through domain knowledge or expert judgment, ensure that more significant criteria carry greater influence in the classification process. Using the decision matrix and assigned weights, performance scores for each alternative are calculated. We represent the weight vector as \(W=(,,....,)\), where \(\)is the weight assigned to criterion j. The performance score for an alternative is the weighted sum of scores across all criteria, providing a numerical representation of the sentiment expressed in the text. These weights ensure that more significant criteria carry greater influence in the sentiment classification process. With the decision matrix X and weight vector W in place, the performance score Si for each sentiment alternative i is calculated using matrix multiplication.

Where S is a vector representing the sentiment scores for all sentiment alternatives, and each element \(\) represents the score for alternative i, denoted by Eq. 1. Mathematically, this operation represents the weighted sum of the scores across all criteria for each alternative. In this way, we evaluate the performance score of each alternative.

Game Theory in Sentiment Analysis

Game theory is a field within mathematics and economics that studies strategic interactions among rational decision-makers. It offers a framework to analyze how individuals or entities make choices when their decisions influence each other’s outcomes. The formalization and extensive development of game theory were carried out by mathematician John von Neumann and economist Oskar Morgenstern in their landmark 1944 publication, “Theory of Games and Economic Behaviour.” However, the roots of game theory extend to earlier contributions by mathematicians such as Émile Borel and John Nash. Notably, John Nash made substantial contributions to the field with his concept of the Nash equilibrium, a condition in which no player has an incentive to change their strategy unilaterally. We explore different components of game theory in this section and how they can be applied to sentiment analysis. Implementing game theory in sentiment classification involves adapting its theoretical concepts to the practical aspects of designing sentiment analysis models. Here’s a theoretical explanation of how game theory can be implemented in sentiment classification.

1) Player Representation

In sentiment classification, players represent text or reviews. Each player aims to maximize their performance in accurately classifying sentiments. Let P represent the set of players involved in sentiment classification. Each player corresponds to a sentiment analysis classifier.

2) Dominant Strategy

Strategies in game theory correspond to the actions or decisions players can take. In sentiment classification, strategies are the approaches classifiers use to analyze and classify text sentiment as positive, negative, or neutral. We denote S as the strategy set available to the player, where each strategy s represents a specific approach used by player P to analyze and classify text sentiment, such as employing different feature sets, algorithms, or pre-processing techniques. A dominant strategy yields the highest payoff for a player regardless of other players’ strategies.

3) Payoff Determination

Payoffs in sentiment classification represent the performance achieved by players based on their chosen strategies, incorporating factors like computational efficiency, performance score of Multi-criteria Decision Making (MCDM), or model interpretability. Let ui represent the payoff function for player i when selecting strategy si, considering the strategies chosen by other players, represented by s− i​. The payoff could be defined as a combination of factors, such as the MCDM performance score.

4) Game Formulation

The game is formulated as a strategic interaction where each player selects a strategy to classify sentiment based on text features like context score, emotion score, and word count score. A payoff matrix is constructed based on each player’s performance (Ri) given the combination of strategies chosen by all players (Ri and Rj). This matrix captures the relationship between the chosen strategies and their corresponding performance scores (Pi).

5) Nash Equilibrium

Nash equilibrium represents a stable state in the game where no player has an incentive to deviate from their chosen strategy unilaterally. Identifying Nash equilibria helps understand the game’s potential outcomes and the strategies players are likely to converge on. Strategies at the Nash equilibrium may not necessarily yield the optimal solution, but they represent a state where no player can improve its performance by changing its strategy alone.

$$(,s_}^) \geqslant (s_^,s_}^)$$

(2)

The Nash equilibrium implies the players’ dominant strategies, and the players’ dominant strategies imply each player’s best response. The inequality Eq. (1) states that the payoff obtained by player i from sticking to its current strategy si* in the Nash equilibrium is greater than or equal to the payoff it would receive from any alternative strategy si while others are still playing optimally. This condition ensures that player i has no incentive to unilaterally change its strategy, since doing so would not yield a higher payoff. Therefore, si* is considered a Nash equilibrium strategy for player i.

