2.1 Literature review on M&A

Recently, the literature on mergers and acquisitions (M&A) has expanded significantly, exploring various directions. Given the global scale of these trillion-dollar processes, it has been crucial to conduct studies with an economic focus. For instance, research on risks through success and failure cases [19] and assessments of efficiency indicators for acquiring organizations [20] have been fundamental. The study by Ding et al. [21] examines the role of discriminatory protections, highlighting how regulatory environments and protectionist policies can influence the strategies and outcomes of cross-border M&As. Additionally, [22] investigates the impact of the COVID-19 pandemic on global M&A activity. [23] provide significant insights into the dynamics of M&A activities in emerging markets, especially during uncertain times. Through a comprehensive literature review, they explore how mergers and acquisitions can be utilized as strategic tools for achieving business sustainability amidst fluctuating economic conditions.

A prominent concept in M&A research is Absorptive Capacity [24], which refers to the acquiring company’s ability to identify and acquire externally generated knowledge from the target firm. This concept is closely related to the ideas of Relatedness and Similarity in Economic Complexity and emphasizes how the similarity between the acquirer and target companies is crucial for successful integration [25]. This “capacity” can depend on various factors. For example, geographical distance has been found to negatively influence the probability of M&A occurrences [26,27], while similarities in ownership or industrial sector can increase such probabilities [28,29]. Analyzing a large set of acquisitions using a similarity measure introduced by [30,31] establish a statistically significant correlation between M&A occurrence and industry relatedness.

A notable section of M&A studies focuses on the patenting activities of involved firms, centring on technological relatedness. [32] introduce a measure of technological similarity between acquirer and target firms, revealing an inverse parabolic relationship between technological similarity and innovation performance post-acquisition. Many subsequent authors develop different measures of technological relatedness and investigate their links to post-acquisition performance [33–39]. However, results regarding the inverse parabolic behaviour between relatedness and performance are inconclusive due to the lack of a standardized and recognized method for robust performance and relatedness measurements [37,40].

While the majority of M&A literature focuses on correlating relatedness measures with successive performances, [41] highlights the importance of forecasting as a crucial test for the validity of relatedness assessments. Notable forecast exercises involve [42], which uses an ensemble learning algorithm to predict future acquisitions based on relative features between companies and patent data, and [43], where a tree-based machine learning algorithm is trained on a large set of M&A features encompassing financial, geographical, industrial, and patent data of firms for M&A prediction. In [44], the authors use patent documents to derive a technological profile of companies and predict possible deals, while in [45] SEC annual reports are analyzed. Other possibilities include the use of machine learning to determine the key factors to predict an M&A [46] (but see also [47]). Also the effects of social networks have been analyzed [48]. Finally, in [14], the authors develop a method to predict future acquisitions by assuming that companies deal more frequently with technologically related ones. Following this line of research, and in contrast with the pure forecast exercises discussed above, the aim of this paper is to understand which features of the technological activity of firms are important for a deal to actually occur. For this reason, we adopt a measure of technological similarity and we modify it to take into account features such as the respective size of firms (in terms of patenting activity) and technology rarity. Note that a specific comparison of the prediction performances of the above mentioned approaches would require an investigation of the union of the different datasets they use.

In the next section, we will illustrate the methodological framework we adopt to build our measure of technological, size and rarity-aware, similarity.

2.2 Literature review on relatedness

Economic Complexity (EC) is a conceptual framework that studies the knowledge intensity of an economy or an industrial sector by using network-based and machine-learning approaches. One line of research stems from the concept of relatedness [3], which measures the similarity between economic activities or the affinity between an economic actor or a geographical area and such activities, for instance exporting a product or patenting in a specific technological field [49,50,62]. Measures such as Product Space [4] and the Taxonomy Network [5] have become essential tools for policymakers and economists to understand and predict economic dynamics [51].

Innovation, usually measured from patent data, serves as a critical driver of economic complexity. The correlation between a country’s patenting activity and its industrial and scientific development has been extensively documented [52]. Moreover, the future trajectory of a firm can be predicted by the economic or technological value of its early patents [53].

For all these reasons, the concept of technology forecasting plays a crucial role in the field of economic complexity applied to innovation [54–58].

