Recommendations provided by group recommendation systems are often complex and dificult for average users to comprehend. To address this issue, many existing approaches incorporate explanations alongside recommendations. However, as group recommendation models become increasingly sophisticated, ofering clear and understandable explanations becomes more challenging. In response, we propose a system that delivers counterfactual, model-agnostic explanations. Counterfactual explanations focus exclusively on items with which the group has interacted, enhancing their interpretability. Additionally, model-agnostic explanations mitigate the intricacies of complex group recommendation models by treating them as black boxes, thereby simplifying the process and making the recommendations more accessible to users. We propose two heuristic approaches to produce fair group counterfactual explanations, i.e., explanations that consider all group members' input and are not focused on individual group members. We conduct an evaluation of our methods utilizing the MovieLens dataset, highlighting our proposed methods' efectiveness in generating counterfactual explanations for groups within a practical and eficient timeframe.
Recommendation systems are widely used across various
applications, helping users navigate vast amounts of
information by providing personalized suggestions. One
common example is movie recommendations, where the system
suggests films based on individual preferences. However,
for many users, the underlying processes—whether for
individual or group recommendations—remain opaque and
dificult to understand. This is particularly true for group
recommender systems, which aggregate the preferences of
multiple users to suggest relevant items. As the
complexity of these systems increases, the challenge of explaining
how recommendations are generated becomes more
dificult, leading to potential mistrust or frustration, especially
when the system fails to meet user expectations [
To generate explanations for recommendations, systems typically rely on information from all users within the system. This requires access to system data to clarify why a particular item is recommended. However, service providers are often concerned with maintaining client privacy and security, meaning that third parties cannot access or be informed about this sensitive data. One solution to this challenge is the use of counterfactual explanations, which only leverage the interactions of the specific user requesting the explanation, addressing privacy concerns.
Counterfactual explanations have been previously
explored [
Since exhaustively searching through all combinations of items is prohibited, we propose two heuristic approaches. We leverage utility scores (Section 2.2) to quickly parse through a large number of items. The first approach, Sliding Window (Section 3.1), aggregates all utility scores for the items and orders them in a list. We apply a sliding window on this list, aiming to quickly find a large subset of items that, if they were removed from the group members’ interactions, then the item in need of explanation is no longer suggested to the group, i.e., a counterfactual explanation.
The second approach, Four Quadrant (Section 3.2), allocates the items based on their utility scores in a fourCEUR
ceur-ws.org quadrant graph, where the axis of the graph represents the four utility metrics: Item Intensity, Item Rating, Item Popularity, and Item Relevance (for more details please refer to Section 2.2). We only examine items that are present within a circular region. If no explanation is found, then we will increase the radius of the circle.
The main contributions of our work are the following: • We introduce the notion of counterfactual explanations for group recommendation, which, to our knowledge, represents the first work in this field. • We present group counterfactual utility metrics that measure the likelihood of an item being part of a counterfactual explanation. • We propose two heuristic algorithms to find fair counterfactual explanations for groups, Sliding Window and Four Quadrant Algorithms. • We evaluate our proposed methods using the realworld recommendation dataset, MovieLens, focusing on two diferent group types: those with high engagement levels with the system and those with moderate engagement.
The rest of the paper is structured as follows. Section 2 describes the group counterfactual explanation problem, and Section 3 details two heuristic methods for producing such explanations. Section 4 presents the experimental evaluation and analyzes the results. Section 5 presents the related work. Finally, Section 6 concludes this paper.
(, ) ∈ ℝ tory.
Interacted Items. Let be all the users involved in the system. Let be the set of items that the user ∈ interacted with, and = ⋃ ∀∈ be all the items that the has group members ⊆
have interacted with. Furthermore, + denote the rating assigned to item by the user . Note that, in our work the presence of ratings is not mandaRecommendations. Additionally, ( turns the recommended items list , i.e., ( group recommendation system which, given the set , re ) = .
For this work, we consider the group recommender system a black box. This allows us to freely use any group recommender system without considering their limitations. ) denote the Group Counterfactual Explanation. In a recommendation system for individual users, a counterfactual explanation about a recommended item is defined as a set of items that if the user had not interacted with (e.g., rated, clicked, liked), then item would not have been recommended. A similar concept can be defined in group recommendation systems.