Combining MCDM and Game Theory Frameworks for Sentiment Analysis

Sentiment classification in real reviews is seldom influenced by a single factor; rather, it arises from heterogeneous cues, including contextual polarity, emotional intensity, and lexical evidence, which often produce conflicting signals (e.g., mixed opinions, polarity shifts, or sarcastic contrasts). The EDGT-ST framework addresses this multi-evidence scenario by conceptualizing sentiment tagging as a Multi-Criteria Decision Making (MCDM) problem, wherein each sentiment class serves as an alternative and each extracted feature functions as a decision criterion, with the associated scores indicating the performance of each alternative relative to these criteria. The EDAS-based ranking system robustly aggregates these cues by comparing alternatives against an average solution, thereby mitigating bias from uncertainty or disproportionate feature dominance. Moreover, while many sentiment cues might competitively affect the ultimate decision—where prioritizing one signal (e.g., mood) may diminish another (e.g., context)—we incorporate a non-cooperative game-theoretic model to elucidate their strategic interaction. In this framework, sentiment alternatives are regarded as players, criteria serve as strategies, and scores from EDAS constitute the payoffs, facilitating equilibrium-based reasoning to yield a stable final sentiment tag. Consequently, integrating MCDM with game theory enhances interpretability and resilience while reflecting human-like decision-making in the assessment of conflicting or competing emotional evidence as shown in Fig. 2.

Fig. 2The alternative text for this image may have been generated using AI.

Mapping to components of MCDM and game theory in the sentiment tagging scenario

Step 1: MCDM Performance Evaluation.

Let N denote the set of classifiers participating in sentiment classification. For each review \( \in N\), MCDM techniques are applied to evaluate its performance having multiple criteria \((,.........,)\). The performance score \(\) of each review \(\) is calculated as an aggregated function of its performance on each criterion. \(=f(,.........,)\).

Step 2: Game Formulation.

The sentiment classification game is defined by a tuple \(G=\langle N,(),()\rangle\)Where N represents the set of reviews, Si denotes the strategy set for review \(\), consisting of possible sentiment classification strategies (positive, negative, neutral), \(\) representing the performance score of the player \(\) obtained from MCDM.

Step 3: Game theory Payoff matrix incorporating MCDM Scores.

The payoff matrix P captures the interactions among reviews and their respective performance scores. Here \(}\) represents the payoff obtained by the reviewer \(\)when it chooses strategy \(\) and reviewers \(\) chooses strategy \(\). Here \(}\) is the performance score obtained from Eq. 3.

$$P=\left[ } }}&}}&&}} \\ }}&}}&&}} \\ \vdots & \vdots & \ddots & \vdots \\ }}&}}&&}} \end} \right]$$

(3)

Step 4: Strategic Decision Making.

For each player \(\)select a strategy \( \in S_\) to maximize its payoff, considering the strategies chosen by other reviewers and their associated performance scores. The strategic decision-making process aims to identify the best response strategy for each classifier given the strategies chosen by other player and their associated performance scores.

Step 5: Nash Equilibrium.

Nash equilibrium is a set of strategy profiles \((s_^,s_^,....,s_^)\)where no player has an incentive to unilaterally deviate from its chosen strategy. Formally, a strategy profile \((s_^,s_^,....,s_^)\) is a Nash equilibrium for every player \(\), using Eq. ??⁢?⁢?⁢?=mode(⁢?1,?2,...??) 5. We use this concept of nash equilibrium as an asset to generate the sentiment tag for the text. The strategies corresponding to the nash equilibrium in the normal form representation is the ultimate sentiment tag to the text. By combining MCDM with game theory in this manner, we can systematically evaluate players’ performance and model strategic interactions among players, leading to the development of more effective sentiment analysis systems. In our setting, a Nash equilibrium represents a stable, unbiased and mathematically justified sentiment tag where no competing sentiment cue (context, emotion, or word-count) can independently change its strategy to obtain a better outcome, ensuring a consistent final decision.

Step 6: Iterative Process.