On the other side, the concept of ’Relatedness’ in EC literature has been particularly instrumental in understanding the dynamics of mergers and acquisitions (M&A). Studies have leveraged this concept to predict the likelihood of M&A by examining the technological and product similarities between firms [14,25–27,42,43,59], providing a nuanced view of strategic business decisions. Recent advances have applied EC principles to predictive modelling [7,41,60,61]. However, applying EC, and the concept of Relatedness in particular, to M&A prediction is an innovative frontier that remains underexplored. The intersection of EC measures and recommender systems has opened new research avenues. Adapting algorithms from recommender systems, researchers have begun to address the unique challenges of M&A prediction within the EC framework, conceptualizing firms as both the users and items of a recommendation engine. Despite these advancements, significant gaps remain in the literature, particularly concerning the application of EC to M&A predictive modelling using patent data. This review highlights the need for a more granular approach to consider the nuances of firm-level data and the specificities of technological sectors. This is the main objective of this paper.

We conclude this section by stressing the need for scientific validation to confirm or falsify the proposed assessments of relatedness. Many different (or similar) measures are available in the economic complexity field, and even small variations can produce very different outputs [40]. As a consequence, an out-of-sample forecasting scheme has been introduced [41] to compare relatedness assessments by using the assumption that economic actors will, on average, engage more frequently in more affine, or related, activities. The results point out the need to use tree-based machine learning methods to quantify relatedness [62]. Sapling Similarity [7] extremely simplifies these approaches, making them fully interpretable and explainable, and keeping the accuracy of machine learning forecasts. In this paper, we will follow the same approach, by suitably modifying Sapling Similarity for M&A predictions and by verifying the goodness of our approach by checking the out-of-sample forecasting performance of this in comparison with other possible measures. This empirical verification is key to validating the ability of our measure to capture significant economic features.

4.1 Data

The study utilized a combination of four distinct databases to gather relevant data. Information regarding patent classifications and technological areas was sourced from the PATSTAT database (www.epo.org/searching-for-patents/business/patstat). The AMADEUS database (https://login.bvdinfo.com/R0/AmadeusNeo) facilitated the linkage of patents with their respective companies. Additionally, details regarding M&A were collected using the Zephyr (https://www.bvdinfo.com/it-it/le-nostre-soluzioni/dati/greenfield-investment-and-ma/zephyr) and Crunchbase (www.crunchbase.com) databases. This section will elaborate on the dataset preprocessing techniques applied and detail the construction of the database that facilitated the development of similarity indices and forecasting models.

4.2 Methods

In this section, we describe the various similarity metrics used to assess the similarity of companies from their patents. The starting point is the matrix whose element M_ft is the number of patents firms f submitted in the technological sector t during year Y. In the following, we will omit the year specification to lighten the notation. This matrix is the representation of a bipartite firm-technology network, and the matrix elements quantify the weights of such a network. Given an acquirer firm A and a target firm T, we want to compute B_AT, the likelihood of firm A acquiring firm T; different methods will provide different estimations of B_AT. We will then use these assessments to predict M&A deals.

4.2.2 Cosine similarity.

The first measure we introduce is cosine similarity, also known as Jaffe similarity in the context of M&A, as it was introduced in 1986 by Jaffe to quantify the productivity of manufacturing R&D [67]. Since then, in the work of Valentini and Dawson [68], the cosine similarity measure was applied in the M&A context. More recently, [14] conducted a comparative analysis of various methodologies, including machine learning algorithms, for estimating the likelihood of a deal. They concluded that the most effective approach involves assessing the similarity in technological portfolios between the two companies, utilizing cosine similarity as the metric. In our notation, the equation for cosine similarity reads

(1)

4.2.3 Sapling similarity for unweighted bipartite networks.

Albora et al. [7] have recently introduced a new metric of similarity between nodes in unweighted bipartite networks: the Sapling Similarity. In this paper, the bipartite network under study will be the one that connects companies to the technology sector mentioned in their patents. The idea behind the Sapling Similarity metric is to extract the main ingredients of tree-based machine learning models, since they outperform other approaches in Relatedness estimations (see [62]), to allow full interpretability and explainability while preserving the prediction performance. Sapling Similarity is an indicator ranging from –1 to 1 that reflects how similar the technological portfolios of two firms are. Given firms A and T, while cosine similarity evaluates the angle between their technological portfolio vectors, Sapling Similarity examines how the information that T is active in a generic technological class C alters our prior probability that A is active in C. If this probability increases, then the Sapling Similarity between A and T is positive; if it decreases, then the similarity is negative. In order to visualize how Sapling Similarity works, in Fig 1 we present what is referred to as a ‘decision sapling’ in [7].

