The problem of Rank Aggregation (RA) is the combination of multiple ordered lists of entities, as base rankers, to achieve a better ranking in the form of an aggregated ranker [1]. In one classification, RA can be divided into two categories [2]. In the first category, a score is assigned to each entity in each ordered list, and these scores are used for aggregating the lists, known as score-based aggregation. In the second category, only the rank of each entity in the ordered lists is used, known as order-based aggregation. RA has a rich history in information retrieval, marketing and advertising, sports competitions, applied psychology, social choice (political elections), and so on. In recent years, with the rapid development of technology, RA has faced new challenges in fields such as building search engines for web page ranking and identifying highly expressed genes in genomic studies, which is the main focus of this paper.

For an important biological question, large amounts of genomic data are often collected from various laboratories or research groups over time. Such data are inherently noisy due to the heterogeneity of different sources, including experiments, platforms, as well as different preprocessing procedures. For example, suppose data are available for gene copy number variation from Single-Nucleotide Polymorphism (SNP) arrays, gene expression from cDNA arrays, and protein-binding data from ChIP-seq; the goal is to find candidate genes whose function may be altered at a specific disease stage. Since the data come from different technological platforms that measure different aspects of the biological system, models that use raw data for all available information cannot be exploited.

To aggregate such data, researchers often use powerful meta-analytic approaches based on ranked lists [3], [4] which, without reverting to raw-data modelling, yield more reliable results and also increase the validity and reproducibility of individual studies [5]. For many such approaches, a core task is to perform RA. For analysing data from different biological experiments, an advantage of using a rank-aggregation-based method is that ranks are invariant to transformations and normalisations as long as the corresponding ranks are preserved. In other words, transformation and normalisation are usually not necessary for rank-based methods [6]. In practice, given the varying quality of base rankers, the RA problem can become even more challenging. For example, in biological studies, some laboratories collect or analyse data more efficiently than others.

In some cases, some base rankers may be meaningless or even misleading. Assigning different weights to base rankers, as mentioned in [2], [5], [7], appears to be the only available approach that accounts for the varying quality of base ranks. However, these approaches have a clear limitation: they lack a systematic, principled strategy for designing an appropriate weighting scheme for a practical problem, and they require strong initial assumptions and high computational complexity. Supervised RA may help to learn good weights, as presented in [8], but it requires a good training dataset, which is often unavailable. In most conventional rank-aggregation methods, it is assumed that base rankers (such as gene-analysis algorithms or outputs from different laboratories) are merely passive sources that provide their outputs independently and without interaction.

However, in many real-world applications, especially in biological and medical studies, rankers (or ranking sources) may use different data sources, different assumptions, and even diverse scientific objectives. This heterogeneity can lead to conflict, inconsistency, or purposeful alignment between rankers. In such situations, ignoring their interactive nature may lead to inefficient integration and inaccurate results.

Game theory is a branch of applied mathematics and economics that studies mathematical models of strategic interactions among rational agents. Applications of this theory in ensemble learning, expert opinion aggregation, multi-agent systems, and information-source combination show that modelling interactions among agents can lead to more stable solutions than classical non-interactive methods [9].

In this paper, we model RA within a game-theoretic framework. In this approach, each ranker is considered a strategic agent that tries to adjust its ranks so as to have the greatest impact on the final ranking while maintaining scientific accuracy. This competition or interaction among rankers can be modelled as a non-cooperative game, in which each player, considering the strategies of the others, makes the optimal decision. In the proposed model, the utility function of each ranker is designed to depend on three key factors: 1) the similarity of its provided ranking to the final ranking (as a measure of influence), 2) the degree of agreement with other rankers (alignment), and 3) the cost of deviating from consensus. By defining this utility and analysing the dynamic behaviour of the rankers, a Nash equilibrium can be reached, which represents the final ranking. Thus, the final aggregation results from a stable, optimal interaction among data sources rather than a mere statistical combination of outputs. This model is not only theoretically innovative in the fields of artificial intelligence and game theory, but also, practically, for integrating heterogeneous results in genomic and biological studies, it provides a transparent, analysable, and adaptable structure.

The remainder of this paper is organised as follows. In Section 2, a review of the literature on the rank-aggregation problem and previous related research is presented. Then, in Section 3, the details of the proposed method are described. Section 4 is devoted to the details of the conducted experiments and the presentation of the obtained results. Finally, in Section 5, a summary of the paper's content will be provided.