The game can be played iteratively; in each interaction, a two-person game is played between players. Each player will use three strategies, and during play, we reach a stable point after which the player cannot improve their performance. That point is called the nash equilibrium of the game. In each interaction, we get at least one nash equilibrium. So if there are (N) players, then there will be (N-1) interactions between players. So for each iteration between two players, there can be more than (N-1) nash equilibrium. This implies we get more than (N-1) sentiment tags for each player. Then we consider that sentiment tag, which occurs the most times in the nash equation of the player. In this way, in the end, we generate the sentiment tag of the text.

Let N be the number of players. For each pair of players (i, j) where \(i \ne j\), let Eij represent the set of nash equilibria resulting from their interaction. Each equilibrium in Eij corresponds to a particular sentiment tag. For each player i, we determine the sentiment tag Ti that appears most frequently across all Eij given by Eq. (4).

$$=}sentiment}tags}in}}}$$

(4)

This means Ti is the sentiment tag that occurs the most frequently in the set of equilibria for player I interacting with all other players j. The overall sentiment tag T for the review is then determined by finding the mode of the sentiment tags \(,,...,\) given by Eq. ( 5).

This formula encapsulates the process of determining the sentiment tag for each player and ultimately for the entire text based on nash equilibria outcomes in the interactions between players.

Rationale of the EDAS MCDM

In the proposed framework, we adopt the Evaluation based on Distance from Average Solution (EDAS) as the multi-criteria decision-making core instead of other popular methods such as TOPSIS or VIKOR. EDAS assigns each alternative separate positive and negative distances from the average value of each criterion, and then aggregates these deviations into a single appraisal score. Formally, for a criterion.

\(PDA_=\left\\frac-Av_j\right)}=\lambda\rightarrow FC\\\frac\right)}=\rho=0\rightarrow NFC\end\right.\), \(ND}=\left\} (0,(A - })}}}}=\psi \to FC} \\ (0,(} - A)}}}}=\zeta =0 \to NFC} \end} \right.\)

This distance-from-average formulation makes the final ranking less sensitive to extreme values than approaches that rely on distances to ideal and anti-ideal points, and it is particularly suitable when criteria (context, emotion, and word-count scores) may be noisy, partially conflicting, or very close in magnitude. Within EDGT-ST, EDAS therefore provides more stable and interpretable sentiment appraisals, which is consistent with the empirical findings from our ablation study, where the EDAS-based variant outperforms TOPSIS, VIKOR, MOORA, COPRAS, and PROMETHEE both in effectiveness and efficiency. In this work we adopt EDAS because it evaluates each sentiment alternative via separate positive and negative deviations from the average solution, rather than distances to ideal and anti-ideal points, which makes the aggregated appraisal score less sensitive to outliers and more stable when context, emotion, and word-count criteria fluctuate or have closely competing values. This distance-from-average formulation is well-suited to noisy sentiment data, where different cues may partially contradict each other across reviews.

In EDGT-ST, the three sentiment criteria are normalized and treated as equally informative in the absence of strong domain priors. Therefore, a uniform weight vector provides a conservative choice that avoids privileging any single cue, while EDAS itself handles bias arising from noisy or conflicting criterion values in the ranking step.

Proposed Methodology

We propose an “Evaluation based on Distance from Average Solution and Game theory-based Sentiment Tagger” (EDGT-ST) model in this work. In this section, we discuss the EDGT-ST framework architecture, which comprises three phases, and later we provide a numerical illustrative example based on this model.

EDGT-ST Framework Architecture

This section outlines a three-phase process within the EDGT-ST framework that incorporates five distinct algorithms to perform various tasks across its architecture. In Phase 1, relevant features are extracted to represent the data accurately. Phase 2 involves applying the EDAS MCDM technique to integrate the extracted feature scores, generating an appraisal score that consolidates information from the different features. In Phase 3, this appraisal score serves as a payoff in a game-theoretic model with two players, facilitating decision-making within the game framework. The EDGT-ST framework is structured into five algorithms, each designed to handle specific tasks and contribute to the architecture’s overall functionality, as illustrated in Fig. 3.

Fig. 3The alternative text for this image may have been generated using AI.

EDGT-ST framework architecture

Phase 1: Feature Extraction.