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Fig 1. On the left figure, an example of a decision sapling illustrating the relationship between two firms, A and T, in a scenario featuring 100 technologies.

The node at the bottom indicates the percentage of technologies that are (right) and are not (left) connected to firm A; in this figure, A is connected to 20% of technologies, as denoted by the green value. The upper nodes show how these percentages change when considering only technologies connected to Firm T (right node) or only those not connected to Firm T (left node). On the right figure, we show the value of each box as a function of the number of co-occurrences CO, the degrees k, and the total number of technologies N.

https://doi.org/10.1371/journal.pone.0341010.g001

On the left, we represent a numerical example in which only N = 100 technologies exist and two firms, A and T, are considered. A is connected to (i.e., has patenting activity in) k_A = 20 technologies, and T is connected to k_T = 5 technologies. A and T share CO_AT = 3 technologies. The bottom node indicates the fraction of the technologies that firm A is either connected with or not; notably, firm A has established connections with 20% of the technologies, as denoted by the green value. Upper nodes detail the updates of these proportions when the analysis is narrowed to technologies either connected with or not connected with firm T. In this example, knowing that a technology is connected with T increases the probability that it is connected to A from 20% to 60%, suggesting a positive similarity between A and T.

To quantify this line of reasoning, we need a measure of the polarization in a node. Given p₁ the fraction of positive samples in a decision tree node, and p₀ the fraction of negative samples, the Gini Impurity of the node [69] is defined as:

(2)

This quantity tells us how much the samples in the node are peaked, or concentrated, towards the positive or the negative case: maximum polarization (only positive or negative samples) implies GI = 0 while minimum polarization (equal number of positive and negative samples) means GI = 0.5. So if we consider the lower node of the decision sapling, we have p₀ = 0.8, p₁ = 0.2, and GI^low = 0.32. Analogously we find that the Gini Impurity of the upright node is GI^upr = 0.48 and the one of the upleft node is GI^upl = 0.29. The variation of the Gini impurity is defined as:

(3)

where f^upl and f^upr are the fraction of samples that are respectively in the upper left and the upper right node (in the case of the figure, 0.95 and 0.05).

The variation of the Gini Impurity quantifies how much the information that T patents or not patents in a generic technology field λ is important to understand whether A patents in λ. This is the absolute value of the Sapling Similarity; its sign is positive if (which means that knowing that T is connected to λ increases the probability that also A is connected to it), and negative otherwise.

Using the figure on the right, in which we report the general formulas, we can easily derive an equation for the sapling similarity as a function of the co-occurrences CO_AT (how many technologies firms A and T share), the degrees k_A and k_T (the number of technologies each firm has) and the total number of technologies in the bipartite network N:

(4)

Where:

(5)

4.2.4 Generalization of the sapling similarity for weighted bipartite networks.

In our specific case, the input data is a bipartite network that connects firms to the technology fields, which is weighted; so we need to generalize the Sapling Similarity, which in previous papers has always been used with binary inputs. Our line of reasoning goes as follows. The degree k_F of a firm F can be seen as the maximum number of co-occurrences that this firm can have (indeed, it cannot share with another firm a number of technologies larger than the number of its technologies); on the other hand, we can think of N as the maximum number of co-occurrences that two generic firms can have (that is, the number of co-occurrences between two firms that possess all the technologies).

In the case in which the matrix M takes continuous values, the number of co-occurrences between the two firms A and T is generalized as the scalar product between the two row-vectors of the matrix M that represent the patenting activity of A and T.