Recently, many researchers have used RA in biological or medical problems. The authors of [10], using the RA method (TopKLists), combined data from six independent CRISPR experiments in the context of cancer immunotherapy. Their goal was to identify genes that improve the infiltration of T cells into tumours. The results showed that the RA method can reveal novel key genes such as AAK1, SPRR1B, and EHHADH, which are not observed in any single study alone. A new method called bootGSEA [11] has been introduced, in which, using the bootstrap technique, Gene Set Enrichment Analysis (GSEA) is performed multiple times, and then the results are combined using RA. This method has been used in the analysis of multi-omics data (such as transcriptomics and proteomics) in cancer and spinal muscular atrophy, and it has been shown that its results are more reproducible than traditional methods. A tool called GeneRaMeN [11] has been developed in the R-Shiny environment, which uses RA methods to analyse the results of multiple gene experiments. This tool was applied to the analysis of 24 CRISPR experiments for flaviviruses as well as 13 studies on SARS-CoV-2, and it was able to identify common key genes as well as those specific to particular conditions. This tool offers a novel approach for extracting shared and differentiated information in biological studies. In another study [12], using the RRA method, transcriptomic data from various studies related to periodontitis (gum inflammation) were combined. The final results led to the identification of key genes such as C1QA, BST2, and an important lncRNA (LINC01133). This study demonstrated that RA can reliably integrate results from heterogeneous studies. Another paper [13] performed an RA of miRNAs involved in Mild Cognitive Impairment (MCI). They combined several independent studies using the RRA method. Although no significant markers were obtained from this aggregation, this work was recognised as an example of the challenges of aggregating heterogeneous clinical data and highlighted the importance of careful interpretation of results.

In a recent study [14], data from seven microarrays from patients with Alzheimer's disease were aggregated using RRA. The result was the identification of 473 significantly differentially expressed genes, with the TAC1 gene selected as one of the most important key genes, which was also validated in an animal model of Alzheimer's. This paper is a successful example of the application of RA in identifying novel genes in neurological diseases. These recent studies show that RA methods (especially RRA and TopKLists) are powerful tools for integrating heterogeneous biological data in various studies. From analysing CRISPR data to combining transcriptomic and microRNA results, these methods are increasingly being used in the discovery of key genes, disease pathways, and biomarkers. Given the presence of heterogeneous and sometimes conflicting ranking sources, traditional statistical approaches (such as Borda, RRA, or Stuart) may not be able to adequately model the complex interactions among ranking sources. Therefore, researchers have proposed using game theory to model these interactions. In this framework, each ranker is considered a rational player who chooses a ranking strategy with the aim of maximising its influence on the final ranking. This game can be non-cooperative, where each ranker acts independently to gain more benefit, or cooperative, where a group of rankers may form coalitions to exert greater influence [15]. In both cases, concepts such as Nash equilibrium or the Shapley value are used to analyse the contribution and impact of each agent on the final output [16].

In a new line of research, the application of game theory to RA, particularly for analysing manipulation trends and strategic attacks on ranking systems, has attracted considerable attention. For example, Ma et al. [17] investigated targeted attacks against well-known RA algorithms based on pairwise comparisons, such as HodgeRank and RankCentrality, using a game-theoretic framework. They showed that the behaviour of the attacker and the ranking system can be modelled as a continuous game, and the Nash equilibrium defines the topology of the attack; consequently, the attacker, with complete or incomplete information, can impose a targeted ranking. Another important step in this area was taken by Mai et al. [18], who introduced the concept of sequential manipulation. In this study, an attacker manipulates pairwise comparisons online during the data collection process and uses a distributionally robust game-theoretic framework to model the game equilibrium under uncertainty. Their results show that the attacker's strategies under conditions of incomplete or complete knowledge can lead to equilibria that are skewed in favour of the attacker, and adaptability or strong stability has been demonstrated in this process. These two studies highlight the key role of game theory in understanding the stability, security, and strategic interaction among rankers. The models presented in these studies show that in order to design robust RA methods, one must engage in the analysis of dynamic equilibria and the strategic behaviour of agents—an approach that precisely forms our proposed path for adapting RA methods in the biomedical context with game theory.