In Phase 1, we extract scores from three features (context, emotion, and word count) using algorithms 1, 2, and 3. We extract positive, negative, and neutral scores of each feature as shown in (i), (ii), and (iii) and illustrated in Fig. 4.

Fig. 4The alternative text for this image may have been generated using AI.

Phase 1 of the EDGT-ST framework

(i)

Context Score Computation using VADER \((PC)\), \((NC)\)and \((OC)\): In this work, the context score of each review is computed using the VADER sentiment analyzer because it is well-suited for real-world user-generated text, such as customer reviews and social media content. Unlike traditional lexicon resources, VADER effectively captures context-dependent sentiment cues, including negation handling (e.g., “not good”), intensity modifiers (e.g., “very good”), punctuation and capitalization emphasis (e.g., “great!!!”, “AWFUL”), and informal expressions. Therefore, VADER provides more robust and reliable positive, negative, and neutral context scores, which are further used as input features in the proposed EDGT-ST sentiment tagging framework. The resulting context scores exhibit values that reside within the closed interval [0, 1]. Let us define \((PC)\) the positive polarity, \((NC)\) as the negative polarity and neutral polarity \((OC)\)of the context. By employing VADER, the specific values of \((PC)\), \((NC)\)and \((OC)\) are determined, enabling a comprehensive evaluation of the positive and negative sentiment expressed within the text.

(ii)

Evaluate the Emotion score of reviews ( \((PE)\),\((NE)\),\((OE)\)).

To compute the emotional content of each review, we use the text2emotion Python library via the GetEmotionScores() function. It generates emotion scores for happy (H), angry (A), sad (S), surprised (Sp), and fear (F); however, fear is excluded in this work. The extracted emotions are then mapped into three sentiment-based groups: positive, negative, and neutral using Algorithm 1. Finally, the emotion scores \((PE)\),\((NE)\),\((OE)\)are obtained in the range [0, 1], enabling consistent comparison across reviews. In the earlier formulation, the emotion category surprise was grouped with positive emotions. However, surprise can express either positive or negative sentiment depending on the contextual usage. To address this, the proposed framework evaluates surprise expressions contextually by examining adjacent sentiment-bearing words within the sentence. If surrounding words indicate positive sentiment (e.g., “pleasantly surprised”), the surprise score contributes positively to the emotion score; if negative sentiment cues are detected (e.g., “surprised by how bad it was”), the contribution is treated as negative. This contextual weighting allows the emotion score computation to better reflect the semantic polarity of surprise expressions.

Algorithm 1The alternative text for this image may have been generated using AI.

The emotion extractor produces fine-grained emotion categories (e.g., happy, angry, sad, fear, surprise). However, the objective of EDGT-ST is to generate the final sentiment label in three polarity classes: positive, negative, and neutral. Hence, we map emotions into three polarity groups to ensure alignment with standard sentiment benchmarks and to support consistent integration with the proposed MCDM-based aggregation and game-theoretic decision layer. This simplification also improves robustness to short, noisy reviews, where emotion signals may coexist or appear weak, and a direct one-to-one emotion classification may introduce instability. Therefore, emotions are grouped based on their dominant polarity contribution, enabling EDGT-ST to retain emotional influence while maintaining a stable 3-class sentiment decision.

iii) Evaluate Wordcount \((PW)\),\((NW)\)and \((OW)\).

Algorithm 2 dynamically calculates the sentiment word-count of a text by expanding a standard static lexicon to include domain-specific slang and informal terms. First, it identifies candidate slang words by finding the nearest embedding neighbours of strongly polar seed words from the base lexicon. To ensure accuracy, we instantiate MM as the open-source Mistral-7B-Instruct model, which acts as a polarity validator: for each candidate slang term, it predicts a sentiment label and confidence, and the term is added to the expanded lexicon only if this label agrees with the embedding-based polarity and the confidence exceeds a high threshold. Finally, the algorithm evaluates the tokenized review using this expanded lexicon—flipping the polarity of any word preceded by a negation—and normalizes the aggregated positive, negative, and neutral counts to produce the final output vector. Because large language models such as Mistral-7B-Instruct may encode social and linguistic biases, we combine agreement and high-confidence filtering with manual inspection of a random subset of added terms to reduce the risk of introducing harmful or skewed vocabulary into the expanded lexicon.