Given a group ⊆ , and an item ∈ , which will be referred to as the item of interest, an explanation for the appearance of the item in the group recommendation list , is a set of items, that if the group members had not interacted with, then the item of interest would not have appeared in the group recommendation list .
Formally, let be a recommendation list, with ∈ ; an item set ⊆ is an explanation, if ∉ ∖ . (1) (2) (3) (4) Counterfactual Cost. Producing the recommendations list , is associated with a recommender cost , e.g., the response time and/or service cost. As a result, a counterfactual is also associated with a cost, denoted as , which corresponds to the overall recommender cost required to find . Hence, given that we need to invoke the recommender times in order to find , the counterfactual cost of is: = ∑ 1 where represents the cost of a single recommender system call. For simplicity, in our evaluation we define = 1 to encapsulate how many times we called on the recommender system.
In order to access the ”quality” of the counterfactual, we exploit several metrics related to: clarity, engagement, preferences, popularity, and relevance.
Counterfactual Minimality. We use the notion of
minimality (a.k.a. sparsity) to access the clarity of a
counterfactual, i.e., the shorter the explanations, the easier it
is to understand [
Note that, in our problem the adopted minimality metric
also captures the actionability (a.k.a. feasibility) of a
counterfactual [
Item Intensity. Item Intensity describes how many of the group members had interacted with an item. The intensity of an item reflects the degree to which it has been interacted with by individual members of the group. This characteristic provides insights into the extent of engagement that group members have had with a particular item, which in turn sheds light on the potential significance of that item within the context of the group. Higher intensity values suggest that the item has garnered substantial attention from group members. Hence, we want the explanation items to have high intensity. intensity of an item int(, ) ∈ (0, 1] is defined as:
Formally, given a group ⊆ and an item ∈ , the int(, ) = ∑∀∈
(, ) || where (, )
returns 1 if user has interacted with item (i.e., ∈ ), and 0 otherwise.
Item Rating. This metric builds upon Item Intensity metric by incorporating the collective evaluation of an item by all group members. Specifically, the Item Rating for a group and an item , denoted as rate(, ) , is computed as the mean of the ratings assigned to an item by members of the group . rate(, ) = ∑∀∈
(, ) ||
This metric ofers a clear representation of the group’s overall sentiment toward the item, providing an aggregated view of the group’s preferences. Note that, in case that items ratings are not available the metric is omitted.
Item Popularity. Item Popularity refers to an item’s overall appeal or recognition within all systems users . Item Popularity for an item , denoted as pop( , ) is quantified by aggregating the ratings the item has received across all users. pop( , ) = ∑ (, ) ∀∈
Popularity serves as a proxy for how widely appreciated or commonly engaged with an item is, regardless of individual user preferences. It is a model-agnostic metric, allowing its straightforward adoption across diferent recommendation systems and environments. Note that, in case that items ratings are not available the metric is omitted.
Item Relevance. Item Relevance measures how pertinent an item in is. This is based on the predicted score for the item of interest assigned to group members from the recommender system. Intuitively, the interacted items of a user with a high predicted score for are more influential in shaping the group recommendation list. Consequently, these items are more likely to be included in the counterfactual explanation. To compute item relevance rel(, , ) in a model-agnostic manner, we treat each individual user as a group and query the group recommender for suggestions based on their preferences and interactions. where rel(, , ) = ∑ ℎ(, , ) ∀∈ ℎ(, , ) = { 0 predScore( , ) if ∈ otherwise The function predScore( , ) returns the predicted score of the item of interest for user , where is the recommendation list resulted when the group recommender is invoked using as input only ’s interacted items, i.e., .
Counterfactual Utility. Given a counterfactual the utility () is defined as a combination of counterfactual minimality, and items intensity, rating, popularity, and relevance metrics: (5) (6) (7) where function aggregates counterfactual minimality with item-based scores, and function aggregates the items scores for all items in .
Next, we formally define the the Group Counterfactual Explanation problem (GCF).