(6)

To simplify the following equations we introduce the matrix Λ whose elements are defined in Eq 6. Since the elements of M take continuous values without an upper limit, in principle the theoretical maximum value of is infinite. However, we can take this value from the empirical counterpart: so we consider as the equivalent of k_A and as the equivalent of k_T. With the same reasoning, we can say that the continuous equivalent of N is . So with these changes, the equation of the sapling similarity in the continuous case reads:

(7)

Where:

(8)

5.1 Pair, target, and acquirer prediction

In this paper, we aim to compare the effectiveness of various methods in predicting future M&A deals between companies. We estimate the probability of the possible M&As occurring between companies in a given year Y using only past data; so the prediction scores are recalculated in each year. To assess the quality of the methodologies, we examine their ability to predict M&As between companies. In practice, this translates into three distinct binary classification exercises:

1. Pair Prediction: For this exercise, the 547 pairs of companies that undergo M&A are labeled as positive events, while negative labels are assigned to randomly generated pairs of companies. For each M&A that occurs in year Y, 200 negative pairs are generated, ensuring that each pair is unique and not among the 547 actual M&A pairs. The best-performing model is the one that accurately distinguishes the true M&A pairs from the randomly generated ones.
2. Target Prediction: Here, for each of the 547 actual acquirers, 200 negative targets are generated ensuring that these do not coincide with the real targets and that there are no repetitions. The optimal model is identified as the one that can effectively differentiate the actual target of the single acquirers from those randomly generated.
3. Acquirer Prediction: Similar to the target prediction, this exercise involves generating 200 negative acquirers for each of the 547 true targets. The model’s task is to identify the true acquirer of each target.

In all three types of experiments, for every single M&A deal, 200 negative company pairs are generated, resulting in a class imbalance of 1:200 in the binary classification exercise. To quantify the performance of the models in these three types of experiments, we use standard performance indicators for binary classification [71,72]. For pair prediction, we employ the following three indicators:

• Best F1 score (best F1) [41,62,73]: This score is computed by finding the threshold that maximizes the F1 score [74], that is defined as the harmonic mean of precision and recall;
• Area under the Precision-Recall Curve (AUC PR) [75,76]; the area under the curve on the precision-recall plane. This area is derived by varying the threshold that determines the score above which predictions are classified as positive.
• Precision at 500 (prec@500); To compute this metric, we evaluate the top 500 scoring elements. prec@500 measures the proportion of these top 500 elements that are true positives (accurately predicted positively) out of the total 500 elements examined.

For target and acquirer prediction, we utilize:

• Best F1 score (best F1), as for pair prediction;
• Hit Ratio at 5 (HR 5) [77]: it measures the proportion of times that the relevant item (the true acquirer in the acquirer prediction exercise and the true target in the target prediction exercise) appears in the list of the top 5 recommendations;
• Mean Average Precision (mAP): the mean of the average precision [78] across all acquirers (target prediction exercise) or targets (acquirer prediction exercise);

Each experiment is repeated 20 times, with negative M&As being regenerated each time. The final score is an arithmetic mean of these 20 repetitions.

5.2 Results

The prediction performances of the different methods are compared in Fig 2, which shows nine bar plots arranged in a 3x3 grid. Each row corresponds to a different prediction exercise: pair prediction, target prediction, and acquirer prediction, respectively. Within each row, three distinct performance indicators are utilized for evaluation: In order Best F1 Score, Area Under Precision Recall Curve, and Precision at 500 for pair prediction; Best F1 Score, Mean Average Precision, and Hit Ratio at 5 for both target and acquirer predictions. The bar plots feature eight bars, grouped in pairs, with pink bars representing predictions made using an unweighted network and green bars for those made using a weighted network. Each pair of bars corresponds to a variant of sapling similarity, incorporating modifications (1) and (2), introduced in the methods section, aimed at refining the prediction accuracy for this particular case study. The current state of the art is cosine similarity measured using the weighted network (red dashed line).

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Fig 2. Performance of the various Sapling Similarity variants in predicting M&A deals in the three cases: pair, target, and acquirer prediction.