Algorithm 2The alternative text for this image may have been generated using AI.

Phase 2: MCDM approach to integrate all three features.

We follow Algorithm 3 to calculate the appraisal score, which will serve as the payoff for the players in the game-theoretic setting. There are many Python libraries available for executing MCDM techniques. Algorithm 4 consists of two criteria. A favorable (FC) is a criterion prioritized to achieve the best possible outcome. The criteria are deemed non-favourable (NFC) when minimum values are desired. We looked at two factors, Emotion, context, and word count, and found them to be FC.Given that not a single criterion meets the definition of NFC, NFC must be equal to zero. Subsequently, we place weights on (W). Given that all criteria are weighted equally, W = 0.5 was selected. The EDAS (Evaluation based on Distance from Average Solution) Technique involves calculating an appraisement score used as a payoff in game theory for sentiment analysis. The algorithm takes positive, negative, and neutral scores of context \((PC)\), \((NC)\), \((OC)\)and Emotion \((PE)\),\((NE)\),\((OE)\)scores, and word count scores \((PW)\),\((NW)\)and \((OW)\) as inputs and outputs the appraisement score (ASi) of each review. Initially, a decision matrix is constructed with three alternatives (positive, negative, neutral) and three criteria (context, emotion, and word count). The matrix D contains scores \((})\)​ representing the three alternatives for the three criteria. Next, the average value for each criterion (\(A\)) is calculated as the mean of all alternatives’ scores for that criterion. The Positive Distance from Average (\(PD}\)) and Negative Distance from Average (\(ND}\)) are then computed. The algorithm then calculates the weighted sum for (\(PD}\)) i.e. (\(S\)) and (\(ND}\)) i.e., (\(S\)), where wj is the weight for three criteria, set equally at 0.5 for all criteria. These sums are normalized by dividing by their respective maximum values to obtain \(NS\) and \(NS\). Finally, the appraisement score (ASi) for each alternative. The resulting appraisement scores are used to rank the alternatives, with the highest (ASi) indicating the best alternative. This score serves as a payoff in a non-cooperative game theory framework for sentiment analysis, ensuring that context, emotion, and word count scores are fairly weighted and normalized for comparison as illustrated in Fig. 5.

Fig. 5The alternative text for this image may have been generated using AI.

Phase 2 of the EDGT-ST framework

Algorithm 3The alternative text for this image may have been generated using AI.

Appraisement Score calculation

For each review, the EDAS procedure produces three appraisement scores \(A},A},A}\)which are subsequently used to derive a unique sentiment decision. To avoid unstable tags when appraisal values are equal or very close, we implement a deterministic tie-handling scheme. First, the class with the highest \(A\) is selected; if two or more classes differ by less than a small margin ε (set to 0.01 in our experiments), we prioritize the alternative with the higher context score, reflecting the primary role of overall sentence polarity in sentiment perception. If the corresponding context scores are also within ε, we next compare emotion scores and, if needed, word-count scores as secondary and tertiary criteria. This hierarchical tie-breaking rule yields a reproducible mapping from EDAS appraisement scores to sentiment classes and ensures that the payoffs passed to the subsequent game-theoretic layer induce a well-defined ranking of sentiment alternatives, even under ties or near-ties in the MCDM stage.

Phase 3: Non-Cooperative Game Model for Sentiment Tagging.

Algorithm 5 aims to determine the sentiment tag for reviews using a non-cooperative game model. It starts by taking as input the appraisal scores for two reviews, R1 and R2, which are the evaluated performance metrics for each review based on positive, negative, and neutral sentiments. Table 2 shows the three strategies and corresponding appraisement scores \(\,,\}\)of R1 and \(\,,\}\) R2 for positive, negative, and neutral, respectively. The output of the algorithm is the final sentiment classification for R1 and R2as illustrated in Fig. 6.

Fig. 6The alternative text for this image may have been generated using AI.