Group Counterfactual Explanation Problem (GCF). Given a group ; a group interacted items ; a group recommended items list ; a item of interest ; a recommender cost ; and a budget in terms of recommender cost; our goal is to find a group counterfactual explanation ∗, such that the explanation utility ( ∗) is maximized and the counterfactual cost ∗ is lower than the budget .
∗ = arg max () s.t. ∗ ≤
Computational Complexity. In order to solve the GCF problem we have to examine the power set of the group interacted items , i.e., all the possible group interacted items subsets. That is, we have to examine 2| |−1 sets (empty set is omitted). Considering that the recommender cost for each set is , the computational complexity is ( ⋅ 2 | |).
In this work, in addition to counterfactual utility, we also examine the notion of fairness in group counterfactual explanations. Specifically, we define fairness in terms of how equally group members have interacted with the items presented in the explanation. The goal is for each member of the group to have engaged with the same set of items, ensuring that no individual is singled out, and the explanation reflects the experiences of the entire group.
Group Counterfactual Fairness. Given a group ⊆ and an explanation , the fairness of the explanation , fair(, ) ∈ (0, 1] is defined as follows. The higher the value, the fairer the explanation. where fair(, ) = ∑∀∈ (, )
|| (, ) =
Equation 9 calculates the extent of group members that have interacted with at least one item in the explanation. Ideally, we want all group members to have interacted with at least one item, i.e., fair(, ) = 1 . On the other hand, if fair = 1/|| , it suggests that the explanation is focused on one member of the group, leaving the others unaccounted for.
Due to the large search space (i.e., 2| |), finding an optimal solution is computationally prohibitive even for a small number of users. To cope with the complexity of the GCF problem, we design two eficient heuristic for finding group counterfactual explanations.
The first approach we introduce employs a sliding window technique applied to an ordered list of items. The Sliding Window (SW) algorithm operates as follows: for each item that has been interacted with by at least one group member, we compute the item-based metrics included in the utility formula (Eq. 8). These scores are aggregated into a final score for each item, enabling the ordering of items from highest to lowest score.
Once the items have been ordered, we apply a sliding window of fixed size to the items list. For instance, with a window size of 5, the window initially covers items 1 through 5, then shifts one position forward to cover items 2 through 6, and so on, until the entire list has been processed. At each step, the items within the current window are temporarily excluded from the group interacted items, and the group recommender system is invoked to compute the new recommendations. If the item of interest is not included in () = ( mini( ), ∀∈ (int( ), rate( ), pop( ), rel( ))) (8)
A B C
Group Recommender
➊ Metrics Calculator int, rate, pop, rel
sort sliding window Sliding Window Algorithm Process ➋ ordered list ➌ Sliding Window Algorithm starts by calculating and aggregating the four Item metrics: Item Intensity, Item Rating, Item Popularity and Item Relevance (Step 1). Since the size of the window is set to 2 we select the top 2 items in the produced ordered list (Step 2) and remove them from the group members’ interactions (Step 3). Lastly we call on the group recommender system again with the altered interaction lists (Step 4) and check if the target item has been removed from the group recommendation list (Step 5). the recommendation list, the items included in the window correspond to a valid counterfactual explanation.
When the window items correspond to a counterfactual and in order to improve minimality, we search for a potentially smaller set of items that also qualifies as a counterfactual. To this end, we conduct an exhaustive search across all possible combinations of items within the window, evaluating which subset of items best generates the counterfactual explanation. This search is performed in an ascending order based on the number of items being considered, starting with individual items and progressively expanding to larger combinations. In this manner, the first valid counterfactual explanation that is discovered will be the smallest possible subset of items, thus ensuring that the explanation is both compact and precise. We have selected to keep the window size small in order to minimize the number of possible combinations that we have to examine.
Sliding Window Algorithm Example. Figure 1 demonstrates the sliding window process. Assume a group of three members, A, B, and C, each has their own interacted items sets: , , and , respectively. These interacted items list are given to a group recommender system which outputs a group recommendation list . Let’s assume that the item of interest (i.e., the item in need of explanation) is 10. 1 We consider all the interacted items of the group members and calculate the item-based scores for each item. 2 Then, we aggregate these scores into a final score and order the list in descending order based on that score. 3 Next, apply a sliding window of size 2, which will initially remove from group members interacted items lists the items included in the window, i.e., the first two items, 2 and 4. In our example,
′ is items list initially we have
= 1, 2, 3 and the updated interacted ′ = 1, 3. 4 Finally, we invoke the recom mender given as input the updated interacted items lists ′ , ′ and ′. 5 The resulted group recommendation list
do not include the item of interest 10, so the items with the
′ window { 2, 4} corresponds to a group counterfactual.