The variants, reported on the x-axis, are described in the Methods section. We report in pink the case in which the input network binary, and in green, the weighted case. The MASS algorithm corresponds to the green SS(1+2) variant. The red line shows the performance of cosine similarity measured using the weighted network. All improvements enhance the prediction performances, MASS being the outperforming algorithm.

https://doi.org/10.1371/journal.pone.0341010.g002

The modifications introduced to the base equation of Sapling Similarity consistently yield improvements. This enhancement aligns with the rationale behind their introduction: smaller target companies are generally the focus of acquisitions, addressed by modification (1), and the attractiveness of a small company possessing a rare technology, as considered in modification (2).

Generally, predictions utilizing a weighted network outperform those based on an unweighted network, validating the effectiveness of extending Sapling Similarity to incorporate the case of weighted networks. However, in target prediction scenarios, the distinction between weighted and unweighted networks is minimal. This suggests that for large companies acquiring smaller ones, the mere possession of a technology by the target is significant, irrespective of whether it is the target’s main technology or not. In contrast, understanding the importance of technologies to the acquiring company is crucial for accurately predicting the acquirer.

The MASS approach, incorporating strategic modifications to the already well-performing Sapling Similarity, consistently outperforms the previous state-of-the-art, that is cosine similarity (the red dashed line in the plots), which previous studies shown to perform better than a number of other approaches [14]. This superiority across all metrics and prediction exercises underlines the method’s robustness and its capacity to set a new benchmark in the field of M&A prediction.

Given that target prediction and acquirer prediction utilize the same performance indicators, a direct comparison reveals that acquirer prediction exercises generally achieve higher performance metrics. This indicates that predicting the acquirer in a M&A scenario is more straightforward than identifying the target.

5.3 Acquisitions with low co-occurrences

In the previous section, our investigation has shown the robustness of the MASS approach as the state-of-the-art approach for forecasting M&As between companies from their patenting activity. However, a pertinent inquiry arises regarding the limitations of this method.

Indeed, not all companies exhibit the same propensity for patenting, leading to a significant number of firms with very few technologies - or links in the bipartite network language. This variance can largely be attributed to diverse patenting policies across different sectors. Specifically, certain industries are naturally inclined to generate fewer patents than others. Consequently, some sectors tend to have limited patenting activity, leading to a higher likelihood of M&A deals occurring between firms with zero co-occurrences. Despite its prowess, methods like MASS struggle to predict these instances due to their reliance on shared technologies for similarity calculation.

Among the 547 M&A instances in our dataset, 123 involve pairs of companies with no direct technology overlap, highlighting a substantial subset where Sapling Similarity for M&A’s predictive power is limited. This is where machine learning comes to help.

In Fig 3, we compare the performances of MASS and LightGCN across two different testing scenarios: the entire dataset of 547 M&As (above). and the subset of 123 M&As between companies with zero co-occurrences (below). The radar plots schematically represent the prediction ability of the two approaches in the three different exercises, the largest area being relative to a higher performance. We note that in generating random pairs of companies for negative test cases, care was taken to ensure that none of the pairs matched the 547 actual M&A instances, even when only 123 M&As are considered. Within these plots, the vertices correspond to the three performance metrics utilized in our evaluation. The resulting shapes formed by connecting these vertices thus visually encapsulate the comparative performance of MASS and LightGCN across these metrics.

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Fig 3. Comparative analysis of MASS and LightGCN performances across different M&A prediction scenarios.

The top row of radar plots presents the performance metrics for both methods when the test is conducted on the entire dataset of 547 M&As. Here, MASS outperforms. The bottom row depicts the same exercises, but the test focuses exclusively on the subset of 123 M&As between companies with zero co-occurrences (0-CO). In this case, LightGCN is able to capture the hidden, higher-order similarities between companies.

https://doi.org/10.1371/journal.pone.0341010.g003

The graphical representation clearly illustrates that while MASS outperforms LightGCN across the board when considering the full dataset, its supremacy diminishes in scenarios involving companies with no technology overlap, i.e. with zero co-occurrences (0-CO). In these cases, LightGCN emerges as the superior predictive tool. This shift can be attributed to LightGCN’s ability to leverage the broader structure of the bipartite network, capturing latent similarities between companies beyond direct technological co-occurrences. By effectively utilizing this comprehensive network information, LightGCN demonstrates a pronounced advantage in predicting M&As among companies that, on the surface, share no common technological ground.