Phase 1 of the EDGT-ST framework

The first step is to generate a Normal Form Matrix for the players R1 and R2, as illustrated in Table 3. This involves constructing a matrix that represents each player’s payoffs for each strategy: positive (P), negative (N), or neutral (O). The notation used includes R1 and R2 for the reviews (players in the game), and P, N, and O for the positive, negative, and neutral strategies, respectively. The appraisement scores represent the numerical values given to each review for each sentiment category. Next, the algorithm computes the dominant strategies for R1 (DR1) and R2 (DR2). A dominant strategy is the best response for a player regardless of the other player’s choice. The dominant strategies for each player are identified using the notation DR1 for the dominant strategy of player R1 and DR2 for that of player R2. Following this, the algorithm finds the Nash Equilibrium (NE), which occurs when no player can benefit by unilaterally changing their strategy. The Nash Equilibrium is the intersection of the dominant strategies of both players, denoted as NE = DR1∩DR2​, indicating the equilibrium state. Finally, the algorithm determines the sentiment tags based on the Nash Equilibrium. The strategies corresponding to the Nash Equilibrium are the sentiment tags for the reviews. If multiple equilibria exist, the algorithm can select the equilibrium that best represents the sentiment using predefined criteria or additional analysis. The notations used include R1andR2 to represent the reviews, P, N, and O for the strategies (positive, negative, and neutral sentiments), appraisement scores for the performance values, DR1and DR2 for the dominant strategies, and NE for the Nash Equilibrium, which is the set of strategies where no player can gain by changing their strategy unilaterally.

Algorithm 4The alternative text for this image may have been generated using AI.

Determine sentiment tag for assessment

Table 2 Normal Form Representation of game showing payoffs corresponding to different strategies evaluated from the appraisement scores

Although Algorithm 4 is illustrated using pairwise interactions between reviews, this design does not restrict EDGT-ST to short or sentiment-homogeneous texts. For longer reviews containing multiple polarity shifts, conflicting cues are already encoded in the context, emotion, and word-count scores aggregated over the entire sequence, so each review enters the game with a payoff profile that reflects all local sentiment fluctuations. The iterative pairwise game then acts as a consensus mechanism over these global payoffs, where the dominant and equilibrium strategies capture the prevailing sentiment orientation rather than each individual clause. By operating on fixed 3 × 3 payoff structures independent of review length, the pairwise formulation preserves computational efficiency while still accommodating reviews with several internal sentiment reversals, which would otherwise make higher-order multi-player games intractable at scale.

We examine the Nash equilibrium to determine the tag for a review as shown in Table 2. To determine an evaluation’s sentimental category, we use Algorithm 5. The feasibility and effectiveness of the proposed method are analyzed based on the multi-criteria decision-making problem used. In this study, we enable the play of the non-cooperative game between two players. To play the game, payoffs are required, so we evaluate them using the performance score, calculated with the EDAS technique of MCDM.

Iteration Reduction Heuristic (Early Stopping)

Phase 3 applies iterative game-theoretic updates to obtain a stable equilibrium strategy for sentiment tagging. In the proposed framework, the pairwise game is applied locally at the review level as a fixed-size decision mechanism over sentiment alternatives, rather than as a global interaction among all reviews in the corpus. Therefore, to improve scalability, we incorporate an early stopping criterion. Let p(t) denote the strategy probability (equilibrium state) at iteration t. The iterative process is terminated when the convergence condition ∣p(t + 1) − p(t)∣<ϵ holds for K consecutive iterations, where ϵ =10− 3 and K = 3, or when the maximum iteration limit Tmax=50 is reached. This stopping rule reduces unnecessary computations while preserving the final stable sentiment decision obtained at equilibrium.