In Four Quadrants Algorithm (Quad) we conceptualize the item-based metrics used in the utility score (Eq. 8), as the axes of a quadrant graph, where each axis corresponds to one of the metric, as illustrated in Figure 2. This representation enables the pairing of the metrics into four pairs: 1. Popularity - Intensity 2. Popularity - Relevance 3. Relevance - Rating 4. Rating - Intensity
To position the items on the graph, we use their
respective scores for each metric. Each item is thus represented
four times, corresponding to its metrics scores along the
respective axis of the graph. Given that the metrics have
significantly diferent numerical ranges, such as popularity
being represented by a large number and Item Intensity
being a score constrained within the range (0,1], it is essential
to normalize all the values to a common scale. Consequently,
we apply a normalization process to ensure that each score
is transformed into the range [
We examine items located within a circular region centered at the origin of the quadrant graph (i.e., the point representing the normalized value 0 for each characteristic). The radius of this circle, denoted as , determines the area of interest. Items that are considered most relevant, such as those with higher popularity, should ideally be positioned closer to the center of the graph. To achieve this, we adjust the item scores by subtracting each score from 1, thereby making higher scores closer to the center. This transformation ensures that more significant values, such as higher popularity scores, will correspond to positions closer to the origin of the quadrant graph.
In a manner similar to the previous approach, we remove from consideration those items that fall within the circular region defined by the radius . After adjusting the dataset accordingly, we call the group recommender with the modiifed information to determine if the item of interest remains in the group recommendation list. If the item of interest is still included, we incrementally increase the radius and repeat the process. This procedure continues until the item of interest is no longer recommended, at which point we have identified a subset of items that provide a counterfactual explanation.
When a counterfactual explanation is identified, we follow an approach akin to the previous one, wherein we systematically explore all possible combinations of items within the subset. This search is conducted in ascending order of the combination size to find the smallest possible subset that serves as the counterfactual explanation.
However, as previously noted, performing an exhaustive search becomes computationally prohibitive when the number of items is large. To mitigate this, we leverage the fact that items can appear multiple times within the circular region. Specifically, instead of evaluating all items, without the loose of generality, we restrict our search to those items that appear at least twice within the circle. This restriction significantly reduces the search space, enabling us to perform the exhaustive search in a feasible manner. For example, in Figure 2, we have a circle with = 0.2 , for which only two items, 2 and 4 are present twice. So, we will only examine all possible combinations of those two items, namely: [{ 2}, { 4}, { 2, 4}].
In this section, we present the experimental results, evaluating the proposed algorithms: Sliding Window (SW) and Four Quadrants (Quad), varying several parameters and comparing them against a baseline method. In the first experiment (Sect. 4.2) we examine the counterfactual cost produced by the algorithms. In the next experiment (Sect. 4.3) we evaluate the group counterfactual fairness, and finally examine the sizes of the explanations (Sect. 4.4).
Methods Parameters. For SW algorithm, we examine two window sizes (i.e., percentages based on the size of the list of items with which at least one group member has interacted). Particularly, we consider two percentages: 10% (SW01) and 5% (SW005). Furthermore, in SW evaluation, we examine two fixed window sizes: ten (SW10) and twenty (SW20). Regarding Quad algorithm, we conduct experiments varying the number of times items appear inside the circle, setting this parameter to 1, 2, 3, and 4, denoted as Quad1, Quad2, Quad3, and Quad4, respectively. The starting value of circle radius is 0.1. For comparison, we also consider the Baseline approach, where we exhaustively examine all possible combinations of items that at least one group member has interacted with.