Traditional RA methods usually assume that base rankers are independent and passive sources that provide their rankings without mutual interaction and solely based on the available data. However, in many biological and medical applications, rankers can represent algorithms, laboratories, or sources that interact with each other or strive to maximise their influence on the final outcome. In such situations, the assumption of independence among rankers is inefficient and may lead to incorrect or unstable integration of results. In order to model the interaction among rankers more realistically, this paper presents a novel framework based on game theory. In this framework, each ranker is modelled as a strategic agent that attempts to determine its rankings in such a way as to have the greatest impact on the final ranking while maintaining credibility and accuracy. This problem is modelled as a non-cooperative game in which agents, with relative awareness of others' strategies, seek to maximise their own utility.

The proposed model specifically analyses the strategic interaction between rankers and employs concepts such as Nash equilibrium to find a stable state in which no ranker has an incentive to change its strategy. Furthermore, for the practical implementation of this model, a utility function combining agreement and disagreement, along with an iterative optimisation algorithm, is used. In the following, the precise definition of the game, its elements, the utility function, and the solution process of the model are presented.

To demonstrate the efficiency of the proposed weighted RA method based on game theory, named MYRA, for solving biological problems, three datasets, Breast1, microRNA [5], and Prostate [19], have been used. The Breast dataset consists of three lists of DNA microarray results (differentially expressed genes) obtained from three breast cancer studies: TransBig [20], MDCC [21], and Pusztai [22], with the aim of finding genes that are differentially expressed between breast cancer and normal breast tissues. Despite their common goal, these three studies were conducted in different laboratories and also differ in the materials and DNA microarray systems used, which makes it difficult to perform an integrated analysis of the data, given that the raw values are not directly comparable. Instead, we apply a high-level meta-analytic method such as the proposed RA to combine the top-k gene lists from each of the three studies. In this dataset, only the genes that are common to all studies are considered; therefore, each of the three lists has a length of n=k=917.

https://rdrr.io/rforge/TopKLists/man/breast.html Table 1 presents the gene rankings obtained by the three breast cancer studies, where the first column shows the gene ranks and the next three columns show the gene symbols obtained in the three studies.

The microRNA dataset pertains to an RA problem in the biological domain, aiming to aggregate results from several DNA microarray target prediction experiments. Today, there are several computational algorithms for predicting the genes whose messenger RNA (mRNA) are targets of DNA microarrays, but their results may differ widely [5]. Therefore, it can be useful to aggregate and filter the disparate results so that researchers can have greater confidence in the validity of these targets before conducting expensive experiments. Since the sensitivity and specificity of such algorithms are largely unknown, it seems more reasonable to regard each list obtained from each algorithm merely as a ranked list.

The microRNA dataset includes three top-60 lists of DNA microarray target genes for miR-155 obtained from miRanda [23], TargetScan [24], and PicTar [25], shown with their rankings in Table 2.

The Prostate dataset includes five top-25 lists of DNA microarray results obtained from five prostate cancer studies named Dhanasekaran [23], Luo [24], Welsh [25], Singh [26], and True [27], to find genes that are differentially expressed between prostate tumours and normal prostate tissues. Table 3 presents the gene rankings obtained by the five prostate studies, where the first column shows the gene ranks, and the next five columns show the gene symbols obtained in the five studies.

A primary task in many genetic studies, such as the two datasets mentioned, is to identify genes associated with certain complex human diseases. Given this, a good criterion for evaluating the performance of RA methods should assess the ability of each method to correctly rank the relevant genes. Suppose only a limited number of resources are available for further investigation of the studied genes; then, the ranking of genes that are at the top of the list is very important, whereas the order of genes near the bottom of the list is almost insignificant. In other words, instead of considering criteria such as the AUC measure, which can only evaluate full lists, in biological studies we need criteria that can also evaluate partial top-k lists [1]. Such criteria include Kendall Distance (KD), Spearman Distance (SD), Kendall Correlation (KC) (the ratio of the number of concordant pairs minus the number of discordant pairs to the total number of element pairs), and SC (applying Pearson correlation to the ranks of elements of two lists instead of to the values of the two lists). One advantage of correlation measures over distance measures is that they lie within a fixed range. Both KC and SC fall within the range [-1, 1]. To use these criteria for evaluating aggregated rankers, the value of these measures is calculated between the aggregated ranker and each of the base rankers individually, and the sum is divided by the number of base rankers to obtain the final value of the criteria. Ultimately, the smaller the final value of the distance measures and the larger (closer to one) the final value of the correlation measures, the better the aggregated ranker.