Scalability and Complexity Analysis

The scalability and computational complexity of EDGT-ST are designed to ensure efficiency without compromising interpretability. Although the game-theoretic component is conceptually illustrated through pairwise interactions, the framework does not construct a global joint game over the entire dataset. Instead, each review is processed independently, where the pairwise non-cooperative game functions as a fixed-size local decision module to derive the final sentiment label from the computed appraisal scores. For a review of length L, the feature extraction stage—including contextual, emotional, and word-count computations—requires a single pass over the tokens, resulting in a time complexity of O(L). The EDAS module operates on a fixed 3 × 3 decision matrix, and the game-theoretic tagging stage similarly relies on a fixed 3 × 3 payoff matrix. Consequently, both components introduce constant-time overhead per review, i.e., O(1). For a dataset comprising N reviews, the total runtime) O(∑i₌₁ᴺ Li) which simplifies to O(N) under the assumption of bounded average review length. This linear scalability enables EDGT-ST to efficiently handle large-scale review corpora while avoiding the combinatorial complexity associated with higher-order multi-player game formulations.

Numerical Illustration

We will analyze the four-star review. The context scores and rating scores of the written review comment are initially calculated in accordance with the procedures outlined in the preceding steps. The context, emotion, and word count scores of the review are presented in Table 3, which serves as a decision matrix. I then proceed to evaluate p1 and p2 according to algorithm 1.

R1 (2 Stars)

“Very nice monitor! Great vibrant color and clarity A very usable 16:10 aspect that works much better for applications than the common 16:9 that is great for movies, but not much else.

R2 (4 Stars):

“This Blu Ray player is defective. It won’t play Blu rays from universal studios or other companies like Sony. The WIFI connection is pretty much non-existent. I tried everything humanly possible in order to get the wireless connection on this Player to no avail. All of the wifi compatible electronic devices in my apartment connect easily except this Player. I will be purchasing a Samsung BD J7500 and will stay away from panasonic once and for all!!!”.

Initially, we compute the context score of English reviews utilizing the SWN presented in Table 3. Subsequently, we evaluate the text characteristics that yield the results presented in Table 3. Subsequently, we implement the EDAS method (Algorithm 4) on these evaluations. Table 4 presents the mean values of the three choices (positive, negative, and neutral) across three criteria (context, emotion, and word count). Table 4 assesses the positive and negative deviations from the mean, as well as PDA and NDA.

Table 3 Scores evaluated using phase 1 for R1Table 4 Scores evaluated using phase 1 for R2

Now we allow playing the non-cooperative game between the appraisal scores of two reviews (R1 and R2), as shown in Table 5. Table 6 shows the 3 strategies, positive, negative, and neutral, and each player’s payoff corresponding to the three strategies is carried out with appraisement scores of R1 and R2 evaluated in Table 5. As we have two players (R1 and R2) and three strategies (P, N, O), we enable them to play the non-cooperative game following Algorithm 5.

Table 5 Appraisement score of both reviews R1 and R2Table 6 Normal form representation of the non-cooperative game

Table 6 presents the payout results derived from the EDAS-based evaluation scores for the two participants in the EDGT-ST model. The payoffs are organized in a normal-form matrix, with each row indicating Player 1’s reward and the matching column entry reflecting Player 2’s payoff. Equation (3) is used to ascertain the prevailing strategy of both players. The intersection of dominating strategies yields the Nash Equilibrium, a stable strategic condition in which neither party can improve their payoff by unilaterally changing their strategy. Employing Definition 1 from Sect. 3.1, the Nash Equilibrium is calculated and used to deduce the emotion choice. The ultimate emotion tag is then derived using Algorithm 2. The Nash Equilibrium in this instance is (0.854, 0.791), with the corresponding strategies being (N, P), signifying that review R1 is classified as Neutral (N) and review R2 as Positive (P). Figure 7. elucidates the tag deduction procedure for R1 and R2, whilst Table 7 encapsulates the Nash Equilibria derived from the game-theoretic study in EDGT-ST as shown in Table 7.

Table 7 Nash equilibrium of the non-cooperative gameFig. 7The alternative text for this image may have been generated using AI.

Deduced tag of each review from the non-cooperative game model

Although Sect. 3.5 illustrates the procedure using two reviews for clarity, EDGT-ST is designed for large-scale sentiment tagging. The feature extraction steps (context, emotion, and word-count scoring) require a single pass over the review tokens, resulting in O(L) time per review, where L is the review length. Since the MCDM and game-theoretic computations operate on a fixed 3 × 3 structure, their overhead is constant per review, making the overall method efficient and scalable.