Dataset. We utilized the MovieLens 100K dataset [
Group Recommendation System. The experiments were conducted in Python, using the Surprise library to create collaborative filtering models and the SciPy library for optimization and batch searches. We employed the k-nearest neighbors (KNN) technique with user-based collaborative ifltering to generate recommendations for individual users. consists of the ten items with the highest aggregated scores. For simplicity, the item of interest, i.e., the item we want to explain, is always the first one in .
The group recommendation model was retrained several times as new batches of movies were removed. This iterative process of batch elimination and model retraining was essential in simulating realistic scenarios where recommendations adapt based on user feedback or potential counterfactual situations.
We first analyzed the computational cost required to find a counterfactual explanations. This evaluation was conducted across 40 groups, evenly split between moderate- and highengagement groups (20 each). The cost was quantified as the average number of calls to the group recommender system required to find a counterfactual explanation. Figures 3 and 4 illustrate these costs for high- and moderate-engagement groups, respectively.
For high-engagement groups, the Baseline approach was excluded due to its prohibitively high cost. This high cost arises from the fact that the Baseline approach will first examine individual items and then item combinations, making the approach impractical since the list for all possible items we want to examine, , is quite large. The size of is approximately 2.000 since we have five members, and each one has given more than 400 ratings.
Sliding Window Algorithm (SW). The Sliding Window algorithm emerged as the most eficient in minimizing the cost of counterfactual explanation generation. Its performance varied between moderate- and high-engagement groups. For moderate-engagement groups (Fig. 4), the best configuration was SW10, which uses a fixed window size of 10. This configuration is efective because the smaller size of the item list in these groups ensures that ten items are suficient to identify the counterfactual explanation eficiently.
In contrast, for high-engagement groups (Fig. 3), the best approach was SW005, where the window size is dynamically set to 5% of the total number of items in . This dynamic adjustment accommodates the significantly larger sized lists in high-engagement groups, often more than three times the size of those in moderate groups. In such cases, a fixed window size of 10 proved insuficient, requiring additional calls to the recommender system to locate the
It is worth noting that larger window sizes, such as those used in SW01 and SW20, reduced the time to locate a counterfactual window, i.e., a subset of items that can produce a counterfactual explanation, but resulted in higher overall costs. This was due to the inclusion of numerous items in the window, necessitating additional calls to the recommender system to find the minimal counterfactual explanation. Four Quadrants Algorithm (Quad). The Four Quadrants algorithm exhibited a diferent cost profile. Among its configurations, Quad 1 incurred the highest cost for both moderate- and high-engagement groups. This can be attributed to the algorithm’s consideration of all items within the circular region without any filtering, resulting in an extensive initial set of items to examine.
Costs decreased as stricter criteria were applied, such as requiring items to appear multiple times within the circular region before being considered. This observation supports our hypothesis that filtering items based on repeated inclusion is essential for cost reduction. Both Quad 3 and Quad 4 demonstrated comparable performance, efectively ifltering out outlier items, such as those with low popularity or those rated by a single group member, thereby reducing computational overhead.
Discussion. The Sliding Window algorithm demonstrated superior eficiency in identifying counterfactual explanations with lower number of calls to the group recommender system. To optimize performance, the window size must balance being large enough to locate the counterfactual window quickly while avoiding excessive inclusion of items, which would increase costs. The dynamic configuration of SW005 proved particularly efective, adapting to varying group engagement levels.
The Four Quadrants Algorithm, while less adaptable, exhibited consistent performance across group types: high and moderate engagement. Quad 3 and Quad 4 were especially efective in filtering out irrelevant items, ensuring steady computational costs across diferent engagement levels. These results highlight the trade-ofs between adaptability and reliability, underscoring the need for tailored algorithmic designs to accommodate varying group dynamics in counterfactual explanation generation. In this experiment, we examine the impact of metrics and methods on achieving counterfactual fairness. Figures 5 and 6 present the fairness metric defined in Equation 9, which measures the proportion of group members who have interacted with at least one item in the counterfactual explanation. Higher fairness metric values indicate better performance in ensuring inclusivity across group members. Ideally, all group members would have interacted with every item in the explanation. However, achieving that is challenging due to the diversity of interactions within the group.