To evaluate the proposed method, MYRA, it is applied to the Breast, microRNA, and Prostate datasets, and the obtained values of the four evaluation criteria, KD, SD, KC, and SC, are compared with the results of conventional RA methods, namely the Borda methods (Mean, MED, GEO, and L2), Markov chain-based methods (MC1, MC2, and MC3), and distribution-based methods (RRA and Stuart). The aggregated rankers obtained by each of the aforementioned methods, along with their evaluation criterion values, are shown for the Breast dataset in Table 4, for the microRNA dataset in Table 5, and for the Prostate dataset in Table 6. For the implementation and execution of the proposed method (RAA), the software Matlab R2019b1 was used on a computer with 8 GB of RAM and a five-core 2.4 GHz processor. For running the RRA and Stuart methods, the ready-made software packageRobustRankAggreg2 in Matlab R2019b was used, and for running the Borda methods (Mean, MED, GEO, and L2) and the Markov chain-based methods (MC1, MC2, and MC3), the ready-made package TopKLists3 in R software was used.

In the tables, the best values of the criteria are shown underlined.

As can be seen in the tables, the aggregated ranker generated by the proposed method MYRA has smaller distance measure values (KD, SD) compared to the aggregated rankers of the other methods, meaning that the distance between the MYRA aggregated ranker and the base rankers is smaller, as is the difference between their ranks. On the other hand, it has larger correlation measure values (KC, SC) compared to the other aggregated rankers, meaning that the similarity between the MYRA aggregated ranker and the base rankers is greater and they are more correlated with each other. Another advantage of the proposed method over other methods is that it estimates the accuracy of each base ranker as its weight. These estimated accuracies can be used in biological problems to assess the reliability of various algorithms, experiments, platforms, and studies, such that those with much lower accuracy can be omitted from future research, and low-accuracy experiments and platforms can be improved. Table 7 shows the estimated accuracy (weight) by the proposed method MYRA for each of the studies used in the Breast, microRNA, and Prostate datasets.

RA has been widely used in various fields. RA is the process of combining several ranked lists into an aggregated ranker, which plays an important role in integrating information from different genomic studies addressing similar biological problems. In fields such as genetic and biological studies, researchers often generate several long ranked lists, but their true rank values are usually unavailable, and the quality of base rankers varies widely; in contrast, fields such as marketing and sociology generate many short ranked lists for which true rank values are usually available. Moreover, in biological studies, partial lists are more common than full lists due to the high heterogeneity of genomic studies and limited resources. Such inherent characteristics of data in the biological studies domain demand specific methods over others. Methods such as Markov chain, Stuart, CEMC, and RRA were initially developed for genetic experiment data. Also, methods such as Borda have been used for this domain. In this research, we present a novel framework for RA in biological and biomedical applications, based on non-cooperative game theory. Unlike traditional methods that consider rankers as independent and static sources, in the proposed model, rankers are modelled as strategic agents that interact and compete with each other with the aim of maximising their own utility. In this framework, a utility function is defined for each ranker, which is a combination of similarity to the final ranking, the degree of agreement or conflict with other rankers, and the cost of producing the ranking. Using an iterative process based on best-response dynamics, the weight of each ranker is dynamically updated, and the final aggregated ranking is obtained as the stable outcome of a strategic game. The implementation of the proposed method for aggregating the results obtained from three breast cancer studies, aggregating the target prediction results of DNA microarrays by three different algorithms, and aggregating the results obtained from five prostate cancer studies demonstrates the efficiency and superiority of the proposed method over other conventional ranking methods, because the aggregated ranker resulting from it has lower distance measures and higher correlation measures with the base rankers. Overall, the game theory-based approach not only improves the accuracy of RA under conditions of source heterogeneity but also enables the analysis of interactions among sources and the design of mechanisms resistant to manipulation and biased ranking. Future development of this model could include the use of cooperative games, Markov dynamic games, and multi-agent reinforcement learning to enhance further dynamism and adaptability.

The authors sincerely appreciate the valuable contributions and support provided throughout this research.

No specific funding was received for this research.

The data used in this study are available from the corresponding author upon reasonable request