The Sliding Window algorithm demonstrated superior performance compared to the other approaches, primarily due to its direct consideration of the number of group members interacting with an item. This is captured through the item intensity metric (Equation 3), where the score for an item increases with the number of group members who have interacted with it. By ranking items based on their combined scores, the Sliding Window algorithm prioritizes items with high intensity—those that multiple group members have interacted with—early in the counterfactual explanation generation process. This prioritization inherently improves fairness, as it ensures greater representation across the group.
In contrast, the Four Quadrants algorithm struggles to achieve comparable fairness levels. This arises from the way item scores are scaled. High-intensity items, such as those interacted with by four out of five group members, receive an item intensity score of 0.8. To align with the algorithm’s scoring framework, this value is reversed by subtracting it from 1, resulting in a score of 0.2. Consequently, these items are not considered, as their scores fall above the initial radius threshold value, which is set to 0.1.
Moreover, for an item to be included in the counterfactual explanation, it must meet additional criteria, such as achieving suficient counterfactual utility scores. For example, the item must exhibit both high popularity within the system and relevance within the group. These stringent requirements further limit the Four Quadrants algorithm’s ability to prioritize high-intensity items, ultimately leading to lower fairness scores compared to the Sliding Window algorithm.
Figures 7 and 8 showcase the size (number of items) of the explanations generated by the algorithms. Under our deifned group counterfactual utility, we want the explanations provided to be minimal, i.e., to consist of as few items as possible. The Four Quadrant algorithm manages to find explanations with the least amount of items. This demonstrates the algorithm’s ability to identify the most “efective” explanations. We quantify these explanations as efective since they manage to alter the initial group recommendation list by removing the least amount of interactions from the group members.
We can observe that the explanation size does not increase with the number of interactions among the group. The sizes remain in the same order for both high and moderateengagement groups as presented in Figures 7 and 8. However, the cost of finding these items is higher the more interactions the group has. As shown in Figures 3 and 4, to find counterfactual explanations for high-engagement groups, the algorithms require more calls to the group recommender system, indicating the diference in search space size.
Group recommendations have a significant research
background [
In the aggregation stage, a group recommendation system
can take into account diferent criteria. For example, [
[
In [
Counterfactual explanations have been a potent tool to improve recommendation systems’ transparency and reliability in recent years. These explanations clarify why certain recommendations were made by showing how little adjustments to the input data could have a significant impact on the results. This method is especially relevant to group recommender systems, in which a recommendation is generated by combining the preferences of several individuals. For the group’s recommendations to be understood by all members and how their preferences influenced their final decision, transparency is extremely crucial.
By altering image embeddings to ascertain whether visual
aspects impact recommendations, CAVIAR [
Additionally, Stratigi et al. [
A recent study [
MACER [
These investigations laid the groundwork for expanding counterfactual explanations into the field of group recommender systems. However, the majority of research to date has been conducted on individual recommendations. In contrast, we examine how to produce counterfactual explanations through the lens of group recommendations, something that, to our knowledge, has not been researched up to now. Group recommendations introduce a new layer of complexity to the problem, and it is essential to reconcile the varying preferences of group members with transparent, implementable counterfactual explanations that guarantee equity and satisfaction for each individual concerned.
This paper proposes two heuristic approaches to finding counterfactual explanations for group recommendations. Since many probable combinations of items can constitute a group counterfactual explanation, we define group counterfactual utilities to help us locate an explanation in a reasonable time. Our two proposed algorithms utilize these utility scores in a diferent manner. The Sliding Window algorithm aggregates the scores and orders the items in a descending order. Then, it applies a sliding window on the list and checks if the items in the window are a counterfactual explanation. If they are, the algorithm exhaustively searches all combinations of items in the window to find the most concise explanations, i.e., the explanation consisting of the least number of items. The Four Quadrant algorithm arranges the items based on their utility scores in a four-quadrant graph. Then, check the items that are located inside a circular area. Similarly to Sliding Window, when it finds a subset of items that produce a counterfactual explanation, it exhaustively searches for the minimal explanation. We evaluate our proposed methods utilizing a real work dataset, MovieLens, for various methods’ parameters. Our results show that the Sliding Window can produce the explanations faster. However, the Four Quadrant algorithm is not overly afected by the increase in the size of the items it needs to search.